Mathmatics for Computer Graphics

Mathmatics for Computer Graphics

A r

Mathematics for Computer Graphics

Contents

Spaces Vectors and Linear combination Affine Combination Coordinate-reference Frames Points and Lines Parametric Representation Linear Transformation Homogeneous Coordinates


Vector Spaces

A nonempty set V of vectors

Vectors have magnitude and direction.

Addition + Scalar Multiplication · u+v = v+u V · (u+v)+w = u+(v+w)

· u+0 = u · u+(-u) = 0 · cu V ,(c is scalar) · c(u+v) = cu+cv · (c+d)u = cu+du · c(du) = (cd)u · 1u = u

Directed Line Segments

… but, have no position !

Identical Vectors


Affine Spaces

e3

e1

e2 e3

e1e2

Basis vectors located at the origin

Introducing the concepts of “points”,which represents “the location”

Truncated plane (No Origin): Vector Space

P 기준의 새 좌표계 설정: Affine Space

P

Contain the necessary elements for building geometric models

Points 와 그에 종속된 Vector 들로 표현


Euclidean Spaces

Affine spaces have no concepts of how far apart two points are, or of what the length of a vector is

To Support a measure of distance between points,such as,..Inner Product!

)()( QPQPQP


Points and Vectors

Point : 좌표계의 한 점을 차지 , 위치표시 Vector : 두 position 간의 차로 정의

Magnitude 와 Direction 으로도 표기

),(),( 121212 yx VVyyxxPPV

V

P2

P1

x1 x2

y1

y222

yx VVV

x

y

V

V1tan


Vectors (계속 )

3 차원에서의 Vector

Vector Addition and Scalar Multiplication

222zyx VVVV

||cos,

||cos,

||cos

V

V

V

V

V

V zyx

1coscoscos 222

V

x

z

y

),,( 21212121 zzyyxx VVVVVVVV

),,( zyx VVVV


Scalar Product

|V2|cos

V2

V1

0,cos|||| 2121 VVVV

Dot Product, Inner Product 라고도 함

For Cartesian reference frame,

zzyyxx VVVVVVVV 21212121

Some PropertiesCommutative

Distributive1221 VVVV

3121321 )( VVVVVVV


Scalar Product (계속 )

V2

V1

Edge 사이의 사잇각 Polygon 의 면적

(x0,y0,z0)

(x1,y1,z1)(x2,y2)

(x0,y0)

(x2,y2,z2)

(x1,y1)


Vector Product

0,sin|||| 2121 VVuVV

V1

V2

V1 V2

),,( 21212121212121 xyyxzxxzyzzy VVVVVVVVVVVVVV

※ ux,uy,uz 를 각 축의 단위 vector 라 하면 ,

zyx

zyx

zyx

VVV

VVV

uuu

VV

222

11121

Properties

AntiCommutative

Not Assotiative

Distributive

)( 1221 VVVV

321321 )()( VVVVVV

)()()( 3121321 VVVVVVV


Vector Product (계속 )

Shading, Reflection Model 평면사이의 위치관계


Vector Spaces

A nonempty set V of vectors

Addition + Scalar Multiplication

· u+v = v+u V · (u+v)+w = u+(v+w)

· u+0 = u · u+(-u) = 0

· cu V ,(c is scalar)

· c(u+v) = cu+cv · (c+d)u = cu+du

· c(du) = (cd)u · 1u = u


Linear Combination

Consist of scalar and vectors

nn2211 v...vv

v

v

v

: real number : nonnegative

Ex) Single Vector 의 linear combination


Span

v1

v2

v1 +v2

Ex) Span{v1,v2}

두 Vector 의 linear combination

Span Vector set 의 모든 가능한 linear combination 을 지칭


Linearly Independence

A set of vectors B={v1,…,vp} in V is said to be linearly independent if the vector equation

has only the trivial solution c1=0,…,cp=0.

