Research ArticleOn the Octonionic Inclined Curves in the 8-DimensionalEuclidean Space

Oumlzcan BektaG and Salim Yuumlce

Department of Mathematics Faculty of Arts and Sciences Yildiz Technical University 34220 Istanbul Turkey

Correspondence should be addressed to Ozcan Bektas obektasyildizedutr

Received 22 September 2014 Accepted 8 December 2014 Published 23 December 2014

Academic Editor Hongyong Zhao

Copyright copy 2014 O Bektas and S Yuce This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We describe octonionic inclined curves and harmonic curvatures for the octonionic curves We give characterizations for anoctonionic curve to be an octonionic inclined curve And finally we obtain some characterisations for the octonionic inclinedcurves in terms of the harmonic curvatures

1 Introduction

Hamilton [1] revealed quaternion in 1843 as an explanation ofgroup construction and also performed to mechanics inthree-dimensional space For quaternions same features areprovided as complex numbers with the discrimination thatthe commutative rule is not effective in their case The octo-nions [2 3] form the widest normed algebra after the algebraof real numbers complex numbers and quaternions Theoctonions are also known as Cayley Graves numbers and alsohave an algebraic structure defined on the eight-dimensionalreal vector space in such a way that two octonions can beadded multiplied and divided with the fact that multi-plication is neither commutative nor associative Inclinedcurves in Euclidean 119899 space were studied by Ozdamar andHacısalihoglu [4] The Serret-Frenet formulae for an octo-nionic curves inR7 andR8 are given by Bektas and Yuce [5]But to our knowledge there has been no study on the octo-nionic inclined curves in the eight-dimensional Euclideanspace Such a study is the object of this paper Our main aimin the present work is to study the differential geometry of asmooth curve in the eight-dimensional Euclidean space

2 Preliminaries

The octonions can be thought of as octal of real numbersOctonion is a real linear combination of the unit octonionse0 e1 e2 e3 e4 e5 e6 e7 where e

0is the scalar or real

element it may be assimilated with the real number 1 Thatis every real octonion (in this study we use octonion insteadof real octonion since two concepts are the same) A can beexpressed in the manner 119860 = sum

7

119894=0119886119894119890119894 Hence an octonion

can be decomposed in terms of its scalar (119878119860) and vector (V

119860)

parts as 119878119860= 1198860and V

119860= sum7

119894=1119886119894119890119894 Addition and extraction

of octonions are made by adding and quarrying corre-sponding terms and thereby their factors like quaternionsMultiplication is more complex Multiplication is distributiveover addition so the product of two octonions can becalculated by summing the product of all the terms again likequaternions The product of each term can be given bymultiplication of the coefficients and a multiplication table ofthe unit octonions like this one [3 6]

times 1198900

1198901

1198902

1198903

1198904

1198905

1198906

1198907

1198900

1198900

1198901

1198902

1198903

1198904

1198905

1198906

1198907

1198901

1198901

minus1198900

1198903

minus1198902

1198905

minus1198904

minus1198907

1198906

1198902

1198902

minus1198903

minus1198900

1198901

1198906

1198907

minus1198904

minus1198905

1198903

1198903

1198902

minus1198901

minus1198900

1198907

minus1198906

1198905

minus1198904

1198904

1198904

minus1198905

minus1198906

minus1198907

minus1198900

1198901

1198902

1198903

1198905

1198905

1198904

minus1198907

1198906

minus1198901

minus1198900

minus1198903

1198902

1198906

1198906

1198907

1198904

minus1198905

minus1198902

1198903

minus1198900

minus1198901

1198907

1198907

minus1198906

1198905

1198904

minus1198903

minus1198902

1198901

minus1198900

(1)

Most of diagonal elements of the table are antisymmetricmaking it almost a skew symmetric matrix except for theelements on the main diagonal the row and the column for

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 218638 8 pageshttpdxdoiorg1011552014218638

2 Mathematical Problems in Engineering

which e0is an operand The table can be epitomized by the

relations [7] 119890119894119890119895= minus120575

1198941198951198900+ 120576119894119895119896119890119896 where is a completely

antisymmetric tensor with value+1when 119894119895119896 = 123 145 176

246 257 347 365 and 1198900119890119894= 1198901198941198900

= 119890119894 with 119890

0the scalar

element and 119894 119895 119896 = 1 7The overhead declaration though is not unique but is only

one of 480 possible declarations for octonion multiplicationThe others can be acquired by permuting the nonscalarelements so can be noted to have different bases Alternatelythey can be acquired by fixing the product rule for a fewterms and deducing the rest from the other properties of theoctonions The 480 different algebras are isomorphic so arein practice identical and there is rarely a need to considerwhich particular multiplication rule is used [8 9] We denotethe set of octonion

119874 = 119860 | 119860 =

7

sum

119894=0

119886119894119890119894 1198900119890119894= 1198901198941198900= 119890119894

1198900= +1 119890

119894119890119895= minus1205751198941198951198900+ 120576119894119895119896119890119896

(2)

where 119894119895119896 = 123 145 176 246 257 347 365 120575119894119895is Kro-

necker delta 120576119894119895119896

is completely antisymmetric tensor withvalue +1 O is spanned by +1 and the +1 imaginary units1198901 1198902 1198903 1198904 1198905 1198906 1198907each with square minus1 so thatO = RoplusR7

[10] Then octonions are isomorphic to R8 [11] Just as thecomplex numbers and quaternions could be used to describeR2 and R4 the octonions may be used to describe points inR8 using the obvious identification [12] Let 119860 = sum

7

119894=0119886119894119890119894

be an octonion If 1198860= 0 then 119860 is called a spatial (pure)

octonion (or 119860 is called a spatial (pure) octonion whenever119860+119860 = 0) Before we define the octonionic product we giveinformation about vector product in R7 Seven-dimensionalEuclidean space and three-dimensional Euclidean space arethe only Euclidean spaces to have a vector product We knowthat we can express an octonion as the sum of a real part 119878

119860

and a pure part 119881119860in R7 So we get 119860 = 119878

119860+ V119860 This

guides to a vector product on R7 described by V119860and V119861

=

V119860V119861+ ⟨V119860V119861⟩ [13] Moreover this is given [13] by V

119860=

(1198861 1198862 1198863 1198864 1198865 1198866 1198867) and V

119861= (1198871 1198872 1198873 1198874 1198875 1198876 1198877)

Let 120577 = (1198861 1198862 1198863 1198864 1198865 1198866 1198867) and (119887

1 1198872 1198873 1198874 1198875 1198876 1198877)

then

120577 = (11988621198873minus 11988631198872+ 11988641198875minus 11988651198874+ 11988671198876minus 11988661198877

11988631198871minus 11988611198873+ 11988641198876minus 11988661198874+ 11988651198877minus 11988671198875

11988611198872minus 11988621198871+ 11988641198877minus 11988671198874+ 11988661198875minus 11988651198876

11988651198871minus 11988611198875+ 11988661198872minus 11988621198876+ 11988671198873minus 11988631198877

11988611198874minus 11988641198871+ 11988631198876minus 11988661198873+ 11988671198872minus 11988621198877

11988611198877minus 11988671198871+ 11988621198874minus 11988641198872+ 11988651198873minus 11988631198875

11988621198875minus 11988651198872+ 11988631198874minus 11988641198873+ 11988661198871minus 11988611198876)

(3)

The vector product in R7 has all the properties expectJacobi identity So generally for pure octonions V

119860and (V119861and

V119862) +V119861and (V119862andV119860) +V119862and (V119860andV119861) = 0 or equivalently

the triple vector product identity fails V119860and (V119861and V119862) =

⟨V119860V119862⟩V119861minus⟨V119860V119861⟩V119862The identitieswhich it does satisfy

are as follows

Theorem 1 LetV119860V119861 andV

119862be spatial octonions and let 120579

be the angle betweenV119860andV

119861 If 119888 is an random real number

then the succeeding identicalnesses run for vector products inR7 [13] Consider the following

(i) V119860and (V119861+ V119862) = V119860and V119861+ V119860and V119862

(ii) V119860and V119860= 0

(iii) V119860and V119861= minusV119861and V119860

(iv) ⟨V119860V119860and V119861⟩ = 0

(v) V119860and V119861 = V

119860V119861 sin 120579

(vi) ⟨V119860and V119861V119862⟩ = ⟨V

119861and V119862V119860⟩ = ⟨V

119862and V119860V119861⟩

(vii) V119860and (V119860and V119861) = ⟨V

119860V119861⟩V119860minus ⟨V119860V119860⟩V119861

If we take widely information about cross product inR7 we can read the references [14] Now we can describeoctonion productTheoctonionic product of two octonions isserved as follows [15]

119860 times 119861 = 119878119860119878119861minus ⟨V119860V119861⟩ + 119878119860V119861+ 119878119861V119860+ V119860and V119861

forall119860 119861 isin 119874

(4)

where we have used the dot and cross products in 1198777 119860 is

called conjugate of 119860 and described as noted below

119860 = 119878119860minus V119860= 1198860minus

7

sum

119894=1

119886119894119890119894 (5)

wherewe have used the conjugates of basis elements as 1198900= 1198900

and 119890119895= minus119890119895(119895 = 1 7) The inner product of octonions

qualifies as follows

⟨119860 119861⟩ 119874 times 119874 997888rarr R

(119860 119861) 997888rarr ⟨119860 119861⟩ =

1

2

(119860 times 119861 + 119861 times 119860) =

7

sum

119894=0

119886119894119887119894

(6)

Hence it is called the octonionic inner product The norm ofan octonion 119860 is defined by

119860 =radic119860 times 119860 = radic

7

sum

119894=0

119886119894 (7)

If 119860 = 1 then119860 is called a unit octonionThe only octonionwith norm 0 is 0 and every nonzero octonion has a uniqueinverse namely [16]

119860minus1

=

119860

1198602 (8)

For all the normed division algebras the norm provides theidenticalness [16]

119860 times 119861 = 119860 119861 (9)

Mathematical Problems in Engineering 3

If we take 119899 = 7 8 in the study named ldquoA characterization ofinclined curves in Euclidean 119899 spacerdquo we can obtain thefollowing definitions

Definition 2 Let 120574 119868 rarr R7 be a curve in R7 with the arclength parameter 119904 and let 119906 be a unit constant vector of R7For all 119904 isin 119868 if

⟨1205741015840(119904) 119906⟩ = cos120593 = constant 120593 =

120587

2

(10)

then the curve is called an inclined curve in R7 where 1205741015840(119904)is the unit tangent vector to the curve 120574 at its point 120574(119904) and120593 is a constant angle between the vectors 1205741015840 and 119906 [4]

We can give same definition in R8

Definition 3 Let 120574 119868 rarr R7 be a curve in R7 with an arclength parameter 119904 and let 119906 be an unit constant vector Lettn1n2n3n4n5n6 3 le 119903 le 7 be the Frenet 7-frame of 120574at its point 120574(119904) If the angle between 120574

1015840(119904) and 119906 is 120593 = 120593(119904)

we define the function

119867119894 119868 997888rarr R 3 le 119894 le 119903 minus 2 (11)

by

⟨119899119894+1

(119904) 119906⟩ = 119867119894(119904) cos120593 (12)

as the harmonic curvature with order 119894 of the curve 120574 at itspoint 120574(119904) We define also119867

0= 0 [4]

We can give same definition in R8Now we are going to give some definitions and theorems

about octonionic curves in R7 and R8

Definition 4 The seven-dimensional Euclidean space R7 isconsubstantiated by the space of spatial real octonions 119874

119875=

120574 isin 119874 | 120574 + 120574 = 0 in an obvious manner Let 119868 = [0 1] be aninterval inR and let 119904 isin 119868 be the parameter along the smoothcurve

120574 119868 sub R 997888rarr 119874119875

119904 997888rarr 120574 (119904) =

7

sum

119894=1

120574119894(119904) 119890119894

(13)

Then the curve is called spatial octonionic curve or octo-nionic curve in R7 [5]

Theorem 5 The seven-dimensional Euclidean space R7 isconsubstantiated by the space of spatial real octonions 119874

119875=

120574 isin 119874 | 120574 + 120574 = 0 in an obvious manner Let 119868 = [0 1] be aninterval in R and let 119904 isin 119868 be the parameter along the smoothcurve

120574 119868 sub R 997888rarr 119874119875

119904 997888rarr 120574 (119904) =

7

sum

119894=1

120574119894(119904) 119890119894

(14)

Let tn1n2n3n4n5n6 be the Frenet trihedron of thedifferentiable Euclidean space curve in the Euclidean spaceR7Then Frenet equations are

t1015840 (119904) = 1198961(119904)n1 (119904)

n10158401 (119904) = minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904)

n10158402 (119904) = minus1198962(119904)n1 (119904) + 119896

3(119904)n3 (119904)

n10158403 (119904) = minus1198963(119904)n2 (119904) + 119896

4(119904)n4 (119904)

n10158404 (119904) = minus1198964(119904)n3 (119904) + 119896

5(119904)n5 (119904)

n10158405 (119904) = minus1198965(119904)n4 (119904) + 119896

6(119904)n6 (119904)

n10158406 (119904) = minus1198966(119904)n5 (119904)

