Graphics

cgvr.korea.ac.kr Graphics Lab @ Korea University

Mathematics for Computer Graphics

고려대학교 컴퓨터 그래픽스 연구실

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Contents

Complex Numbers Pure imaginary number Complex conjugate Representation with polar coordinates

Quaternions Definition Addition, scalar multiplication, multiplication, division,

and inverse 3D Rotations

Nonparametric Representations Parametric Representations

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Complex Numbers

Real Part + Imaginary Part:

Addition and Subtraction

Scalar Multiplication

Multiplication

iyxz

zyzx Im,Re

yxz ,

y

x

z

imaginary axis

real axis

21212211 , , , yyxxyxyx

1111 , , kykxyxk

122121212211 , , , yxyxyyxxyxyx

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Pure Imaginary Number & Complex Conjugate

Imaginary Unit:

Complex Conjugate

Modulus or absolute value

Division

1 ,0i

0 ,11 ,01 ,02 i

1i

iyxz iyxz

22 yxzzz

22

22

211222

22

212122

22

2211

22

21

2

1 , , ,

yx

yxyx

yx

yyxx

yx

yxyx

zz

zz

z

z

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Representation with Polar Coordinates

Euler’s Formula

Complex Multiplication and Division

nth Roots

rθ

z=(x, y)

imaginary axis

real axis

sincos irz

irez sincos iei

2121

2

1

2

12121 , ii e

r

r

z

zerrzz

1 , ,2 ,1 ,0,2

sin2

cos

nkn

ki

n

krz nn

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Quaternions

One Real Part + Three Imaginary Part

Properties:

Addition and Scalar Multiplication

kcjbiasq 1222 kji

jikki

ikjjk

kjiij

dkcdjbdiadsdq 11111

2121212121 cckbbjaaissqq

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Ordered-Pair Notation

Scalar ‘s’ + Vector “v = (a, b, c)”

Addition: Multiplication

Magnitude

Inverse

v ,sq 212121 , vv ssqq

211221212121 , vvvvvv ssssqq

vv 22sq

v ,12

1 sq

q 0 ,111 qqqq

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

3D Rotation

For a 3D Point (α, β, γ) A unit quaternion its conjugate

Rotating (α, β, γ) by angle 2θ about the axis parallel to (a, b, c)

For

Rq is a 3D Rotation about (a, b, c) by 2θ

zyxwq , , , zyxwq , , ,

, , ,0 , , ,0 qq

qpqpRq

cbaq , ,sin ,cos

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Nonparametric Representations

Definition Object descriptions directly in terms of the

coordinates of the reference frame Ex. implicit expression: Ex. explicit Expression:

Advantages Useful in describing objects

Disadvantages Change the independent variable as derivatives

Implicit vs. Explicit Equations

0 , , zyxf yxfz ,

22222 , xryryx

cgvr.korea.ac.kr

CGVR

Graphics Lab @ Korea University

Parametric Representations

3D Curves with Parameter u

Ex.

3D Surfaces with Parameter u, v

Ex.

uzuyuxu , ,P

1002sin2cos uuzuruyurux

vuzvuyvuxvu , , , , , , P

1,0

cos,

2sinsin,

2cossin,

vu

urvuz

vurvuy

vurvux