0...2211 ppvcvcvc

uw

v

u

w

v

Linearly dependent,w in Span{u,v}

Linearly independent,w not in Span{u,v}

(6,2)(3,1)

Linearly dependent, Collinear

(6,2)(3,2)

Linearly independent, Not collinear


Basis

어떤 linearly independent vector set B 에 의해 vector space H 가 span 될 때 , 이 vector set B 를 H 의 basis 라 한다 .

1

0

0

u,

0

1

0

u,

0

0

1

u 321

z

x

yu1 u2

u3

The standard basis for R3

H = Span{b1, … ,bp}


Orthogonal Basis

Orthogonal basis

- 각 원소 vector 들이 서로 직교 할 때

- ,1 kk uu for all k,0 kj uu for all j k

-

kk

kkkkkk vv

vycvyvvc

yvcvcvc pp ...2211

z

x

yu1 u2

u3

Orthonormal basis

※ Orthonormal basis 각 원소 vector 들이 서로 직교 하면서 동시에 단위 vector 로 이루어져 있을 때

kkpp vyvvcvcvc )...( 2211


Affine Space

The Extension of the Vector Space

Geometric operation 들이 의미를 갖는 공간

Points 와 그에 종속된 vector 들로 표현

Truncated plane (No Origin): Vector Space

P 기준의 새 좌표계 설정: Affine Space

P


Affine Combination

Linear combination of points in an affine space make no se

nse (Operations on the points are limited)

Using Parametric Representation

)( 12 PPt

)( 121 PPtP

1P

2P

)( 121 PPtPP

Line segment joining P1and P2, if 0 t 1

21)1( tPPtP

2211 PPP Affine combination of two points

1, 21


Affine Combination

nnPPPP ...2211

2P

3P1P

P

)( 133 PP

)( 122 PP

One Example of Affine Combination

Generalized form

1..., 21 n

33211 PPPP 1, 321

gives a point in the triangle.

)()( 1331221 PPPPPP

2

1,

4

1321


Euclidean Transformation

One-to-One Correspondence between the points of such space and the set of all real-number pair or triples

Transforming Only by rotation and translation

w

y

xtR

w

y

x

T 1|

|

0'

'

')0,0(0

T

y

x

t

tt

Action of the euclidean group on five points

cossin

sincosR


Affine Transformation

w

y

xtA

w

y

x

T 1|

|

0'

'

'

projectiveaffineeuclidean

Action of the affine group on five points

Parallel lines are preseved.

R is Replaced by a general non-singular transformation matrix A


General Linear Transformation

w

y

x

t

t

t

t

t

t

t

t

t

w

y

x

33

23

13

32

22

12

31

21

11

'

'

'

It preserves the collinearity of points.

For general non-singular linear transformations T,

We get general linear or projective group of transformations.

Action of the projective group on five points


Coordinate Reference Frames

Coordinate Reference FramesCartesian coordinate system

x,y,z 좌표축사용 , 전형적 좌표계

Non-Cartesian coordinate system– 특수한 경우의 object 표현에 사용 .

– Polar, Spherical, Cylindrical 좌표계등


2 D Coordinate System

Two-dimensional Cartesian Reference Frames

a) b)

y

xy

x


Polar Coordinates

가장 많이 쓰이는 non-Cartesian System

Elliptical coordinates, hyperbolic, parabolic plane coordinates 등 원 이외에 symmetry 를 가진 다른 2 차 곡선들로도 좌표계 표현 가능 .

sin,cos ryrx

x

yyxr 122 tan,

rs

r


Why Polar Coordinates?

