Chun-Yuan Lin

Mathematics for Computer Graphics

112/04/211 CG

Coordinate Reference Frames

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See the powerpoint: Coordinate Reference Frames.ppt

Points and Vectors (1)

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There is a fundamental difference between the concept of a geometric point and that of a vector.A point is a position specified with coordinate

values in some reference frame. (depend on the choice for the frame of refernece)

A vector has properties that are independent of any particular coordinate system.

Point Properties

Frame A

Frame B

x

y P

22 yxL

Points and Vectors (2)

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Vector PropertiesWe can define a vector as the difference between two

point positions.

Vx and Vy are the projection V onto the x and the y axes.We can obtain these same vector components using two

other point positions in the same coordinate reference frames.

A vector has no fixed position within a coordinate system.We can describe a vector as a directed line segment that

has two fundamental properties: magnitude and direction.

P2

P1

V

),(),( 121212 yx VVyyxxPPV

Points and Vectors (3)

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Magnitude:

We can specify the vector direction in various ways, such as

A vector has the same magnitude and direction within a single coordinate system.

If we transform the vector to another reference frame, the value for its components and direction within that reference frame may change.

For a three-dimensional Cartesian vector representation

22yx VVV

x

y

V

V1tan

),,( zyx VVVV 222

zyx VVVV

Points and Vectors (4)

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We can give the vector direction in terms of the direction angles, α, β, γ.

The values cosα, cos β, cos γ are called the direction cosines of the vector.

Vectors are used to represent any quantities that have the properties of magnitude and direction. (force and velocity)

Vγ

β

α

z

y

x

V

VxcosV

VycosV

Vzcos

1coscoscos 222

Points and Vectors (5)

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Vector Addition and Scalar Multiplication

V2

V1

V2

V1

V1+V2

),,( 21212121 zzyyxx VVVVVVVV

),,( zyx sVsVsVsV

Points and Vectors (6)

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Scalar Product of two Vectors

This multiplication scheme is called the scalar product or dot product. (inner product)

is the projection of vector V2 in the direction of V1.

In addition to the coordinate-independent form of the scalar product.

0,cos2121 VVVV

V2V1

θ

cos2V

zzyyxx VVVVVVVV 21212121

Points and Vectors (7)

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The scalar product of two vectors is zero if and only if the two vectors are perpendicular (orthogonal)

1221 VVVV

3121321 )( VVVVVVV

Points and Vectors (8)

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Vector Product of Two Vectors

0,sin2121 VVuVVV2

V1

V1 × V2

u

Cross product

),,( 21212121212121 xyyxzxxzyzzy VVVVVVVVVVVVVV

zyx

zyx

zyx

VVV

VVV

uuu

VV

222

11121

Points and Vectors (9)

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1221 VVVV

321321 VVVVVV

3121321 VVVVVVV

Matrices (1)

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A matrix is a rectangular array of quantities, called the elements of the matrix.

We identify matrices according to the number of rows and number of columns. When the number of rows is the same as the number of columns, this matrix is called a square matrix.

63.100.046.5

00.201.060.3

22 xe

xex

x

321 aaa

z

y

x

rcrr

c

c

mmm

mmm

mmm

M

...

............

...

...

21

22221

11211 An r by c matrix

Row vectorColumn vector

Matrices (2)

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The matrix representation for a three-dimensional vector in Cartesian coordinates as

We use this standard matrix representation for both points and vectors.

z

y

x

v

v

v

V

Scalar Multiplication and Matrix Addition

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654

321M

181512

9633M

0.41.60.2

2.35.30.1

0.41.60.2

2.35.30.1

654

321

Matrix Multiplication(1)

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The product of two matrices is defined as a generalization of the vector dot product.

ABC

n

kkjikij baC

1

2822

3826

43

48223812

47253715

4)1(203)1(10

43

21

82

75

10

Matrix Multiplication(2)

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32

6

5

4

321

18126

15105

1284

321

6

5

4

AB≠BA

A(B+C)=AB+AC

Matrix TransposeThe transpose MT of a matrix is obtained by

interchanging rows and columns.

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63

52

41

654

321T

c

b

a

cba T

(M1M2)T=M2TM1

T

Determinant of a MatrixIf we have a square matrix, we can combine the

matrix elements to produce a single number called the determinant of the matrix.

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211222112221

1211det aaaaaa

aaA

BAAB detdetdet

Matrix InverseWith square matrices, we can obtain an inverse

matrix if and only of the determinant of the matrix is nonzero.

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IMMMM 11

Identity matrix