Mathematics for Computer Graphics

(Appendix A)

2001. 1. 10Won-Ki Jeong

A-1. Coordinate Reference Frame

2D Cartesian reference frame

x

y

x

y

2D Polar Coordinate reference frame

r

r

x

y

x

yyxr

ryrx

122 tan,

sin,cos

s

r

P

radianr

rr

sradian

22

360

3D Cartesian reference frame

Right-handed v.s left-handed

y

xz

y

x

z

Right-handed Left-handed

3D curvilinear coordinate systems

General curvilinear reference frame Orthogonal coordinate system

Each coordinate surfaces intersects at right angles

axisX1

axisX 2

axisX 3

Cylindrical-coordinate

: vertical cylinder

: vertical plane containing z-axis

: horizontal plane parallel to xy-plane

z

constant

zz

y

x

sin

cos

Transform to Cartesian coordinator

z

x axis

y axis

z axis

),,( zP

Spherical-coordinate

x axis

y axis

z axis

),,( rP

: sphere

: vertical plane containing z-axis

: cone with the apex at the origin

r

constant

cos

sinsin

sincos

rz

ry

rx

Transform to Cartesian coordinator

r

Solid angle

3D Angle defined on a sphere(steradian)

r

A

2r

ASteradian :

Total solid angle :

44

2

2

2

r

r

r

Asteradian

A-2. Points & Vectors

Point Position in some reference frame Distance from the origin depends on the

reference frame

PFrame B

Frame Ax

y

AO

BO

Vector Difference between two point positions Properties : Magnitude & direction

Same properties within a single coordinate system

Magnitude is independent from coordinate frames

12 PPV

),(

),( 1212

yx VV

yyxx

22yx VVV Magnitude :

x

y

V

V1tanDirection :

3D vector

Magnitude

Directional angle

222zyx VVVV

x

y

z

V

V

V

V

V

V zyx cos,cos,cos

1coscoscos 222

Vector addition &scalar multiplication

Addition

Scalar multiplication

),,( 21212121 zzyyxx VVVVVVVV

1V

2V 2V

1V

21 VV

),,( 1111 zyx aVaVaVaV

Vector multiplication

Scalar product(inner product)

zzyyxx VVVVVV 212121

cos2121 VVVV Commutative :

Distributive :

1221 VVVV

3121321 )( VVVVVVV

Orthogonal : 021 VV

1V

2V

cos2V

Vector product(Cross product)

),,( 212121212121 xyyxzxxzyzzy VVVVVVVVVVVV

sin2121 VVuVV

zyx

zyx

zyx

VVV

VVV

uuu

222

111

Noncommutative :

Nonassociative :

Distributive :

1221 VVVV

3121321 )( VVVVVVV 3121321 )( VVVVVVV

21 VV 2V

1V

uRight-handed rule!

A-3. Basis vectors and the metric tensor

Basis of vector space Linearly independent axis vectors

Orthonormal basis Orthogonal : Normalized : Orthonormal = Orthogonal + Normalized Orthonormal basis of 3D Cartesian reference frame

1u

2u 3u

1u

2u

3u

)1,0,0()0,1,0()0,0,1( zyx uuu

kjallforuu kj 0

1 kk uu

Metric tensor

Tensor Generalization of a vector with rank & dim. that satisfy c

ertain transformation properties n-th rank with dim m : m-dimensional space which has

n indices Rank 0: scalar, rank 1: dim m vector

rank 2 : vector which has m2 componentMetric tensor Definition : The tensor for

Distance metric Used as transformation equation Component of differential vector operators (gradient, di

vergence, and curl)

kjjk uug

Example of metric tensor

Cartesian coordinate system

Polar coordinates

0

1jkg

If j = k

otherwise

22222112

2111

2 dxgdxdxgdxgdsPythagorean theorem :

22

21

2 dxdxds

22

22

21 dxdxdxds For 3D Cartesian coordinate

system :