(15)

where 119896119894 1 le 119894 le 6 curvature functions

We may state Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

[

t1015840

n10158401n10158402n10158403n10158404n10158405n10158406

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

0 1198961

0 0 0 0 0

minus1198961

0 1198962

0 0 0 0

0 minus1198962

0 1198963

0 0 0

0 0 minus1198963

0 1198964

0 0

0 0 0 minus1198964

0 1198965

0

0 0 0 0 minus1198965

0 1198966

0 0 0 0 0 minus1198966

0

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

tn1n2n3n4n5n6

]

]

]

]

]

]

]

]

]

]

(16)

This is the Serret-Frenet formulae for the spatial octonioniccurve 120574 in R7 [5]

tn1n2n3n4n5n6 1198961 1198962 1198963 1198964 1198965 1198966 is the Frenetapparatus for spatial octonionic curve 120574 in R7

Remark 6 What has been achieved in this theorem isreputable in local differential geometryWe have done this fortwo especial goals

(1) to designate the demonstration for the Serret-Frenetformulae and Frenet apparatus of the curve 120574 in R7Wewill roll the outcomes of this theorem comprehen-sively in the next theorem

(2) to indicate how octonions are to be used in designat-ing curvature numbers of curves in general

Definition 7 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let 119868 = [0 1] be aninterval inR and let 119904 isin 119868 be the parameter along the smoothcurve

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(17)

Then the curve is called octonionic curve [5]

4 Mathematical Problems in Engineering

Theorem 8 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(18)

be a smooth curve in R8 described over 119868 Let the parameter 119904be selected that T = 120573

1015840(119904) = sum

7

119894=01205741015840

119894(119904)119890119894has unit magnitude

Let TN1N2N3N4N5N6N7 be the Frenet elements of 120573Then the Frenet equations are

T1015840 (119904) = 119870 (119904)N1 (119904)

N10158401 (119904) = minus119870 (119904)T (119904) + 1198961(119904)N2 (119904)

N10158402 (119904) = minus1198961(119904)N1 (119904) + (119896

2minus 119870) (119904)N3 (119904)

N10158403 (119904) = minus (1198962minus 119870) (119904)N2 (119904) + 119896

3(119904)N4(119904)

N10158404 (119904) = minus1198963(119904)N3 (119904) + (119896

4minus 119870) (119904)N5 (119904)

N10158405 (119904) = minus (1198964minus 119870) (119904)N4 (119904) + 119896

5(119904)N6 (119904)

N10158406 (119904) = minus1198965(119904)N5 (119904) + (119896

6+ 119870) (119904)N7 (119904)

N10158407 (119904) = minus (1198966+ 119870) (119904)N6 (119904)

(19)

where N1 = t times T N2 = n1 times T N3 = n2 times T N4 = n3 times TN5 = n4 times T N6 = n5 times T and N7 = n6 times T 119870 = T1015840(119904)

We may express Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

T1015840N10158401N10158402N10158403N10158404N10158405N10158406N10158407

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

0 119870 0 0 0 0 0 0

minus119870 0 1198961

0 0 0 0 0

0 minus1198961

0 (1198962minus 119870) 0 0 0 0

0 0 minus (1198962minus 119870) 0 119896

30 0 0

0 0 0 minus1198963

0 (1198964minus 119870) 0 0

0 0 0 0 minus (1198964minus 119870) 0 119896

50

0 0 0 0 0 minus1198965

0 (1198966+ 119870)

0 0 0 0 0 0 minus (1198966+ 119870) 0

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

TN1N2N3N4N5N6N7

]

]

]

]

]

]

]

]

]

]

]

(20)

This is the Serret-Frenet formulae for octonionic curve 120573 inR8 [5]

3 Octonionic Inclined Curves andHarmonic Curvatures

Definition 9 Let 120574(119868) be spatial octonionic curve with an arclength parameter 119904 and let 119880 be an unit and constant spatialoctonion For all 119904 isin 119868 let ⟨1205741015840(119904) 119880⟩ be a constant defined by

⟨1205741015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(21)

Then 120574(119868) is called spatial octonionic inclined curve

Definition 10 120574(119868) octonionic curve is given by arc lengthparameter 119904 Let tn1n2n3n4n5n6 be the Frenet trihe-dron in the point 120574(119904) of the curve 120574 and let 119880 be unit andconstant spatial octonion such that angle120593(119904) is between 120574

1015840(119904)

and 119880

119867119894 119868 997888rarr R 1 le 119894 le 5 (22)

be a function defined by

⟨119899119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(23)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120574(119904) of the 120574 spatial octonionic curve with respect to 119906

Definition 11 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 such that 119880 is a unit and constantspatial octonion for every 119904 isin 119868

⟨1205731015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(24)

Then curve 120573 is called octonionic inclined curve in 119874

Definition 12 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 Let TN1N2N3N4N5N6N7 bethe Frenet apparatus and let119880 be unit and constant such thatangle 120593(119904) is between T1015840(119904) and 119880 Let

119867119894 119868 997888rarr R 1 le 119894 le 6 (25)

be a function defined by

⟨119873119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(26)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120573(119904) of the 120573 octonionic curve with respect to 119880

Theorem 13 Let 120574 119868 rarr R7 be spatial octonionic inclinedcurve given by arc length parameter 119904 Curvatures in the point

Mathematical Problems in Engineering 5

120574(119904) of curve 120574 are 119896119894(119904) 120577119894= 1119896

119894(119904) and 119867

119894(119904) 1 le 119894 le 6 are

harmonic curvatures they are

1198671=

1198961

1198962

1198672=

1198671015840

1

1198963

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894) 120577119895+1

2 le 119895 le 5

(27)

Proof Let 120593 be an angle between the unit and constant spatialoctonion119880 and t(119904) Such that tn1n2n3n4n5n6 Frenetapparatus in the point 120574(119904) we obtain that

⟨t (119904) 119880⟩ = cos120593 (28)

Here differentiatingwith respect to 119904 we find that ⟨t1015840(119904) 119880⟩ =

0 By the aid of (15) we obtain that ⟨n1(119904) 119880⟩ = 0 Ifderivative of this function with respect to 119904 is taken we findthat ⟨n10158401(119904) 119880⟩ = 0 Here using (15)

⟨minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904) 119880⟩ = 0 (29)

is obtained Thus if (21) and (23) are used

(minus1198961(119904) + 119896

2(119904)1198671) cos120593 = 0 cos120593 =

120587

2

(30)

is found Thus

1198671=

1198961(119904)

1198962(119904)

(31)

By the aid of (15) we obtain that ⟨n2(119904) 119880⟩ = 1198671cos120593

If derivative of this function with respect to 119904 is taken and(15) (21) and (23) are used we find that 119867

2= 1198671015840

11198963 For

the higher harmonic curvatures let us differentiate (23) withrespect to 119904 for 119895 then ⟨119899

1015840

119895+1(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 By the aid

of (15) ⟨minus119896119895(119904)ni(119904) + 119896

119895+1(119904)ni+2(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 we get

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894)120577119895+1

2 le 119895 le 5

Theorem 14 Let 120574 119868 rarr R7 be a spatial octonionic inclinedcurve Such that 120574(119904) = sum

7

119894=1120574119894(119904)119890119894

120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894 (32)

obtained from 120574 octonionic curve is an octonionic inclinedcurve

Proof Let 120573 119868 rarr 119874 be an octonionic curve given by arclength parameter 119904

Let TN1N2N3N4N5N6N7 be the Frenet appara-tus and let 119880 be unit and constant spatial octonion If we useDefinition 11 we get the following statement

⟨1205731015840(119904) 119906⟩ = ⟨T (119904) 119880⟩

T (119904) = 119878T(119904) + T(119904) 119880 = 119878119880+ 119880

(33)

where

T (119904) = 119878T(119904) + T(119904) (34)

We notice that⟨T (119904) 119880⟩

=

1

2

(T (119904) times 119880 + 119880 times T (119904))

=

1

2

[(119878T(119904) + T(119904)) times 119880] + [119880 times (119878T(119904) minus T(119904))]

(35)

Since 119880 is spatial octonion then 119880 = 119880 119880 = minus119880 Here

we can account for the product of octonion

⟨T (119904) 119880⟩

=

1

2

[119878T(119904) sdot 0 minus ⟨T(119904) minus119880⟩ + 119878T(119904)

sdot (minus119880) + 0 sdot T(119904) + T(119904) and (minus119880)]

+ [0 sdot 119878T(119904) minus ⟨119880 T(119904)⟩ + 119878T(119904)

sdot 119880 + 0 sdot T(119904) + 119880 and (minusT(119904))]

=

1

2

[⟨T(119904) 119880⟩ minus 119878T(119904)119880 minus T(119904) and 119880

+ ⟨119880 T(119904)⟩ + 119878T(119904)119880 minus 119880 and T(119904)]

= ⟨T(119904) 119880⟩

(36)

and so ⟨1205731015840(119904) 119880⟩ = cos120593 is obtainedThen 120573 curve is octon-

ionic inclined curve

Theorem 15 Let 120573 119868 rarr 119874 be an octonionic inclined curvegiven by arc length parameter 119904 Such that119870

119894(119904) are curvatures

in the point 120573(119904) 120575119894(119904) = 1119870

119894(119904) 1 le 119894 le 7 are curvature radii

and119867119894(119904) 1 le 119894 le 6 are harmonic curvatures they are

1198671=

1119888119906119903V1198861199051199061199031198902119888119906119903V119886119905119906119903119890

1198672=

1198671015840

1

(1198962minus 119870) (119904)

1198673=

1198671015840

2+ (1198962minus 119870)119867

1

1198963

1198674=

1198671015840

3+ 11989631198672

(1198964minus 119870) (119904)

1198675=

1198671015840

4+ (1198964minus 119870)119867

3

1198965

1198676=

1198671015840

5+ 11989651198674

(1198966+ 119870) (119904)

(37)

where 1198701(119904) = 119870 119870

2(119904) = 119896

1 1198703(119904) = 119896

2minus 119870 119870

4(119904) = 119896

3

1198705(119904) = 119896

4minus 119870 119870

6(119904) = 119896

5 1198707(119904) = 119896

6+ 119870

Proof 120573 119868 rarr 119874 curve is given by regular octonionic119880 is an unit and a constant spatial octonion and suchthat TN1N2N3N4N5N6N7 is Frenet apparatus in thepoint 120573(119904)

⟨T (119904) 119880⟩ = cos120593 = constant (38)

6 Mathematical Problems in Engineering

is written If derivative with respect to 119904 of this equationis taken we obtain that ⟨T1015840(119904) 119880⟩ = 0 Here using (19)⟨119870(119904)N1(119904) 119880⟩ = 0 is found Because of 119870(119904)N1(119904) = 0 wewrite as ⟨N1(119904) 119880⟩ = 0 Thus ⟨N10158401(119904) 119880⟩ = 0 is obtainedHere using (19)

minus119870 (119904) ⟨T (119904) 119880⟩ + 1198961(119904) ⟨N2 (119904) 119880⟩ = 0 (39)

is found In addition from (26) for 119894 = 1 we obtain that

⟨N2 (119904) 119880⟩ = 1198671(119904) cos120593 (40)

By taking (24) and (40) into consideration

(minus119870 (119904) + 1198961(119904)1198671(119904)) cos120593 = 0 cos120593 = 0

1198671(119904) =

119870 (119904)

1198961(119904)

=

1curvature2curvature

(41)

is found On the other hand if derivative of (40) with respectto 119904 is taken

⟨N10158402 (119904) 119880⟩ = 1198671015840

1(119904) cos120593 (42)

is found Here using (19)

minus1198961(119904) ⟨N1 (119904) 119880⟩ + (119896

2minus 119870) (119904) ⟨N3 (119904) 119880⟩ = 119867

1015840

1(119904) cos120593

(43)

is found In addition from (26) for 119894 = 2 we obtain that

⟨N3 (119904) 119880⟩ = 1198672(119904) cos120593 (44)

By taking (24) and (44) into consideration

(1198962minus 119870) (119904)119867

2(119904) cos120593 = 119867

1015840

1(119904) cos120593 (45)

is obtained Thus

1198671(119904) =

1198671015840

1(119904)

(1198962minus 119870) (119904)

(46)

If derivative of (44) with respect to 119904 is taken

⟨N10158403(119904) 119906⟩ = 119867

1015840

2(119904) cos120593 (47)

is found Here using (19)

minus (1198962minus 119870) (119904) ⟨N2 (119904) 119880⟩ + 119896

3(119904) ⟨N4 (119904) 119880⟩ = 119867

1015840

2(119904) cos120593

(48)

is found In addition from (26) for 119894 = 3 we obtain that

⟨N4 (119904) 119880⟩ = 1198673(119904) cos120593 (49)

