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Two Methods of Creating Beautiful

Mathematical Curves

Ref: http://en.wikipedia.org/wiki/Functional-theoretic_algebra

Based on Functional Theoretic Algebras

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)1( )0( : )1( )0(:

)1( and )0(

)(),();()()( f

]1,0[:function continuousA

int

CurveOpenCurveClosed

tytxtittI

C

sPoEnd

FormParametric

Curve

Basic Definitions

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Closed CurvesExamples

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Examples of Closed Curves(Loops)

1

)1(1)0(2sin)(2cos)(

)2sin()2()( 1

atLoop

FormParametric

uuttyttx

titCostueUnit Circl

0 1

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055

11/4 1/2 3/4

1 2sin4cos)(2cos4cos)(

2cos

Cos2-Rhodonea 2

atLooptttytttx

rFormParametric

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1at Loopt)t)sin(2cos(6yt)t)cos(2cos(6x

3cosCos3-Rhodonea 3

r

0

1

1/21/4

3/4

(-1/2, 1/2)

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1at Loop

cardioid theas Take1at loop a be will1-c

t)t))sin(2cos(2(1(t)1-t)t))cos(2cos(2(1(t)

2at loop a is (1)2(0)

t)t))sin(2cos(2i(1t)t))cos(2cos(2(1(t)cos1: 4

ccc

rCardioid

01

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1at Loop

umDoubleFoli theas Take1at loop a be will1

t)t)sin(2t)sin(4cos(24(t)t)t)cos(2t)sin(4cos(241)(

0at loop a is (1)0(0)

t)t)sin(2t)sin(4cos(24it)t)cos(2t)sin(4cos(24(t)2sincos4: 5

t

rFoliumDouble

0

1

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1at Loop

Folium Double theas Take1at loop a be will2

t)(2sin))t2cos(21((t)2-t)cos(2))t2cos(21()(

3at loop a is (1)3(0)

t)(2sin))t2cos(21(it)cos(2))t2cos(21((t)cos21: 6

t

rPascalofLimacon

0

1

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1at loop a is

)sin()3sin3(cosi)cos()3sin3(cos(t)(1)1(0)

3sin3cos:Egg rooked 7

rC

0

1

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Nephroid theas Take1at loop a be will2

2iat loop a is (1)2(0)

t)](12cos)t4cos(i[3t)(12sin)t4sin(3(t))cos(6-)3cos(2y

),sin(6)-3sin(2x:ephroid 8

i

i

N

0

1

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Method1

n-Star Transformation

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1, ,

1 ]1,0[,1)( lg ]1,0[

],1,0[ , ]1,0[

Pr

)1()0()1()0(

HIfatloopsofsetH

ttebydefinedunitywithebraAecommutativnonaisCThen

CIfCincurvescontinuousofsetC

CurvesofoductTheoreticFunctional

Ref: Sebastian Vattamattam, Non-Commutative Function Algebras, Bulletin of Kerala Mathematics Association, Vol. 4, No. 2(2007 December)

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, 1 ,

])[()( [0,1], tIf1] [0,[x]-x

int ][, int

1

n

oftiontransformaStarnthe

calledisnHIfatloopacurvenancalledis

ntntt

xegergreatestthexRxIfegerpositivean

atloopa

n

CurvenDefining

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EXAMPLES

Of

n-Star Transform

ation

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n-Star Transformation of Unit Circle

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n-Star Transformation of Double Folium

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n-Star Transformation of Rhodonea-Cos2

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n-Star Transformation of Crooked Egg

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n-Star Transformation of Nephroid

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Examples

of

Open Curves

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ititttSegmentLineA

)1(,1)0(10,1)( :

22

0 1

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iittittPA

1)1(,1)0(10,)12(12)( : 2arabola

0 1

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icicttittc

2)1(,)0(10,2cos2)(

Curve Cosine

0 1

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4)1(,0)0(4sin4)(

sstitts

Sine Curve

0 1

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4)1(,0)0(t)sin(4t4yt)cos(4t4x