cossin

sincos

ruruu

uuu

yx

yxr

20

01

rg

A-4. Matrices

Rows & columns

Matrix multiplication

mnmm

n

n

aaa

aaa

aaa

A

21

22221

11211

Column

row

n

kkjikij bac

1ABC

ACABCBA

BAAB

)(

Properties

Transpose & Determinant

Matrix transpose

Determinant

Large matrix A

63

52

41

654

321T

TTT ABAB )(

n

jjkik

kj AaA1

det)1(det211222112221

1211 aaaaaa

aa

LUA

nnnn dlll

dll

dl

d

L

321

33231

221

1

00

000

0000

n

n

n

n

e

ue

uue

uuue

U

000

00

0

33

2232

113121

))(det(det)det(det ULLUA ))(( 2121 nn eeeddd

Inverse of a matrix

Inverse matrix Determinant is not 0 : Non-singular matrix

Elements of

IAAAA 11

1A

A

A

jkkj

kj

a det

det)1(1

A-5. Complex numbers

Real + Imaginary partiyxyxz ),(

Real axis

Imaginary axis

x

y12 i

)(),( 221121 yxyxzz

),( 2121 yyxx ),)(,( 221121 yxyxzz

),( 12212121 yxyxyyxx

iyxz 22 yxzzz

22

22

211222

22

212122

22

2211

22

21

2

1 ,),)(,(

yx

yxyx

yx

yyxx

yx

yxyx

zz

zz

z

z

Polar form & Euler’s formula

Polar form

Euler’s formula

Real axis

Imaginary axis

x

y

r

),( yxz )sin(cos irz

)sin(cos irrez i )(2121

21 ierrzz

)(

2

1

2

1 21 ier

r

z

z

n

ki

n

krz nn 2

sin2

cos

A-6. Quaternions

Higher dimension complex number

Addition, multiplication, magnitude, & inverse

kcjbiasq 1222 kji

jikkiikjjkkjiij ,,

)()()()( 2121212121 cckbbjaaissqq

),( 211221212121 vvvsvsvvssqq ),( vsq

vvsq 22),(

12

1 vsq

q )0,1(11 qqqq

A-7. Nonparameteric representation

Direct description in terms of the reference frame Surface : or Curve : Useful in the given reference frame

Example (circle)

0),,( zyxf ),( yxfz )(),( xgzxfy

222 ryx 22 xry

Implicit form

Explicit form

A-8. Parameteric representation

Use parameter domain Curve

Ex. Circle

Surface

Ex. Spherical surface

))(),(),(()( uzuyuxuP

0)(),2sin()(),2cos()( uzuruyurux

)),(),,(),,((),( vuzvuyvuxvuP

)cos(),(

)2sin()sin(),(

)2cos()sin(),(

urvuz

vurvuy

vurvux

r : radius of the sphere

u: latitudev: longitude

A-9. Numerical methods

Solving sets of linear equation Matrix form

Cramer’s rule Adequate for a few variables

nnnnnn

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

2211

22222121

11212111

BAX

BAX1

A

Ax k

k det

det kA : matrix A with the kth column replaced

with B

Gauss elimination Elementary Row Operation

Multiply a row through by a nonzero constant Interchange two rows Add a multiple of one row to another row

Make row-echelon form by e.r.o Row-echelon form

First nonzero number of each row is 1(leading 1)

Entire-zero-rows are grouped together at the bottom of the matrix

In any successive non-entire-zero rows, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row

Gauss-Seidel method Start with initial guess and repeatedly calculate su

ccessive approximations until their difference is small

Convergence condition Each diagonal element of a matrix A has a magnitude greater

than the sum of the magnitudes of the other elements across that row

12

121211122

11

131321211

a

xaxaxabx

a

xaxaxabx

nn

nn

Finding roots of nonlinear equation

Object Finding the solution of

Newton-Raphson algorithm Iterative approximation Fast, but it may be fail to converge

0)( xf

10

0 )(

xx

xf

dx

df

)(

)(

0

001 xf

xfxx

0x1x

Initial guess

Bisection method Convergence guaranteed

0x1x0x 2x3x

Evaluating integrals

Rectangle approximation

Polynomial approximation Simpson’s rule

a

b

n

kkk xxfdxxf

1

)()(

ax 0 bxn

a

b

n

kodd

n

kevenkk xfxfbfaf

xdxxf

1

1

2

2

)(2)(4)()(3

)(

nkxxxaxn

abx kk ,......,2,1,,, 10

Monte Carlo method For high-frequency oscillation function or

multiple integrals Use random positions : uniformly

distributedmaxy

miny

maxy

miny

b

a

count

n

nabhdxxf )()(

: # of random points between f(x) and x-axiscountn hryy

abrax

yyh

2min

1

minmax

),(

,

Given two random number r1 and r2 :

Fitting curves to data sets Least-squares algorithm

Fitting a function to a set of data points Ex. 2D linear case

2

1

)(

n

kkk xfyE

xaaxf 10)( 0,010

a

E

a

E

Solve linear equation!

),(,),,(),,( 2211 kk yxyxyx

kx

ky

)( kxf