By taking (40) and (49) into consideration

minus (1198962minus 119870) (119904)119867

1(119904) cos120593 + 119896

3(119904)1198673(119904) cos120593 = 119867

1015840

2(119904) cos120593

(50)

is obtained Thus

1198673(119904) =

1198671015840

2(119904) + (119896

2minus 119870) (119904)119867

1(119904)

1198963(119904)

(51)

Similarly If derivative of (49) and following equations withrespect to 119904 is taken

⟨N5 (119904) 119880⟩ = 1198674(119904) cos120593

⟨N6 (119904) 119880⟩ = 1198675(119904) cos120593

(52)

we get

1198674(119904) =

1198671015840

3(119904) + 119896

3(119904)1198672(119904)

(1198964minus 119870) (119904)

1198675(119904) =

1198671015840

4(119904) + (119896

4minus 119870) (119904)119867

3

1198965(119904)

1198676=

1198671015840

5(119904) + 119896

5(119904)1198674(119904)

(1198966+ 119870) (119904)

(53)

Theorem 16 120574 is a spatial octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 5 be harmonic curvatures

in the point 120574(119904) 120574 is octonionic inclined curve if and only ifsum5

119894=11198672

119894is constant

Proof (rArr) Let 120574 be a spatial octonionic curve given by arclength parameter 119904 Then there is a 119880 unit and constantspatial octonion Therefore

⟨1205741015840(119904) 119880⟩ = cos120593 (54)

is constant for 120574 spatial octonionic inclined curvewith respectto arc length parameter 119904 such that tn1n2n3n4n5n6 isbasis of spatial octonion in the point 120574(119904) spatial octonion 119880

119880 = ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904) (55)

is obtained Since 119880 is a unit

1198802= 119880 times 119880 = 1 (56)

Here using (55)

1198802= (⟨t (119904) 119880⟩ t (119904) +

6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904))

times ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904)

(57)

if we use Definition 10 in the last equation we can write

1 = (cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593)

times cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593

(58)

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

which e0is an operand The table can be epitomized by the

relations [7] 119890119894119890119895= minus120575

1198941198951198900+ 120576119894119895119896119890119896 where is a completely

antisymmetric tensor with value+1when 119894119895119896 = 123 145 176

246 257 347 365 and 1198900119890119894= 1198901198941198900

= 119890119894 with 119890

0the scalar

element and 119894 119895 119896 = 1 7The overhead declaration though is not unique but is only

one of 480 possible declarations for octonion multiplicationThe others can be acquired by permuting the nonscalarelements so can be noted to have different bases Alternatelythey can be acquired by fixing the product rule for a fewterms and deducing the rest from the other properties of theoctonions The 480 different algebras are isomorphic so arein practice identical and there is rarely a need to considerwhich particular multiplication rule is used [8 9] We denotethe set of octonion

119874 = 119860 | 119860 =

7

sum

119894=0

119886119894119890119894 1198900119890119894= 1198901198941198900= 119890119894

1198900= +1 119890

119894119890119895= minus1205751198941198951198900+ 120576119894119895119896119890119896

(2)

where 119894119895119896 = 123 145 176 246 257 347 365 120575119894119895is Kro-

necker delta 120576119894119895119896

is completely antisymmetric tensor withvalue +1 O is spanned by +1 and the +1 imaginary units1198901 1198902 1198903 1198904 1198905 1198906 1198907each with square minus1 so thatO = RoplusR7

[10] Then octonions are isomorphic to R8 [11] Just as thecomplex numbers and quaternions could be used to describeR2 and R4 the octonions may be used to describe points inR8 using the obvious identification [12] Let 119860 = sum

7

119894=0119886119894119890119894

be an octonion If 1198860= 0 then 119860 is called a spatial (pure)

octonion (or 119860 is called a spatial (pure) octonion whenever119860+119860 = 0) Before we define the octonionic product we giveinformation about vector product in R7 Seven-dimensionalEuclidean space and three-dimensional Euclidean space arethe only Euclidean spaces to have a vector product We knowthat we can express an octonion as the sum of a real part 119878

119860

and a pure part 119881119860in R7 So we get 119860 = 119878

119860+ V119860 This

guides to a vector product on R7 described by V119860and V119861

=

V119860V119861+ ⟨V119860V119861⟩ [13] Moreover this is given [13] by V

119860=

(1198861 1198862 1198863 1198864 1198865 1198866 1198867) and V

119861= (1198871 1198872 1198873 1198874 1198875 1198876 1198877)

Let 120577 = (1198861 1198862 1198863 1198864 1198865 1198866 1198867) and (119887

1 1198872 1198873 1198874 1198875 1198876 1198877)

then

120577 = (11988621198873minus 11988631198872+ 11988641198875minus 11988651198874+ 11988671198876minus 11988661198877

11988631198871minus 11988611198873+ 11988641198876minus 11988661198874+ 11988651198877minus 11988671198875

11988611198872minus 11988621198871+ 11988641198877minus 11988671198874+ 11988661198875minus 11988651198876

11988651198871minus 11988611198875+ 11988661198872minus 11988621198876+ 11988671198873minus 11988631198877

11988611198874minus 11988641198871+ 11988631198876minus 11988661198873+ 11988671198872minus 11988621198877

11988611198877minus 11988671198871+ 11988621198874minus 11988641198872+ 11988651198873minus 11988631198875

11988621198875minus 11988651198872+ 11988631198874minus 11988641198873+ 11988661198871minus 11988611198876)

(3)

The vector product in R7 has all the properties expectJacobi identity So generally for pure octonions V

119860and (V119861and

V119862) +V119861and (V119862andV119860) +V119862and (V119860andV119861) = 0 or equivalently

the triple vector product identity fails V119860and (V119861and V119862) =

⟨V119860V119862⟩V119861minus⟨V119860V119861⟩V119862The identitieswhich it does satisfy

are as follows

Theorem 1 LetV119860V119861 andV

119862be spatial octonions and let 120579

be the angle betweenV119860andV

119861 If 119888 is an random real number

then the succeeding identicalnesses run for vector products inR7 [13] Consider the following

(i) V119860and (V119861+ V119862) = V119860and V119861+ V119860and V119862

(ii) V119860and V119860= 0

(iii) V119860and V119861= minusV119861and V119860

(iv) ⟨V119860V119860and V119861⟩ = 0

(v) V119860and V119861 = V

119860V119861 sin 120579

(vi) ⟨V119860and V119861V119862⟩ = ⟨V

119861and V119862V119860⟩ = ⟨V

119862and V119860V119861⟩

(vii) V119860and (V119860and V119861) = ⟨V

119860V119861⟩V119860minus ⟨V119860V119860⟩V119861

If we take widely information about cross product inR7 we can read the references [14] Now we can describeoctonion productTheoctonionic product of two octonions isserved as follows [15]

119860 times 119861 = 119878119860119878119861minus ⟨V119860V119861⟩ + 119878119860V119861+ 119878119861V119860+ V119860and V119861

forall119860 119861 isin 119874

(4)

where we have used the dot and cross products in 1198777 119860 is

called conjugate of 119860 and described as noted below

119860 = 119878119860minus V119860= 1198860minus

7

sum

119894=1

119886119894119890119894 (5)

wherewe have used the conjugates of basis elements as 1198900= 1198900

and 119890119895= minus119890119895(119895 = 1 7) The inner product of octonions

qualifies as follows

⟨119860 119861⟩ 119874 times 119874 997888rarr R

(119860 119861) 997888rarr ⟨119860 119861⟩ =

1

2

(119860 times 119861 + 119861 times 119860) =

7

sum

119894=0

119886119894119887119894

(6)

Hence it is called the octonionic inner product The norm ofan octonion 119860 is defined by

119860 =radic119860 times 119860 = radic

7

sum

119894=0

119886119894 (7)

If 119860 = 1 then119860 is called a unit octonionThe only octonionwith norm 0 is 0 and every nonzero octonion has a uniqueinverse namely [16]

119860minus1

=

119860

1198602 (8)

For all the normed division algebras the norm provides theidenticalness [16]

119860 times 119861 = 119860 119861 (9)

Mathematical Problems in Engineering 3

If we take 119899 = 7 8 in the study named ldquoA characterization ofinclined curves in Euclidean 119899 spacerdquo we can obtain thefollowing definitions

Definition 2 Let 120574 119868 rarr R7 be a curve in R7 with the arclength parameter 119904 and let 119906 be a unit constant vector of R7For all 119904 isin 119868 if

⟨1205741015840(119904) 119906⟩ = cos120593 = constant 120593 =

120587

2

(10)

then the curve is called an inclined curve in R7 where 1205741015840(119904)is the unit tangent vector to the curve 120574 at its point 120574(119904) and120593 is a constant angle between the vectors 1205741015840 and 119906 [4]

We can give same definition in R8

Definition 3 Let 120574 119868 rarr R7 be a curve in R7 with an arclength parameter 119904 and let 119906 be an unit constant vector Lettn1n2n3n4n5n6 3 le 119903 le 7 be the Frenet 7-frame of 120574at its point 120574(119904) If the angle between 120574

1015840(119904) and 119906 is 120593 = 120593(119904)

we define the function

119867119894 119868 997888rarr R 3 le 119894 le 119903 minus 2 (11)

by

⟨119899119894+1

(119904) 119906⟩ = 119867119894(119904) cos120593 (12)

as the harmonic curvature with order 119894 of the curve 120574 at itspoint 120574(119904) We define also119867

0= 0 [4]

We can give same definition in R8Now we are going to give some definitions and theorems

about octonionic curves in R7 and R8

Definition 4 The seven-dimensional Euclidean space R7 isconsubstantiated by the space of spatial real octonions 119874

119875=

120574 isin 119874 | 120574 + 120574 = 0 in an obvious manner Let 119868 = [0 1] be aninterval inR and let 119904 isin 119868 be the parameter along the smoothcurve

120574 119868 sub R 997888rarr 119874119875

119904 997888rarr 120574 (119904) =

7

sum

119894=1

120574119894(119904) 119890119894

(13)

Then the curve is called spatial octonionic curve or octo-nionic curve in R7 [5]

Theorem 5 The seven-dimensional Euclidean space R7 isconsubstantiated by the space of spatial real octonions 119874

119875=

120574 isin 119874 | 120574 + 120574 = 0 in an obvious manner Let 119868 = [0 1] be aninterval in R and let 119904 isin 119868 be the parameter along the smoothcurve

120574 119868 sub R 997888rarr 119874119875

119904 997888rarr 120574 (119904) =

7

sum

119894=1

120574119894(119904) 119890119894

(14)

Let tn1n2n3n4n5n6 be the Frenet trihedron of thedifferentiable Euclidean space curve in the Euclidean spaceR7Then Frenet equations are

t1015840 (119904) = 1198961(119904)n1 (119904)

n10158401 (119904) = minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904)

n10158402 (119904) = minus1198962(119904)n1 (119904) + 119896

3(119904)n3 (119904)

n10158403 (119904) = minus1198963(119904)n2 (119904) + 119896

4(119904)n4 (119904)

n10158404 (119904) = minus1198964(119904)n3 (119904) + 119896

5(119904)n5 (119904)

n10158405 (119904) = minus1198965(119904)n4 (119904) + 119896

6(119904)n6 (119904)

n10158406 (119904) = minus1198966(119904)n5 (119904)

(15)

where 119896119894 1 le 119894 le 6 curvature functions

We may state Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

[

t1015840

n10158401n10158402n10158403n10158404n10158405n10158406

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

0 1198961

0 0 0 0 0

minus1198961

0 1198962

0 0 0 0

0 minus1198962

0 1198963

0 0 0

0 0 minus1198963

0 1198964

0 0

0 0 0 minus1198964

0 1198965

0

0 0 0 0 minus1198965

0 1198966

0 0 0 0 0 minus1198966

0

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

tn1n2n3n4n5n6

]

]

]

]

]

]

]

]

]

]

(16)

This is the Serret-Frenet formulae for the spatial octonioniccurve 120574 in R7 [5]

tn1n2n3n4n5n6 1198961 1198962 1198963 1198964 1198965 1198966 is the Frenetapparatus for spatial octonionic curve 120574 in R7

Remark 6 What has been achieved in this theorem isreputable in local differential geometryWe have done this fortwo especial goals

(1) to designate the demonstration for the Serret-Frenetformulae and Frenet apparatus of the curve 120574 in R7Wewill roll the outcomes of this theorem comprehen-sively in the next theorem

(2) to indicate how octonions are to be used in designat-ing curvature numbers of curves in general

Definition 7 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let 119868 = [0 1] be aninterval inR and let 119904 isin 119868 be the parameter along the smoothcurve

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(17)

Then the curve is called octonionic curve [5]