x x

y y

dxdx

dd

균등하게 분포되지 않은 점들 연속된 점들 사이에 일정간격유지

Polar CoordinatesCartesian Coordinates

예 ) 원의 표현

222 ryx

sin

,cos

ry

rx


3D Cartesian Frames

Three Dimensional Point


3D Cartesian Frames

오른손 좌표계

- 대부분의 Graphics Package 에서 표준

왼손 좌표계

- 관찰자로부터 얼마만큼 떨어져 있는지

나타내기에 편리함

- Video Monitor 의 좌표계


3D NonCartesian System

Cylindrical coordinates

z

P(,,z)

x axis

y axis

z axis

P(r,, )

x axis

y axis

z axis

r

Spherical coordinates

cosx siny

zz

sincosrx sinsinry

cosrz


Points

가장 기본적인 Output Primitive

0 차원으로 크기와 길이 측정 불가

순서쌍 (x,y) 나 vector 형식으로 표기

Raster Scan display 의 한 Pixel 차지

2D or 3D


Lines

Defined as a list of points(PolyLine)

RasterizationStairstep effect(jaggies)


Line Drawing Algorithm

Accomplished by calculating intermediate positions along the line path between two specified endpoint positions

12 xy Bresenham’s Line Algorithm


Why “y=mx” is not good for Graphics Applications

Defects in Nonparametric representation Explicit function 의 경우

1. Can only represent infinite lines, not finite line segments 2. Cannot represent vertical lines(m=) 3. Can only 2D lines, not 3D

Implicit function 의 경우

Redundant representation

Ex) 원호의 표현 :

bmxy

0),( yxf

122 yx x

y


Parametric Line Equation

P1

P2

y = Y1 + t * ( Y2 -Y1 )

x = X1 + t * ( X2 - X1 )

z = Z1 + t * ( Z2 - Z1 )

0.0 t 1.0

p = P1 + t * ( P2 - P1 )p


Parametric Line Equation

P1

P2

x = ( 1 - t ) * X1 + t * X2

0.0 t 1.0

Can Also be thought of as a blending function...

y = ( 1 - t ) * Y1 + t * Y2

z = ( 1 - t ) * Z1 + t * Z2


Linear Blending

q = Q1 + t * ( Q2 - Q1 )

You can linearly blend any two quantities with :

q = ( 1 - t ) * Q1 + t * Q2

Or, if you’d prefer :

Color, Shape, Location, Angle, Scale factors,….


Matrices

Definition A rectangular array of quantities

Scalar multiplication and Matrix Addition

mnmm

n

n

aaa

aaa

aaa

A

...

:::

...

...

21

22221

11211

2221

1211

2221

1211 ,bb

bbB

aa

aaA

22222121

12121111

baba

babaBA

2221

1211

kaka

kakakA


Matrix Multiplication

Definition

PropertiesNot CommutativeAssotiativeDistributiveScalar multiplication

× = (i,j)

j-th column

i-th row

ml

nnm

l

n

kkjikij bac

1

ABC

BAAB

)()( BCACAB

BCABCBA )(

)()()( ABkkBABkA


Translation

y

x

T

TT

y

xP

y

xP ,

'

'',

x

y

x

y

(b)

(a)

P

P’

110

01

'

'y

x

T

T

y

x

y

x

y

x

Ty'y

,Tx'x


Scaling x’ = x · Sx , y’ = y · Sy

y

x

S

S

y

x

y

x

0

0

'

'

x’ x

P2

(xf, yf) : fixed pointx

y (xf,yf)P1

P3

x’ = xf + (x-xf) sx , y’ = yf + ( y- yf) sy

P)y,x(T)s,s(S)y,x(T'P ffyxff

1100

10

01

100

00

00

100

10

01

1

y

x

T

T

S

S

T

T

'y

'x

y

x

y

x

y

x


Rotation

P’ = R P

x = r cos , y = r sin

x’ = r cos ( + ) = r cos cos - r sin sin y’ = r sin ( + ) = r cos sin + r sin cos

x’= x cos - y sin , y’ = x sin + y cos

cossin

sincosR

죄표중심을 회전점으로각 만큼 회전

(x,y)

r

(x’,y’)

r


Homogeneous Coordinates

PTP '

In basic Transformations,

PSP 'PRP '

( Addition )

( Multiplication )