))in(4it)t(cos(44)(40,:

tstSpiralnArchimedia

01

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Method2

n-Curving

Ref: Sebastian Vattamattam, Transforming Curves by n-Curving, Bulletin of Kerala Mathematics Association, Vol. 5, No.1(2008 December)

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Defining n-Curving

. with curved-n called is ),1)](0()1([)(curve,-nan is and curveopen an is

).()( then sin and cos of functions are of partsimaginary and real thesuch that 1,at loop a

n

n

n

n

:

If

nttttα(t)isIf

Curvingn

Theorem

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N-Curving the Line Segment l

ntntttyntntttx

iyxIf

i

nu

titCostu

titttCircleUnittheWith

2sin2cos1)(2sin2cos2)(

)(

1)0()1(

)2sin()2()(

10,1)(

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Unit Circle – Line Segment

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ntntttyttx

iyxIftt

i

nc

cCardioid

titttCtheWith

2sint))cos(2(12cost))cos(2(12)(t)t))sin(2cos(2(1t)t))cos(2cos(2(13)(

)( t)t))sin(2cos(2(1)(1-t)t))cos(2cos(2(1)(

1)0()1(

10,1)(ardioid

N-Curving the Line Segment l

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Cardioid – Line Segment

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).sin(2pintcos(4pint)+).cos(2pintcos(4pint)+1-t=y ).sin(2pintcos(4pint)-).cos(2pintcos(4pint)+1- t=x

)( sin(2pit)cos(4pit).=y

cos(2pit)cos(4pit).=x

1)0()1(2

10,1)(Cos2-RhodoneaWith

iyxIf

i

nc

CosRhodonea

tittt

N-Curving the Line Segment l

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Rhodonea-Cos2 – Line Segment

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).sin(2pintcos(6pint)+).cos(2pintcos(6pint)+1-t=y ).sin(2pintcos(6pint)-).cos(2pintcos(6pint)+1- t=x

)( sin(2pit)cos(6pit).=y

cos(2pit)cos(6pit).=x

1)0()1(3

10,1)(Cos3-RhodoneaWith

iyxIf

i

nc

CosRhodonea

tittt

N-Curving the Line Segment l

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Rhodonea-Cos3 – Line Segment

n = 1n = 2

n = 3n = 10

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).sin(2pintcos(4pint)+).cos(2pintcos(4pint)+1-t=y ).sin(2pintcos(4pint)-).cos(2pintcos(4pint)+1- t=x

)( sin(2pit)cos(4pit).=y

cos(2pit)cos(4pit).=x

1)0()1(2

10,1)(Sin2-RhodoneaWith

iyxIf

i

nc

CosRhodonea

tittt

N-Curving the Line Segment l

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RhodoneaSin2 – LineSegment

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).sin(2pintcos(4pint)+).cos(2pintcos(4pint)+1-t=y ).sin(2pintcos(4pint)-).cos(2pintcos(4pint)+1- t=x

)( t)t)sin(2t)sin(4cos(24(t)

t)t)cos(2t)sin(4cos(241)(1)0()1(

10,1)(

iyxIf

ti

tittt

n

umDoubleFoliWith

N-Curving the Line Segment l

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DoubleFolium – LineSegment

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N-Curving the Line Segment l

nt)nt))sin(22cos(2(1-nt)nt))cos(22cos(2(13-tynt)nt))sin(22cos(2(1-nt)nt))cos(22cos(2(1-t-4 x

)( t)t))sin(2cos(221((t)

2-t)t))cos(2cos(221()(

1)0()1(10,1)(

Pascal ofimacon

iyxIf

tPascalofLimaconi

tittt

n

LWith

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Limacon– LineSegment

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Random

Examples

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Example 1n-Curved Cosine with Rhodonea-cos2

nt)(nt)(t)(tynt)(nt)(t-tx

2sin4cos22cos)()2cos4cos1(2)(

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Example 2n-Curved Cosine with Double Folium

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Example 3n-Curved Archimedean Spiral with Unit Circle

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Example 4n-Curved Archimedean Spiral with Cardioid

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