4 Mathematical Problems in Engineering

Theorem 8 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(18)

be a smooth curve in R8 described over 119868 Let the parameter 119904be selected that T = 120573

1015840(119904) = sum

7

119894=01205741015840

119894(119904)119890119894has unit magnitude

Let TN1N2N3N4N5N6N7 be the Frenet elements of 120573Then the Frenet equations are

T1015840 (119904) = 119870 (119904)N1 (119904)

N10158401 (119904) = minus119870 (119904)T (119904) + 1198961(119904)N2 (119904)

N10158402 (119904) = minus1198961(119904)N1 (119904) + (119896

2minus 119870) (119904)N3 (119904)

N10158403 (119904) = minus (1198962minus 119870) (119904)N2 (119904) + 119896

3(119904)N4(119904)

N10158404 (119904) = minus1198963(119904)N3 (119904) + (119896

4minus 119870) (119904)N5 (119904)

N10158405 (119904) = minus (1198964minus 119870) (119904)N4 (119904) + 119896

5(119904)N6 (119904)

N10158406 (119904) = minus1198965(119904)N5 (119904) + (119896

6+ 119870) (119904)N7 (119904)

N10158407 (119904) = minus (1198966+ 119870) (119904)N6 (119904)

(19)

where N1 = t times T N2 = n1 times T N3 = n2 times T N4 = n3 times TN5 = n4 times T N6 = n5 times T and N7 = n6 times T 119870 = T1015840(119904)

We may express Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

T1015840N10158401N10158402N10158403N10158404N10158405N10158406N10158407

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

0 119870 0 0 0 0 0 0

minus119870 0 1198961

0 0 0 0 0

0 minus1198961

0 (1198962minus 119870) 0 0 0 0

0 0 minus (1198962minus 119870) 0 119896

30 0 0

0 0 0 minus1198963

0 (1198964minus 119870) 0 0

0 0 0 0 minus (1198964minus 119870) 0 119896

50

0 0 0 0 0 minus1198965

0 (1198966+ 119870)

0 0 0 0 0 0 minus (1198966+ 119870) 0

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

TN1N2N3N4N5N6N7

]

]

]

]

]

]

]

]

]

]

]

(20)

This is the Serret-Frenet formulae for octonionic curve 120573 inR8 [5]

3 Octonionic Inclined Curves andHarmonic Curvatures

Definition 9 Let 120574(119868) be spatial octonionic curve with an arclength parameter 119904 and let 119880 be an unit and constant spatialoctonion For all 119904 isin 119868 let ⟨1205741015840(119904) 119880⟩ be a constant defined by

⟨1205741015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(21)

Then 120574(119868) is called spatial octonionic inclined curve

Definition 10 120574(119868) octonionic curve is given by arc lengthparameter 119904 Let tn1n2n3n4n5n6 be the Frenet trihe-dron in the point 120574(119904) of the curve 120574 and let 119880 be unit andconstant spatial octonion such that angle120593(119904) is between 120574

1015840(119904)

and 119880

119867119894 119868 997888rarr R 1 le 119894 le 5 (22)

be a function defined by

⟨119899119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(23)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120574(119904) of the 120574 spatial octonionic curve with respect to 119906

Definition 11 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 such that 119880 is a unit and constantspatial octonion for every 119904 isin 119868

⟨1205731015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(24)

Then curve 120573 is called octonionic inclined curve in 119874

Definition 12 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 Let TN1N2N3N4N5N6N7 bethe Frenet apparatus and let119880 be unit and constant such thatangle 120593(119904) is between T1015840(119904) and 119880 Let

119867119894 119868 997888rarr R 1 le 119894 le 6 (25)

be a function defined by

⟨119873119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(26)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120573(119904) of the 120573 octonionic curve with respect to 119880

Theorem 13 Let 120574 119868 rarr R7 be spatial octonionic inclinedcurve given by arc length parameter 119904 Curvatures in the point

Mathematical Problems in Engineering 5

120574(119904) of curve 120574 are 119896119894(119904) 120577119894= 1119896

119894(119904) and 119867

119894(119904) 1 le 119894 le 6 are

harmonic curvatures they are

1198671=

1198961

1198962

1198672=

1198671015840

1

1198963

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894) 120577119895+1

2 le 119895 le 5

(27)

Proof Let 120593 be an angle between the unit and constant spatialoctonion119880 and t(119904) Such that tn1n2n3n4n5n6 Frenetapparatus in the point 120574(119904) we obtain that

⟨t (119904) 119880⟩ = cos120593 (28)

Here differentiatingwith respect to 119904 we find that ⟨t1015840(119904) 119880⟩ =

0 By the aid of (15) we obtain that ⟨n1(119904) 119880⟩ = 0 Ifderivative of this function with respect to 119904 is taken we findthat ⟨n10158401(119904) 119880⟩ = 0 Here using (15)

⟨minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904) 119880⟩ = 0 (29)

is obtained Thus if (21) and (23) are used

(minus1198961(119904) + 119896

2(119904)1198671) cos120593 = 0 cos120593 =

120587

2

(30)

is found Thus

1198671=

1198961(119904)

1198962(119904)

(31)

By the aid of (15) we obtain that ⟨n2(119904) 119880⟩ = 1198671cos120593

If derivative of this function with respect to 119904 is taken and(15) (21) and (23) are used we find that 119867

2= 1198671015840

11198963 For

the higher harmonic curvatures let us differentiate (23) withrespect to 119904 for 119895 then ⟨119899

1015840

119895+1(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 By the aid

of (15) ⟨minus119896119895(119904)ni(119904) + 119896

119895+1(119904)ni+2(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 we get

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894)120577119895+1

2 le 119895 le 5

Theorem 14 Let 120574 119868 rarr R7 be a spatial octonionic inclinedcurve Such that 120574(119904) = sum

7

119894=1120574119894(119904)119890119894

120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894 (32)

obtained from 120574 octonionic curve is an octonionic inclinedcurve

Proof Let 120573 119868 rarr 119874 be an octonionic curve given by arclength parameter 119904

Let TN1N2N3N4N5N6N7 be the Frenet appara-tus and let 119880 be unit and constant spatial octonion If we useDefinition 11 we get the following statement

⟨1205731015840(119904) 119906⟩ = ⟨T (119904) 119880⟩

T (119904) = 119878T(119904) + T(119904) 119880 = 119878119880+ 119880

(33)

where

T (119904) = 119878T(119904) + T(119904) (34)

We notice that⟨T (119904) 119880⟩

=

1

2

(T (119904) times 119880 + 119880 times T (119904))

=

1

2

[(119878T(119904) + T(119904)) times 119880] + [119880 times (119878T(119904) minus T(119904))]

(35)

Since 119880 is spatial octonion then 119880 = 119880 119880 = minus119880 Here

we can account for the product of octonion

⟨T (119904) 119880⟩

=

1

2

[119878T(119904) sdot 0 minus ⟨T(119904) minus119880⟩ + 119878T(119904)

sdot (minus119880) + 0 sdot T(119904) + T(119904) and (minus119880)]

+ [0 sdot 119878T(119904) minus ⟨119880 T(119904)⟩ + 119878T(119904)

sdot 119880 + 0 sdot T(119904) + 119880 and (minusT(119904))]

=

1

2

[⟨T(119904) 119880⟩ minus 119878T(119904)119880 minus T(119904) and 119880

+ ⟨119880 T(119904)⟩ + 119878T(119904)119880 minus 119880 and T(119904)]

= ⟨T(119904) 119880⟩

(36)

and so ⟨1205731015840(119904) 119880⟩ = cos120593 is obtainedThen 120573 curve is octon-

ionic inclined curve

Theorem 15 Let 120573 119868 rarr 119874 be an octonionic inclined curvegiven by arc length parameter 119904 Such that119870

119894(119904) are curvatures

in the point 120573(119904) 120575119894(119904) = 1119870

119894(119904) 1 le 119894 le 7 are curvature radii

and119867119894(119904) 1 le 119894 le 6 are harmonic curvatures they are

1198671=

1119888119906119903V1198861199051199061199031198902119888119906119903V119886119905119906119903119890

1198672=

1198671015840

1

(1198962minus 119870) (119904)

1198673=

1198671015840

2+ (1198962minus 119870)119867

1

1198963

1198674=

1198671015840

3+ 11989631198672

(1198964minus 119870) (119904)

1198675=

1198671015840

4+ (1198964minus 119870)119867

3

1198965

1198676=

1198671015840

5+ 11989651198674

(1198966+ 119870) (119904)

(37)

where 1198701(119904) = 119870 119870

2(119904) = 119896

1 1198703(119904) = 119896

2minus 119870 119870

4(119904) = 119896

3

1198705(119904) = 119896

4minus 119870 119870

6(119904) = 119896

5 1198707(119904) = 119896

6+ 119870

Proof 120573 119868 rarr 119874 curve is given by regular octonionic119880 is an unit and a constant spatial octonion and suchthat TN1N2N3N4N5N6N7 is Frenet apparatus in thepoint 120573(119904)

⟨T (119904) 119880⟩ = cos120593 = constant (38)

6 Mathematical Problems in Engineering

is written If derivative with respect to 119904 of this equationis taken we obtain that ⟨T1015840(119904) 119880⟩ = 0 Here using (19)⟨119870(119904)N1(119904) 119880⟩ = 0 is found Because of 119870(119904)N1(119904) = 0 wewrite as ⟨N1(119904) 119880⟩ = 0 Thus ⟨N10158401(119904) 119880⟩ = 0 is obtainedHere using (19)

minus119870 (119904) ⟨T (119904) 119880⟩ + 1198961(119904) ⟨N2 (119904) 119880⟩ = 0 (39)

is found In addition from (26) for 119894 = 1 we obtain that

⟨N2 (119904) 119880⟩ = 1198671(119904) cos120593 (40)

By taking (24) and (40) into consideration

(minus119870 (119904) + 1198961(119904)1198671(119904)) cos120593 = 0 cos120593 = 0

1198671(119904) =

119870 (119904)

1198961(119904)

=

1curvature2curvature

(41)

is found On the other hand if derivative of (40) with respectto 119904 is taken

⟨N10158402 (119904) 119880⟩ = 1198671015840

1(119904) cos120593 (42)

is found Here using (19)

minus1198961(119904) ⟨N1 (119904) 119880⟩ + (119896

2minus 119870) (119904) ⟨N3 (119904) 119880⟩ = 119867

1015840

1(119904) cos120593

(43)

is found In addition from (26) for 119894 = 2 we obtain that

⟨N3 (119904) 119880⟩ = 1198672(119904) cos120593 (44)

By taking (24) and (44) into consideration

(1198962minus 119870) (119904)119867

2(119904) cos120593 = 119867

1015840

1(119904) cos120593 (45)

is obtained Thus

1198671(119904) =

1198671015840

1(119904)

(1198962minus 119870) (119904)

(46)

If derivative of (44) with respect to 119904 is taken

⟨N10158403(119904) 119906⟩ = 119867

1015840

2(119904) cos120593 (47)

is found Here using (19)

minus (1198962minus 119870) (119904) ⟨N2 (119904) 119880⟩ + 119896

3(119904) ⟨N4 (119904) 119880⟩ = 119867

1015840

2(119904) cos120593

(48)

is found In addition from (26) for 119894 = 3 we obtain that

⟨N4 (119904) 119880⟩ = 1198673(119904) cos120593 (49)

By taking (40) and (49) into consideration

minus (1198962minus 119870) (119904)119867

1(119904) cos120593 + 119896

3(119904)1198673(119904) cos120593 = 119867

1015840

2(119904) cos120593

(50)

is obtained Thus

1198673(119904) =

1198671015840

2(119904) + (119896

2minus 119870) (119904)119867

1(119904)

1198963(119904)

(51)

Similarly If derivative of (49) and following equations withrespect to 119904 is taken

⟨N5 (119904) 119880⟩ = 1198674(119904) cos120593

⟨N6 (119904) 119880⟩ = 1198675(119904) cos120593

(52)

we get

1198674(119904) =

1198671015840

3(119904) + 119896

3(119904)1198672(119904)

(1198964minus 119870) (119904)

1198675(119904) =

1198671015840

4(119904) + (119896

4minus 119870) (119904)119867

3

1198965(119904)

1198676=

1198671015840

5(119904) + 119896

5(119904)1198674(119904)

(1198966+ 119870) (119904)

(53)

Theorem 16 120574 is a spatial octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 5 be harmonic curvatures

in the point 120574(119904) 120574 is octonionic inclined curve if and only ifsum5

119894=11198672

119894is constant

Proof (rArr) Let 120574 be a spatial octonionic curve given by arclength parameter 119904 Then there is a 119880 unit and constantspatial octonion Therefore

⟨1205741015840(119904) 119880⟩ = cos120593 (54)

is constant for 120574 spatial octonionic inclined curvewith respectto arc length parameter 119904 such that tn1n2n3n4n5n6 isbasis of spatial octonion in the point 120574(119904) spatial octonion 119880

119880 = ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904) (55)

is obtained Since 119880 is a unit

1198802= 119880 times 119880 = 1 (56)

Here using (55)

1198802= (⟨t (119904) 119880⟩ t (119904) +

6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904))

times ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904)

(57)

if we use Definition 10 in the last equation we can write

1 = (cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593)

times cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593

(58)

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

If we take 119899 = 7 8 in the study named ldquoA characterization ofinclined curves in Euclidean 119899 spacerdquo we can obtain thefollowing definitions

Definition 2 Let 120574 119868 rarr R7 be a curve in R7 with the arclength parameter 119904 and let 119906 be a unit constant vector of R7For all 119904 isin 119868 if

⟨1205741015840(119904) 119906⟩ = cos120593 = constant 120593 =

120587

2

(10)

then the curve is called an inclined curve in R7 where 1205741015840(119904)is the unit tangent vector to the curve 120574 at its point 120574(119904) and120593 is a constant angle between the vectors 1205741015840 and 119906 [4]

We can give same definition in R8

Definition 3 Let 120574 119868 rarr R7 be a curve in R7 with an arclength parameter 119904 and let 119906 be an unit constant vector Lettn1n2n3n4n5n6 3 le 119903 le 7 be the Frenet 7-frame of 120574at its point 120574(119904) If the angle between 120574

1015840(119904) and 119906 is 120593 = 120593(119904)

we define the function

119867119894 119868 997888rarr R 3 le 119894 le 119903 minus 2 (11)

by

⟨119899119894+1

(119904) 119906⟩ = 119867119894(119904) cos120593 (12)

as the harmonic curvature with order 119894 of the curve 120574 at itspoint 120574(119904) We define also119867

0= 0 [4]

We can give same definition in R8Now we are going to give some definitions and theorems

about octonionic curves in R7 and R8

Definition 4 The seven-dimensional Euclidean space R7 isconsubstantiated by the space of spatial real octonions 119874

119875=

120574 isin 119874 | 120574 + 120574 = 0 in an obvious manner Let 119868 = [0 1] be aninterval inR and let 119904 isin 119868 be the parameter along the smoothcurve

120574 119868 sub R 997888rarr 119874119875

119904 997888rarr 120574 (119904) =

7

sum

119894=1

120574119894(119904) 119890119894

(13)

Then the curve is called spatial octonionic curve or octo-nionic curve in R7 [5]

Theorem 5 The seven-dimensional Euclidean space R7 isconsubstantiated by the space of spatial real octonions 119874

119875=

120574 isin 119874 | 120574 + 120574 = 0 in an obvious manner Let 119868 = [0 1] be aninterval in R and let 119904 isin 119868 be the parameter along the smoothcurve

120574 119868 sub R 997888rarr 119874119875

119904 997888rarr 120574 (119904) =

7

sum

119894=1

120574119894(119904) 119890119894

(14)

Let tn1n2n3n4n5n6 be the Frenet trihedron of thedifferentiable Euclidean space curve in the Euclidean spaceR7Then Frenet equations are

t1015840 (119904) = 1198961(119904)n1 (119904)

n10158401 (119904) = minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904)

n10158402 (119904) = minus1198962(119904)n1 (119904) + 119896

3(119904)n3 (119904)

n10158403 (119904) = minus1198963(119904)n2 (119904) + 119896

4(119904)n4 (119904)

n10158404 (119904) = minus1198964(119904)n3 (119904) + 119896

5(119904)n5 (119904)

n10158405 (119904) = minus1198965(119904)n4 (119904) + 119896

6(119904)n6 (119904)

n10158406 (119904) = minus1198966(119904)n5 (119904)

(15)

where 119896119894 1 le 119894 le 6 curvature functions

We may state Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

[

t1015840

n10158401n10158402n10158403n10158404n10158405n10158406

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

0 1198961

0 0 0 0 0

minus1198961

0 1198962

0 0 0 0

0 minus1198962

0 1198963

0 0 0

0 0 minus1198963

0 1198964

0 0

0 0 0 minus1198964

0 1198965

0

0 0 0 0 minus1198965

0 1198966

0 0 0 0 0 minus1198966

0

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

tn1n2n3n4n5n6

]

]

]

]

]

]

]

]

]

]

(16)

This is the Serret-Frenet formulae for the spatial octonioniccurve 120574 in R7 [5]

tn1n2n3n4n5n6 1198961 1198962 1198963 1198964 1198965 1198966 is the Frenetapparatus for spatial octonionic curve 120574 in R7

Remark 6 What has been achieved in this theorem isreputable in local differential geometryWe have done this fortwo especial goals

(1) to designate the demonstration for the Serret-Frenetformulae and Frenet apparatus of the curve 120574 in R7Wewill roll the outcomes of this theorem comprehen-sively in the next theorem

(2) to indicate how octonions are to be used in designat-ing curvature numbers of curves in general

Definition 7 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let 119868 = [0 1] be aninterval inR and let 119904 isin 119868 be the parameter along the smoothcurve

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(17)

Then the curve is called octonionic curve [5]

4 Mathematical Problems in Engineering

Theorem 8 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(18)

be a smooth curve in R8 described over 119868 Let the parameter 119904be selected that T = 120573

1015840(119904) = sum

7

119894=01205741015840

119894(119904)119890119894has unit magnitude

Let TN1N2N3N4N5N6N7 be the Frenet elements of 120573Then the Frenet equations are

T1015840 (119904) = 119870 (119904)N1 (119904)

N10158401 (119904) = minus119870 (119904)T (119904) + 1198961(119904)N2 (119904)

N10158402 (119904) = minus1198961(119904)N1 (119904) + (119896

2minus 119870) (119904)N3 (119904)

N10158403 (119904) = minus (1198962minus 119870) (119904)N2 (119904) + 119896

3(119904)N4(119904)

N10158404 (119904) = minus1198963(119904)N3 (119904) + (119896

4minus 119870) (119904)N5 (119904)

N10158405 (119904) = minus (1198964minus 119870) (119904)N4 (119904) + 119896

5(119904)N6 (119904)

N10158406 (119904) = minus1198965(119904)N5 (119904) + (119896

6+ 119870) (119904)N7 (119904)

N10158407 (119904) = minus (1198966+ 119870) (119904)N6 (119904)

(19)

where N1 = t times T N2 = n1 times T N3 = n2 times T N4 = n3 times TN5 = n4 times T N6 = n5 times T and N7 = n6 times T 119870 = T1015840(119904)

We may express Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

T1015840N10158401N10158402N10158403N10158404N10158405N10158406N10158407

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

0 119870 0 0 0 0 0 0

minus119870 0 1198961

0 0 0 0 0

0 minus1198961

0 (1198962minus 119870) 0 0 0 0

0 0 minus (1198962minus 119870) 0 119896

30 0 0

0 0 0 minus1198963

0 (1198964minus 119870) 0 0

0 0 0 0 minus (1198964minus 119870) 0 119896

50

0 0 0 0 0 minus1198965

0 (1198966+ 119870)

0 0 0 0 0 0 minus (1198966+ 119870) 0

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

TN1N2N3N4N5N6N7

]

]

]

]

]

]

]

]

]

]

]

(20)

This is the Serret-Frenet formulae for octonionic curve 120573 inR8 [5]

3 Octonionic Inclined Curves andHarmonic Curvatures

Definition 9 Let 120574(119868) be spatial octonionic curve with an arclength parameter 119904 and let 119880 be an unit and constant spatialoctonion For all 119904 isin 119868 let ⟨1205741015840(119904) 119880⟩ be a constant defined by

⟨1205741015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(21)

Then 120574(119868) is called spatial octonionic inclined curve

Definition 10 120574(119868) octonionic curve is given by arc lengthparameter 119904 Let tn1n2n3n4n5n6 be the Frenet trihe-dron in the point 120574(119904) of the curve 120574 and let 119880 be unit andconstant spatial octonion such that angle120593(119904) is between 120574

1015840(119904)

and 119880

119867119894 119868 997888rarr R 1 le 119894 le 5 (22)

be a function defined by

⟨119899119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(23)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120574(119904) of the 120574 spatial octonionic curve with respect to 119906

Definition 11 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 such that 119880 is a unit and constantspatial octonion for every 119904 isin 119868

⟨1205731015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(24)

Then curve 120573 is called octonionic inclined curve in 119874

Definition 12 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 Let TN1N2N3N4N5N6N7 bethe Frenet apparatus and let119880 be unit and constant such thatangle 120593(119904) is between T1015840(119904) and 119880 Let

119867119894 119868 997888rarr R 1 le 119894 le 6 (25)

be a function defined by

⟨119873119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(26)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120573(119904) of the 120573 octonionic curve with respect to 119880

Theorem 13 Let 120574 119868 rarr R7 be spatial octonionic inclinedcurve given by arc length parameter 119904 Curvatures in the point

Mathematical Problems in Engineering 5

120574(119904) of curve 120574 are 119896119894(119904) 120577119894= 1119896

119894(119904) and 119867

119894(119904) 1 le 119894 le 6 are

harmonic curvatures they are

1198671=

1198961

1198962

1198672=

1198671015840

1

1198963

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894) 120577119895+1

2 le 119895 le 5

(27)

Proof Let 120593 be an angle between the unit and constant spatialoctonion119880 and t(119904) Such that tn1n2n3n4n5n6 Frenetapparatus in the point 120574(119904) we obtain that

⟨t (119904) 119880⟩ = cos120593 (28)

Here differentiatingwith respect to 119904 we find that ⟨t1015840(119904) 119880⟩ =

0 By the aid of (15) we obtain that ⟨n1(119904) 119880⟩ = 0 Ifderivative of this function with respect to 119904 is taken we findthat ⟨n10158401(119904) 119880⟩ = 0 Here using (15)

⟨minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904) 119880⟩ = 0 (29)

is obtained Thus if (21) and (23) are used

(minus1198961(119904) + 119896

2(119904)1198671) cos120593 = 0 cos120593 =

120587

2

(30)

is found Thus

1198671=

1198961(119904)

1198962(119904)

(31)

By the aid of (15) we obtain that ⟨n2(119904) 119880⟩ = 1198671cos120593

If derivative of this function with respect to 119904 is taken and(15) (21) and (23) are used we find that 119867

2= 1198671015840

11198963 For

the higher harmonic curvatures let us differentiate (23) withrespect to 119904 for 119895 then ⟨119899

1015840

119895+1(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 By the aid

of (15) ⟨minus119896119895(119904)ni(119904) + 119896

119895+1(119904)ni+2(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 we get

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894)120577119895+1

2 le 119895 le 5

Theorem 14 Let 120574 119868 rarr R7 be a spatial octonionic inclinedcurve Such that 120574(119904) = sum

7

119894=1120574119894(119904)119890119894

120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894 (32)

obtained from 120574 octonionic curve is an octonionic inclinedcurve

Proof Let 120573 119868 rarr 119874 be an octonionic curve given by arclength parameter 119904

Let TN1N2N3N4N5N6N7 be the Frenet appara-tus and let 119880 be unit and constant spatial octonion If we useDefinition 11 we get the following statement

⟨1205731015840(119904) 119906⟩ = ⟨T (119904) 119880⟩

T (119904) = 119878T(119904) + T(119904) 119880 = 119878119880+ 119880

(33)

where

T (119904) = 119878T(119904) + T(119904) (34)

We notice that⟨T (119904) 119880⟩

=

1

2

(T (119904) times 119880 + 119880 times T (119904))

=

1

2

[(119878T(119904) + T(119904)) times 119880] + [119880 times (119878T(119904) minus T(119904))]

(35)

Since 119880 is spatial octonion then 119880 = 119880 119880 = minus119880 Here

we can account for the product of octonion

⟨T (119904) 119880⟩

=

1

2

[119878T(119904) sdot 0 minus ⟨T(119904) minus119880⟩ + 119878T(119904)

sdot (minus119880) + 0 sdot T(119904) + T(119904) and (minus119880)]

+ [0 sdot 119878T(119904) minus ⟨119880 T(119904)⟩ + 119878T(119904)

sdot 119880 + 0 sdot T(119904) + 119880 and (minusT(119904))]

=

1

2

[⟨T(119904) 119880⟩ minus 119878T(119904)119880 minus T(119904) and 119880

+ ⟨119880 T(119904)⟩ + 119878T(119904)119880 minus 119880 and T(119904)]

= ⟨T(119904) 119880⟩

(36)

and so ⟨1205731015840(119904) 119880⟩ = cos120593 is obtainedThen 120573 curve is octon-

ionic inclined curve

Theorem 15 Let 120573 119868 rarr 119874 be an octonionic inclined curvegiven by arc length parameter 119904 Such that119870

119894(119904) are curvatures

in the point 120573(119904) 120575119894(119904) = 1119870

119894(119904) 1 le 119894 le 7 are curvature radii

and119867119894(119904) 1 le 119894 le 6 are harmonic curvatures they are

1198671=

1119888119906119903V1198861199051199061199031198902119888119906119903V119886119905119906119903119890

1198672=

1198671015840

1

(1198962minus 119870) (119904)

1198673=

1198671015840

2+ (1198962minus 119870)119867

1

1198963

1198674=

1198671015840

3+ 11989631198672

(1198964minus 119870) (119904)

1198675=

1198671015840

4+ (1198964minus 119870)119867

3

1198965

1198676=

1198671015840

5+ 11989651198674

(1198966+ 119870) (119904)

(37)

where 1198701(119904) = 119870 119870

2(119904) = 119896

1 1198703(119904) = 119896

2minus 119870 119870

4(119904) = 119896

3

1198705(119904) = 119896

4minus 119870 119870

6(119904) = 119896

5 1198707(119904) = 119896

6+ 119870

Proof 120573 119868 rarr 119874 curve is given by regular octonionic119880 is an unit and a constant spatial octonion and suchthat TN1N2N3N4N5N6N7 is Frenet apparatus in thepoint 120573(119904)

⟨T (119904) 119880⟩ = cos120593 = constant (38)

6 Mathematical Problems in Engineering

is written If derivative with respect to 119904 of this equationis taken we obtain that ⟨T1015840(119904) 119880⟩ = 0 Here using (19)⟨119870(119904)N1(119904) 119880⟩ = 0 is found Because of 119870(119904)N1(119904) = 0 wewrite as ⟨N1(119904) 119880⟩ = 0 Thus ⟨N10158401(119904) 119880⟩ = 0 is obtainedHere using (19)

minus119870 (119904) ⟨T (119904) 119880⟩ + 1198961(119904) ⟨N2 (119904) 119880⟩ = 0 (39)

is found In addition from (26) for 119894 = 1 we obtain that

⟨N2 (119904) 119880⟩ = 1198671(119904) cos120593 (40)

By taking (24) and (40) into consideration

(minus119870 (119904) + 1198961(119904)1198671(119904)) cos120593 = 0 cos120593 = 0

1198671(119904) =

119870 (119904)

1198961(119904)

=

1curvature2curvature

(41)

is found On the other hand if derivative of (40) with respectto 119904 is taken

⟨N10158402 (119904) 119880⟩ = 1198671015840

1(119904) cos120593 (42)

is found Here using (19)

minus1198961(119904) ⟨N1 (119904) 119880⟩ + (119896

2minus 119870) (119904) ⟨N3 (119904) 119880⟩ = 119867

1015840

1(119904) cos120593

(43)

is found In addition from (26) for 119894 = 2 we obtain that

⟨N3 (119904) 119880⟩ = 1198672(119904) cos120593 (44)

By taking (24) and (44) into consideration

(1198962minus 119870) (119904)119867

2(119904) cos120593 = 119867

1015840

1(119904) cos120593 (45)

is obtained Thus

1198671(119904) =

1198671015840

1(119904)

(1198962minus 119870) (119904)

(46)

If derivative of (44) with respect to 119904 is taken

⟨N10158403(119904) 119906⟩ = 119867

1015840

2(119904) cos120593 (47)

is found Here using (19)

minus (1198962minus 119870) (119904) ⟨N2 (119904) 119880⟩ + 119896

3(119904) ⟨N4 (119904) 119880⟩ = 119867

1015840

2(119904) cos120593

(48)

is found In addition from (26) for 119894 = 3 we obtain that

⟨N4 (119904) 119880⟩ = 1198673(119904) cos120593 (49)

By taking (40) and (49) into consideration

minus (1198962minus 119870) (119904)119867

1(119904) cos120593 + 119896

3(119904)1198673(119904) cos120593 = 119867

1015840

2(119904) cos120593

(50)

is obtained Thus

1198673(119904) =

1198671015840

2(119904) + (119896

2minus 119870) (119904)119867

1(119904)

1198963(119904)

(51)

Similarly If derivative of (49) and following equations withrespect to 119904 is taken

⟨N5 (119904) 119880⟩ = 1198674(119904) cos120593

⟨N6 (119904) 119880⟩ = 1198675(119904) cos120593

(52)

we get

1198674(119904) =

1198671015840

3(119904) + 119896

3(119904)1198672(119904)

(1198964minus 119870) (119904)

1198675(119904) =

1198671015840

4(119904) + (119896

4minus 119870) (119904)119867

3

1198965(119904)

1198676=

1198671015840

5(119904) + 119896

5(119904)1198674(119904)

(1198966+ 119870) (119904)

(53)

Theorem 16 120574 is a spatial octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 5 be harmonic curvatures

in the point 120574(119904) 120574 is octonionic inclined curve if and only ifsum5

119894=11198672

119894is constant

Proof (rArr) Let 120574 be a spatial octonionic curve given by arclength parameter 119904 Then there is a 119880 unit and constantspatial octonion Therefore

⟨1205741015840(119904) 119880⟩ = cos120593 (54)

is constant for 120574 spatial octonionic inclined curvewith respectto arc length parameter 119904 such that tn1n2n3n4n5n6 isbasis of spatial octonion in the point 120574(119904) spatial octonion 119880

119880 = ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904) (55)

is obtained Since 119880 is a unit

1198802= 119880 times 119880 = 1 (56)

Here using (55)

1198802= (⟨t (119904) 119880⟩ t (119904) +

6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904))

times ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904)

(57)

if we use Definition 10 in the last equation we can write

1 = (cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593)

times cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593

(58)

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Theorem 8 The eight-dimensional Euclidean space R8 isassimilated into the space of real octonion Let

120573 119868 sub R 997888rarr 119874

119904 997888rarr 120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894

(18)

be a smooth curve in R8 described over 119868 Let the parameter 119904be selected that T = 120573

1015840(119904) = sum

7

119894=01205741015840

119894(119904)119890119894has unit magnitude

Let TN1N2N3N4N5N6N7 be the Frenet elements of 120573Then the Frenet equations are

T1015840 (119904) = 119870 (119904)N1 (119904)

N10158401 (119904) = minus119870 (119904)T (119904) + 1198961(119904)N2 (119904)

N10158402 (119904) = minus1198961(119904)N1 (119904) + (119896

2minus 119870) (119904)N3 (119904)

N10158403 (119904) = minus (1198962minus 119870) (119904)N2 (119904) + 119896

3(119904)N4(119904)

N10158404 (119904) = minus1198963(119904)N3 (119904) + (119896

4minus 119870) (119904)N5 (119904)

N10158405 (119904) = minus (1198964minus 119870) (119904)N4 (119904) + 119896

5(119904)N6 (119904)

N10158406 (119904) = minus1198965(119904)N5 (119904) + (119896

6+ 119870) (119904)N7 (119904)

N10158407 (119904) = minus (1198966+ 119870) (119904)N6 (119904)

(19)

where N1 = t times T N2 = n1 times T N3 = n2 times T N4 = n3 times TN5 = n4 times T N6 = n5 times T and N7 = n6 times T 119870 = T1015840(119904)

We may express Frenet formulae of the Frenet apparatus inthe matrix form

[

[

[

[

[

[

[

[

[

[

[

[

T1015840N10158401N10158402N10158403N10158404N10158405N10158406N10158407

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

0 119870 0 0 0 0 0 0

minus119870 0 1198961

0 0 0 0 0

0 minus1198961

0 (1198962minus 119870) 0 0 0 0

0 0 minus (1198962minus 119870) 0 119896

30 0 0

0 0 0 minus1198963

0 (1198964minus 119870) 0 0

0 0 0 0 minus (1198964minus 119870) 0 119896

50

0 0 0 0 0 minus1198965

0 (1198966+ 119870)

0 0 0 0 0 0 minus (1198966+ 119870) 0

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

TN1N2N3N4N5N6N7

]

]

]

]

]

]

]

]

]

]

]

(20)

This is the Serret-Frenet formulae for octonionic curve 120573 inR8 [5]

3 Octonionic Inclined Curves andHarmonic Curvatures

Definition 9 Let 120574(119868) be spatial octonionic curve with an arclength parameter 119904 and let 119880 be an unit and constant spatialoctonion For all 119904 isin 119868 let ⟨1205741015840(119904) 119880⟩ be a constant defined by

⟨1205741015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(21)

Then 120574(119868) is called spatial octonionic inclined curve

Definition 10 120574(119868) octonionic curve is given by arc lengthparameter 119904 Let tn1n2n3n4n5n6 be the Frenet trihe-dron in the point 120574(119904) of the curve 120574 and let 119880 be unit andconstant spatial octonion such that angle120593(119904) is between 120574

1015840(119904)

and 119880

119867119894 119868 997888rarr R 1 le 119894 le 5 (22)

be a function defined by

⟨119899119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(23)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120574(119904) of the 120574 spatial octonionic curve with respect to 119906

Definition 11 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 such that 119880 is a unit and constantspatial octonion for every 119904 isin 119868

⟨1205731015840(119904) 119880⟩ = cos120593 = constant 120593 =

120587

2

(24)

Then curve 120573 is called octonionic inclined curve in 119874

Definition 12 120573 119868 sub R rarr 119874 octonionic curve is given byarc length parameter 119904 Let TN1N2N3N4N5N6N7 bethe Frenet apparatus and let119880 be unit and constant such thatangle 120593(119904) is between T1015840(119904) and 119880 Let

119867119894 119868 997888rarr R 1 le 119894 le 6 (25)

be a function defined by

⟨119873119894+1

(119904) 119880⟩ = 119867119894cos120593 = constant 120593 =

120587

2

(26)

Then functions 119867119894are called 119894th Harmonic curvature in the

point 120573(119904) of the 120573 octonionic curve with respect to 119880

Theorem 13 Let 120574 119868 rarr R7 be spatial octonionic inclinedcurve given by arc length parameter 119904 Curvatures in the point

Mathematical Problems in Engineering 5

120574(119904) of curve 120574 are 119896119894(119904) 120577119894= 1119896

119894(119904) and 119867

119894(119904) 1 le 119894 le 6 are

harmonic curvatures they are

1198671=

1198961

1198962

1198672=

1198671015840

1

1198963

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894) 120577119895+1

2 le 119895 le 5

(27)

Proof Let 120593 be an angle between the unit and constant spatialoctonion119880 and t(119904) Such that tn1n2n3n4n5n6 Frenetapparatus in the point 120574(119904) we obtain that

⟨t (119904) 119880⟩ = cos120593 (28)

Here differentiatingwith respect to 119904 we find that ⟨t1015840(119904) 119880⟩ =

0 By the aid of (15) we obtain that ⟨n1(119904) 119880⟩ = 0 Ifderivative of this function with respect to 119904 is taken we findthat ⟨n10158401(119904) 119880⟩ = 0 Here using (15)

⟨minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904) 119880⟩ = 0 (29)

is obtained Thus if (21) and (23) are used

(minus1198961(119904) + 119896

2(119904)1198671) cos120593 = 0 cos120593 =

120587

2

(30)

is found Thus

1198671=

1198961(119904)

1198962(119904)

(31)

By the aid of (15) we obtain that ⟨n2(119904) 119880⟩ = 1198671cos120593

If derivative of this function with respect to 119904 is taken and(15) (21) and (23) are used we find that 119867

2= 1198671015840

11198963 For

the higher harmonic curvatures let us differentiate (23) withrespect to 119904 for 119895 then ⟨119899

1015840

119895+1(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 By the aid

of (15) ⟨minus119896119895(119904)ni(119904) + 119896

119895+1(119904)ni+2(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 we get

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894)120577119895+1

2 le 119895 le 5

Theorem 14 Let 120574 119868 rarr R7 be a spatial octonionic inclinedcurve Such that 120574(119904) = sum

7

119894=1120574119894(119904)119890119894

120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894 (32)

obtained from 120574 octonionic curve is an octonionic inclinedcurve

Proof Let 120573 119868 rarr 119874 be an octonionic curve given by arclength parameter 119904

Let TN1N2N3N4N5N6N7 be the Frenet appara-tus and let 119880 be unit and constant spatial octonion If we useDefinition 11 we get the following statement

⟨1205731015840(119904) 119906⟩ = ⟨T (119904) 119880⟩

T (119904) = 119878T(119904) + T(119904) 119880 = 119878119880+ 119880

(33)

where

T (119904) = 119878T(119904) + T(119904) (34)

We notice that⟨T (119904) 119880⟩

=

1

2

(T (119904) times 119880 + 119880 times T (119904))

=

1

2

[(119878T(119904) + T(119904)) times 119880] + [119880 times (119878T(119904) minus T(119904))]

(35)

Since 119880 is spatial octonion then 119880 = 119880 119880 = minus119880 Here

we can account for the product of octonion

⟨T (119904) 119880⟩

=

1

2

[119878T(119904) sdot 0 minus ⟨T(119904) minus119880⟩ + 119878T(119904)

sdot (minus119880) + 0 sdot T(119904) + T(119904) and (minus119880)]

+ [0 sdot 119878T(119904) minus ⟨119880 T(119904)⟩ + 119878T(119904)

sdot 119880 + 0 sdot T(119904) + 119880 and (minusT(119904))]

=

1

2

[⟨T(119904) 119880⟩ minus 119878T(119904)119880 minus T(119904) and 119880

+ ⟨119880 T(119904)⟩ + 119878T(119904)119880 minus 119880 and T(119904)]

= ⟨T(119904) 119880⟩

(36)

and so ⟨1205731015840(119904) 119880⟩ = cos120593 is obtainedThen 120573 curve is octon-

ionic inclined curve

Theorem 15 Let 120573 119868 rarr 119874 be an octonionic inclined curvegiven by arc length parameter 119904 Such that119870

119894(119904) are curvatures

in the point 120573(119904) 120575119894(119904) = 1119870

119894(119904) 1 le 119894 le 7 are curvature radii

and119867119894(119904) 1 le 119894 le 6 are harmonic curvatures they are

1198671=

1119888119906119903V1198861199051199061199031198902119888119906119903V119886119905119906119903119890

1198672=

1198671015840

1

(1198962minus 119870) (119904)

1198673=

1198671015840

2+ (1198962minus 119870)119867

1

1198963

1198674=

1198671015840

3+ 11989631198672

(1198964minus 119870) (119904)

1198675=

1198671015840

4+ (1198964minus 119870)119867

3

1198965

1198676=

1198671015840

5+ 11989651198674

(1198966+ 119870) (119904)

(37)

where 1198701(119904) = 119870 119870

2(119904) = 119896

1 1198703(119904) = 119896

2minus 119870 119870

4(119904) = 119896

3

1198705(119904) = 119896

4minus 119870 119870

6(119904) = 119896

5 1198707(119904) = 119896

6+ 119870

Proof 120573 119868 rarr 119874 curve is given by regular octonionic119880 is an unit and a constant spatial octonion and suchthat TN1N2N3N4N5N6N7 is Frenet apparatus in thepoint 120573(119904)

⟨T (119904) 119880⟩ = cos120593 = constant (38)

6 Mathematical Problems in Engineering

is written If derivative with respect to 119904 of this equationis taken we obtain that ⟨T1015840(119904) 119880⟩ = 0 Here using (19)⟨119870(119904)N1(119904) 119880⟩ = 0 is found Because of 119870(119904)N1(119904) = 0 wewrite as ⟨N1(119904) 119880⟩ = 0 Thus ⟨N10158401(119904) 119880⟩ = 0 is obtainedHere using (19)

minus119870 (119904) ⟨T (119904) 119880⟩ + 1198961(119904) ⟨N2 (119904) 119880⟩ = 0 (39)

is found In addition from (26) for 119894 = 1 we obtain that

⟨N2 (119904) 119880⟩ = 1198671(119904) cos120593 (40)

By taking (24) and (40) into consideration

(minus119870 (119904) + 1198961(119904)1198671(119904)) cos120593 = 0 cos120593 = 0

1198671(119904) =

119870 (119904)

1198961(119904)

=

1curvature2curvature

(41)

is found On the other hand if derivative of (40) with respectto 119904 is taken

⟨N10158402 (119904) 119880⟩ = 1198671015840

1(119904) cos120593 (42)

is found Here using (19)

minus1198961(119904) ⟨N1 (119904) 119880⟩ + (119896

2minus 119870) (119904) ⟨N3 (119904) 119880⟩ = 119867

1015840

1(119904) cos120593

(43)

is found In addition from (26) for 119894 = 2 we obtain that

⟨N3 (119904) 119880⟩ = 1198672(119904) cos120593 (44)

By taking (24) and (44) into consideration

(1198962minus 119870) (119904)119867

2(119904) cos120593 = 119867

1015840

1(119904) cos120593 (45)

is obtained Thus

1198671(119904) =

1198671015840

1(119904)

(1198962minus 119870) (119904)

(46)

If derivative of (44) with respect to 119904 is taken

⟨N10158403(119904) 119906⟩ = 119867

1015840

2(119904) cos120593 (47)

is found Here using (19)

minus (1198962minus 119870) (119904) ⟨N2 (119904) 119880⟩ + 119896

3(119904) ⟨N4 (119904) 119880⟩ = 119867

1015840

2(119904) cos120593

(48)

is found In addition from (26) for 119894 = 3 we obtain that

⟨N4 (119904) 119880⟩ = 1198673(119904) cos120593 (49)

By taking (40) and (49) into consideration

minus (1198962minus 119870) (119904)119867

1(119904) cos120593 + 119896

3(119904)1198673(119904) cos120593 = 119867

1015840

2(119904) cos120593

(50)

is obtained Thus

1198673(119904) =

1198671015840

2(119904) + (119896

2minus 119870) (119904)119867

1(119904)

1198963(119904)

(51)

Similarly If derivative of (49) and following equations withrespect to 119904 is taken

⟨N5 (119904) 119880⟩ = 1198674(119904) cos120593

⟨N6 (119904) 119880⟩ = 1198675(119904) cos120593

(52)

we get

1198674(119904) =

1198671015840

3(119904) + 119896

3(119904)1198672(119904)

(1198964minus 119870) (119904)

1198675(119904) =

1198671015840

4(119904) + (119896

4minus 119870) (119904)119867

3

1198965(119904)

1198676=

1198671015840

5(119904) + 119896

5(119904)1198674(119904)

(1198966+ 119870) (119904)

(53)

Theorem 16 120574 is a spatial octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 5 be harmonic curvatures

in the point 120574(119904) 120574 is octonionic inclined curve if and only ifsum5

119894=11198672

119894is constant

Proof (rArr) Let 120574 be a spatial octonionic curve given by arclength parameter 119904 Then there is a 119880 unit and constantspatial octonion Therefore

⟨1205741015840(119904) 119880⟩ = cos120593 (54)

is constant for 120574 spatial octonionic inclined curvewith respectto arc length parameter 119904 such that tn1n2n3n4n5n6 isbasis of spatial octonion in the point 120574(119904) spatial octonion 119880

119880 = ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904) (55)

is obtained Since 119880 is a unit

1198802= 119880 times 119880 = 1 (56)

Here using (55)

1198802= (⟨t (119904) 119880⟩ t (119904) +

6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904))

times ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904)

(57)

if we use Definition 10 in the last equation we can write

1 = (cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593)

times cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593

(58)

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

120574(119904) of curve 120574 are 119896119894(119904) 120577119894= 1119896

119894(119904) and 119867

119894(119904) 1 le 119894 le 6 are

harmonic curvatures they are

1198671=

1198961

1198962

1198672=

1198671015840

1

1198963

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894) 120577119895+1

2 le 119895 le 5

(27)

Proof Let 120593 be an angle between the unit and constant spatialoctonion119880 and t(119904) Such that tn1n2n3n4n5n6 Frenetapparatus in the point 120574(119904) we obtain that

⟨t (119904) 119880⟩ = cos120593 (28)

Here differentiatingwith respect to 119904 we find that ⟨t1015840(119904) 119880⟩ =

0 By the aid of (15) we obtain that ⟨n1(119904) 119880⟩ = 0 Ifderivative of this function with respect to 119904 is taken we findthat ⟨n10158401(119904) 119880⟩ = 0 Here using (15)

⟨minus1198961(119904) t (119904) + 119896

2(119904)n2 (119904) 119880⟩ = 0 (29)

is obtained Thus if (21) and (23) are used

(minus1198961(119904) + 119896

2(119904)1198671) cos120593 = 0 cos120593 =

120587

2

(30)

is found Thus

1198671=

1198961(119904)

1198962(119904)

(31)

By the aid of (15) we obtain that ⟨n2(119904) 119880⟩ = 1198671cos120593

If derivative of this function with respect to 119904 is taken and(15) (21) and (23) are used we find that 119867

2= 1198671015840

11198963 For

the higher harmonic curvatures let us differentiate (23) withrespect to 119904 for 119895 then ⟨119899

1015840

119895+1(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 By the aid

of (15) ⟨minus119896119895(119904)ni(119904) + 119896

119895+1(119904)ni+2(119904) 119880⟩ = 119867

1015840

119895minus1cos120593 we get

119867119895= (1198671015840

119895minus1+ 1198671015840

119895minus2119896119894)120577119895+1

2 le 119895 le 5

Theorem 14 Let 120574 119868 rarr R7 be a spatial octonionic inclinedcurve Such that 120574(119904) = sum

7

119894=1120574119894(119904)119890119894

120573 (119904) =

7

sum

119894=0

120574119894(119904) 119890119894 (32)

obtained from 120574 octonionic curve is an octonionic inclinedcurve

Proof Let 120573 119868 rarr 119874 be an octonionic curve given by arclength parameter 119904

Let TN1N2N3N4N5N6N7 be the Frenet appara-tus and let 119880 be unit and constant spatial octonion If we useDefinition 11 we get the following statement

⟨1205731015840(119904) 119906⟩ = ⟨T (119904) 119880⟩

T (119904) = 119878T(119904) + T(119904) 119880 = 119878119880+ 119880

(33)

where

T (119904) = 119878T(119904) + T(119904) (34)

We notice that⟨T (119904) 119880⟩

=

1

2

(T (119904) times 119880 + 119880 times T (119904))

=

1

2

[(119878T(119904) + T(119904)) times 119880] + [119880 times (119878T(119904) minus T(119904))]

(35)

Since 119880 is spatial octonion then 119880 = 119880 119880 = minus119880 Here

we can account for the product of octonion

⟨T (119904) 119880⟩

=

1

2

[119878T(119904) sdot 0 minus ⟨T(119904) minus119880⟩ + 119878T(119904)

sdot (minus119880) + 0 sdot T(119904) + T(119904) and (minus119880)]

+ [0 sdot 119878T(119904) minus ⟨119880 T(119904)⟩ + 119878T(119904)

sdot 119880 + 0 sdot T(119904) + 119880 and (minusT(119904))]

=

1

2

[⟨T(119904) 119880⟩ minus 119878T(119904)119880 minus T(119904) and 119880

+ ⟨119880 T(119904)⟩ + 119878T(119904)119880 minus 119880 and T(119904)]

= ⟨T(119904) 119880⟩

(36)

and so ⟨1205731015840(119904) 119880⟩ = cos120593 is obtainedThen 120573 curve is octon-

ionic inclined curve

Theorem 15 Let 120573 119868 rarr 119874 be an octonionic inclined curvegiven by arc length parameter 119904 Such that119870

119894(119904) are curvatures

in the point 120573(119904) 120575119894(119904) = 1119870

119894(119904) 1 le 119894 le 7 are curvature radii

and119867119894(119904) 1 le 119894 le 6 are harmonic curvatures they are

1198671=

1119888119906119903V1198861199051199061199031198902119888119906119903V119886119905119906119903119890

1198672=

1198671015840

1

(1198962minus 119870) (119904)

1198673=

1198671015840

2+ (1198962minus 119870)119867

1

1198963

1198674=

1198671015840

3+ 11989631198672

(1198964minus 119870) (119904)

1198675=

1198671015840

4+ (1198964minus 119870)119867

3

1198965

1198676=

1198671015840

5+ 11989651198674

(1198966+ 119870) (119904)

(37)

where 1198701(119904) = 119870 119870

2(119904) = 119896

1 1198703(119904) = 119896

2minus 119870 119870

4(119904) = 119896

3

1198705(119904) = 119896

4minus 119870 119870

6(119904) = 119896

5 1198707(119904) = 119896

6+ 119870

Proof 120573 119868 rarr 119874 curve is given by regular octonionic119880 is an unit and a constant spatial octonion and suchthat TN1N2N3N4N5N6N7 is Frenet apparatus in thepoint 120573(119904)

⟨T (119904) 119880⟩ = cos120593 = constant (38)

6 Mathematical Problems in Engineering

is written If derivative with respect to 119904 of this equationis taken we obtain that ⟨T1015840(119904) 119880⟩ = 0 Here using (19)⟨119870(119904)N1(119904) 119880⟩ = 0 is found Because of 119870(119904)N1(119904) = 0 wewrite as ⟨N1(119904) 119880⟩ = 0 Thus ⟨N10158401(119904) 119880⟩ = 0 is obtainedHere using (19)

minus119870 (119904) ⟨T (119904) 119880⟩ + 1198961(119904) ⟨N2 (119904) 119880⟩ = 0 (39)

is found In addition from (26) for 119894 = 1 we obtain that

⟨N2 (119904) 119880⟩ = 1198671(119904) cos120593 (40)

By taking (24) and (40) into consideration

(minus119870 (119904) + 1198961(119904)1198671(119904)) cos120593 = 0 cos120593 = 0

1198671(119904) =

119870 (119904)

1198961(119904)

=

1curvature2curvature

(41)

is found On the other hand if derivative of (40) with respectto 119904 is taken

⟨N10158402 (119904) 119880⟩ = 1198671015840

1(119904) cos120593 (42)

is found Here using (19)

minus1198961(119904) ⟨N1 (119904) 119880⟩ + (119896

2minus 119870) (119904) ⟨N3 (119904) 119880⟩ = 119867

1015840

1(119904) cos120593

(43)

is found In addition from (26) for 119894 = 2 we obtain that

⟨N3 (119904) 119880⟩ = 1198672(119904) cos120593 (44)

By taking (24) and (44) into consideration

(1198962minus 119870) (119904)119867

2(119904) cos120593 = 119867

1015840

1(119904) cos120593 (45)

is obtained Thus

1198671(119904) =

1198671015840

1(119904)

(1198962minus 119870) (119904)

(46)

If derivative of (44) with respect to 119904 is taken

⟨N10158403(119904) 119906⟩ = 119867

1015840

2(119904) cos120593 (47)

is found Here using (19)

minus (1198962minus 119870) (119904) ⟨N2 (119904) 119880⟩ + 119896

3(119904) ⟨N4 (119904) 119880⟩ = 119867

1015840

2(119904) cos120593

(48)

is found In addition from (26) for 119894 = 3 we obtain that

⟨N4 (119904) 119880⟩ = 1198673(119904) cos120593 (49)

By taking (40) and (49) into consideration

minus (1198962minus 119870) (119904)119867

1(119904) cos120593 + 119896

3(119904)1198673(119904) cos120593 = 119867

1015840

2(119904) cos120593

(50)

is obtained Thus

1198673(119904) =

1198671015840

2(119904) + (119896

2minus 119870) (119904)119867

1(119904)

1198963(119904)

(51)

Similarly If derivative of (49) and following equations withrespect to 119904 is taken

⟨N5 (119904) 119880⟩ = 1198674(119904) cos120593

⟨N6 (119904) 119880⟩ = 1198675(119904) cos120593

(52)

we get

1198674(119904) =

1198671015840

3(119904) + 119896

3(119904)1198672(119904)

(1198964minus 119870) (119904)

1198675(119904) =

1198671015840

4(119904) + (119896

4minus 119870) (119904)119867

3

1198965(119904)

1198676=

1198671015840

5(119904) + 119896

5(119904)1198674(119904)

(1198966+ 119870) (119904)

(53)

Theorem 16 120574 is a spatial octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 5 be harmonic curvatures

in the point 120574(119904) 120574 is octonionic inclined curve if and only ifsum5

119894=11198672

119894is constant

Proof (rArr) Let 120574 be a spatial octonionic curve given by arclength parameter 119904 Then there is a 119880 unit and constantspatial octonion Therefore

⟨1205741015840(119904) 119880⟩ = cos120593 (54)

is constant for 120574 spatial octonionic inclined curvewith respectto arc length parameter 119904 such that tn1n2n3n4n5n6 isbasis of spatial octonion in the point 120574(119904) spatial octonion 119880

119880 = ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904) (55)

is obtained Since 119880 is a unit

1198802= 119880 times 119880 = 1 (56)

Here using (55)

1198802= (⟨t (119904) 119880⟩ t (119904) +

6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904))

times ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904)

(57)

if we use Definition 10 in the last equation we can write

1 = (cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593)

times cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593

(58)

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

is written If derivative with respect to 119904 of this equationis taken we obtain that ⟨T1015840(119904) 119880⟩ = 0 Here using (19)⟨119870(119904)N1(119904) 119880⟩ = 0 is found Because of 119870(119904)N1(119904) = 0 wewrite as ⟨N1(119904) 119880⟩ = 0 Thus ⟨N10158401(119904) 119880⟩ = 0 is obtainedHere using (19)

minus119870 (119904) ⟨T (119904) 119880⟩ + 1198961(119904) ⟨N2 (119904) 119880⟩ = 0 (39)

is found In addition from (26) for 119894 = 1 we obtain that

⟨N2 (119904) 119880⟩ = 1198671(119904) cos120593 (40)

By taking (24) and (40) into consideration

(minus119870 (119904) + 1198961(119904)1198671(119904)) cos120593 = 0 cos120593 = 0

1198671(119904) =

119870 (119904)

1198961(119904)

=

1curvature2curvature

(41)

is found On the other hand if derivative of (40) with respectto 119904 is taken

⟨N10158402 (119904) 119880⟩ = 1198671015840

1(119904) cos120593 (42)

is found Here using (19)

minus1198961(119904) ⟨N1 (119904) 119880⟩ + (119896

2minus 119870) (119904) ⟨N3 (119904) 119880⟩ = 119867

1015840

1(119904) cos120593

(43)

is found In addition from (26) for 119894 = 2 we obtain that

⟨N3 (119904) 119880⟩ = 1198672(119904) cos120593 (44)

By taking (24) and (44) into consideration

(1198962minus 119870) (119904)119867

2(119904) cos120593 = 119867

1015840

1(119904) cos120593 (45)

is obtained Thus

1198671(119904) =

1198671015840

1(119904)

(1198962minus 119870) (119904)

(46)

If derivative of (44) with respect to 119904 is taken

⟨N10158403(119904) 119906⟩ = 119867

1015840

2(119904) cos120593 (47)

is found Here using (19)

minus (1198962minus 119870) (119904) ⟨N2 (119904) 119880⟩ + 119896

3(119904) ⟨N4 (119904) 119880⟩ = 119867

1015840

2(119904) cos120593

(48)

is found In addition from (26) for 119894 = 3 we obtain that

⟨N4 (119904) 119880⟩ = 1198673(119904) cos120593 (49)

By taking (40) and (49) into consideration

minus (1198962minus 119870) (119904)119867

1(119904) cos120593 + 119896

3(119904)1198673(119904) cos120593 = 119867

1015840

2(119904) cos120593

(50)

is obtained Thus

1198673(119904) =

1198671015840

2(119904) + (119896

2minus 119870) (119904)119867

1(119904)

1198963(119904)

(51)

Similarly If derivative of (49) and following equations withrespect to 119904 is taken

⟨N5 (119904) 119880⟩ = 1198674(119904) cos120593

⟨N6 (119904) 119880⟩ = 1198675(119904) cos120593

(52)

we get

1198674(119904) =

1198671015840

3(119904) + 119896

3(119904)1198672(119904)

(1198964minus 119870) (119904)

1198675(119904) =

1198671015840

4(119904) + (119896

4minus 119870) (119904)119867

3

1198965(119904)

1198676=

1198671015840

5(119904) + 119896

5(119904)1198674(119904)

(1198966+ 119870) (119904)

(53)

Theorem 16 120574 is a spatial octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 5 be harmonic curvatures

in the point 120574(119904) 120574 is octonionic inclined curve if and only ifsum5

119894=11198672

119894is constant

Proof (rArr) Let 120574 be a spatial octonionic curve given by arclength parameter 119904 Then there is a 119880 unit and constantspatial octonion Therefore

⟨1205741015840(119904) 119880⟩ = cos120593 (54)

is constant for 120574 spatial octonionic inclined curvewith respectto arc length parameter 119904 such that tn1n2n3n4n5n6 isbasis of spatial octonion in the point 120574(119904) spatial octonion 119880

119880 = ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904) (55)

is obtained Since 119880 is a unit

1198802= 119880 times 119880 = 1 (56)

Here using (55)

1198802= (⟨t (119904) 119880⟩ t (119904) +

6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904))

times ⟨t (119904) 119880⟩ t (119904) +6

sum

119894=1

⟨n119894(119904) 119880⟩n

119894(119904)

(57)

if we use Definition 10 in the last equation we can write

1 = (cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593)

times cos120593119905+5

sum

119894=0

119867119894(119904)n119894+1

(119904) cos120593

(58)

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

From octonionic product we have

1 = cos2120593+5

sum

119894=1

1198672

119894(119904) cos2120593 (59)

where5

sum

119894=1

1198672

119894(119904) = tan2120593 = constant (60)

(lArr) In contrast suppose thatsum5119894=1

1198672

119894(119904) is constant for 120574

spatial octonionic curve It is study to show that ⟨1205741015840(119904) 119880⟩ =

cos120593 Therefore there is 120593 angle so that tan2120593 = 119886 Thus wedefine 119880 spatial octonion where

119880 = cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593 (61)

Here we demonstrate that 119906 is a constant Thus if derivativeof (61) with respect to 119904 is taken

1

cos120593119889119880

119889119904

= 1199051015840+

6

sum

119894=2

1198671015840

119894minus1(119904)n119894(119904) +

6

sum

119894=2

119867119894minus1

(119904)n1015840119894(119904)

1

cos120593119889119880

119889119904

= 1199051015840+ 1198671015840

1n2+ 1198671015840

2n3+ 1198671015840

3n4+ 1198671015840

4n5+ 1198671015840

5n6

+ 1198671n10158402+ 1198672n10158403+ 1198673n10158404+ 1198674n10158405+ 1198675n10158406

(62)

is found On the other hand

⟨1198993(119904) 119880⟩ = 119867

2cos120593 997904rArr ⟨119899

1015840

3(119904) 119880⟩ = 119867

1015840

2cos120593 (63)

is obtained Here using (15)

1198671015840

2= minus11989631198671+ 11989641198673

(64)is obtained Similarly

1198671015840

3= minus11989641198672+ 11989651198674

1198671015840

4= minus11989651198673+ 11989661198675

1198671015840

5= minus11989661198674

(65)

Finally we get1

cos120593119889119880

119889119904

= 0 (66)

Thus 119906 is a constant On the other hand

1198802= 119880 times 119880 (67)

1198802= (cos120593119905+

6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593)

times cos120593119905+6

sum

119894=2

119867119894minus1

(119904)n119894(119904) cos120593

= cos2120593 + cos2120593(

5

sum

119894=1

1198672

119894(119904))

= 1

(68)

is obtained Thus

⟨119905 (119904) 119880⟩ =

1

2

(119905 times 119880 + 119880 times 119905)

= cos120593(69)

is found Therefore 120574 is an inclined curve

Theorem 17 120573 is an octonionic curve given by arc lengthparameter 119904 And let 119867

119894 1 le 119894 le 6 be harmonic curvatures in

the point 120573(119904) 120573 is an octonionic inclined curve if and only ifsum6

119894=11198672

119894is constant

Proof The result is straightforward

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W R Hamilton Elements of Quaternions Chelsea PublicationsNew York NY USA 1969

[2] R P Graves Life of Sir William Rowan Hamilton vol 3 ArnoPress New York NY USA 1975

[3] A Cayley ldquoOn Jacobirsquos elliptic functions in reply to the RevB Brownwin and on quaternionsrdquo Philosophical Magazine vol26 pp 208ndash211 1845

[4] E Ozdamar and H H Hacısalihoglu ldquoA characterization ofinclined curves in Euclidean n spacerdquoCommunication de la Fac-ulte des Sciences de LrsquoUniversite drsquoAnkara Series A1 vol 24A pp15ndash23 1975

[5] O Bektas and S Yuce ldquoReal variable Serret Frenet formulaeof an octonion valued function (octonionic curves)rdquo in Pro-ceedings of the 33nd Colloquium on Combinatorics IlmenauGermany November 2014

[6] G Gentili C Stoppato D C Struppa and F Vlacci ldquoRecentdevelopments for regular functions of a hypercomplex variablerdquoin Hypercomplex Analysis I Sabadini M Shapiro and FSommen Eds Trends inMathematics pp 168ndash185 BirkhauserBasel Switzerland

[7] L Sabinin L Sbitneva and I P Shestakov Non-AssociativeAlgebra and Its Applications CRC Press 2006

[8] R Ablamowicz P Lounesto and J M Parra Clifford AlgebrasWith Numeric and Symbolic Computations Birkhauser BostonMass USA 1996

[9] J Schray and C A Manogue ldquoOctonionic representations ofClifford algebras and trialityrdquo Foundations of Physics vol 26 no1 pp 17ndash70 1996

[10] P Lounesto ldquoOctonions and trialityrdquo Advances in AppliedClifford Algebras vol 11 no 2 pp 191ndash213 2001

[11] EUrhammer RealDivisionAlgebras httpwwwmathkudksimmollerundervisningaktuelrap2emil2pdf

[12] D W Aaron ldquoThe structure of 1198646rdquo httparxivorgabs0711

3447v2[13] R Fenn Geometry Springer Undergraduate Mathematics

Series 2007[14] B C S Chauhan and O P S Negi ldquoOctonion formulation

of seven dimensional vector spacerdquo Fundamental Journal ofMathematical Physics vol 1 no 1 pp 41ndash53 2011

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[15] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 1997

[16] C A Manogue and T Dray ldquoOctonions E6 and particle

physicsrdquo Journal of Physics Conference Series vol 254 no 1Article ID 012005 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of