( Multiplication )Only Translation istreated differently

We hope to combine the multiplicative and translational terms for two-dimensional geometric transformations into a single matrix

representation for enabling the composite transformations such as,

PTSRTP )('


Extend the matrix

1100

10

01

1

'

'

y

x

t

t

y

x

y

x

1100

0cossin

0sincos

1

'

'

y

x

y

x

1100

00

00

1

'

'

y

x

s

s

y

x

y

x

We can solve this problem by extending 2 by2 matrix into 3 by 3 matrix

P2

(xf, yf) : fixed pointx

y (xf,yf) P1

P3

1100

10

01

100

00

00

100

10

01

1

y

x

T

T

S

S

T

T

'y

'x

y

x

y

x

y

x

Translation Rotation Scaling

Ex)


Homogeneous Space

y

x

z =3

z =2

z =1

P(3x1,3y1,3)

P(2x1,2y1,2)P(x1,y1,1)

P(hx1,hy1,h)= P(X,Y, h)

3D Representarion of homogeneous space

Any two dimensional point can be represented by one of the points along the ray in 3D space


Point at Infinity

The points with h=0 are called points at infinity,and this will not appear very often.

Ex) [ X Y h ] = [ 4 3 1]

h x* y* X Y

1 4 3 4 3

1/2 8 6 4 3

1/3 12 9 4 3

… … … … …

1/100 400 300 4 3


Point at Infinity(Cont’d)

0

1

yx

yx

Ex) The intersection point of two Parallel lines

00

01

11

11

1

yx

011

011

110

100

100

101

011

011

1001

1

yx

The resulting homogeneous coordinates [1 -1 0] represent the ‘pointof intersection’ for the two parallel lines, i.e. a point at infinity.


Point at Infinity(Cont’d)

Two Lines are intersect each otherif they are not parallel

Non-Homogeneous treatment !!

A

B


Determinant of Matrix For n 2, the Determinant of nn matrix A is,

and for a 2 by 2 matrix,

Ex)

※ if A is a triangular matrix, det A is the product of the entries on the main diagonal of A

n

jjkjk

kj AaA1

det)1(det

211222112221

1211 aaaaaa

aa

20

42det0

00

12det5

02

14det1det A

2)04(0)00(5)20(1

020

142

051

A


Properties of Determinants

Row OperationsLet A,B be a square matrix1) A 의 한 행의 실수배가 다른 row 에 더해져 B 를 만들었다면 , detB = detA2) A 의 두 행이 교환되어 B 를 생성했다면 , detB = -detA3) A 의 한 행이 k 배된 것이 B 라면 , detB = kdetA

Ex)

Column Operations

AAT detdet ))(det(det)det( BAAB

071

982

241

A

15))5(31(

500

230

241

detdet

A

071

500

241

230

500

241

500

230

241


Solving Linear Equations

Linear Equation

where, ajk and bj are known

Using Matrix Equation

11212111 ... bxaxaxa nn

22222121 ... bxaxaxa nn

nnnnnn bxaxaxa ...2211

...

BAX 1BAX

※Coefficient Matrix A 의 역행렬이 존재할 때만 성립


Inverse Matrix

If ad-bc = 0, then A is not invertible.

IAAIAA 11

• Definition

• 2 2 matrix 의 경우

• Properties

TT )A()A(AB)AB(A)A( 1111111

ac

bd

bcadA

11

dc

baA


Inverse Matrix

[ I A-1] 형태의 row reduction 이 존재하지 않으면 , A is not invertible.

• Algorithm for Finding A-1

834

301

210

A

1AI~IA : Row reduction

Ex)

140

001

010

430

210

301

100

001

010

834

210

301

100

010

001

834

301

210

~~IA

21223

142

23729

100

010

001

21223

001

010

100

210

301

143

001

010

200

210

301

//

//

~

//

~~

21223

142

237291

//

//

A


Gaussian Elimination

System 에서 한 행의 실수배를 다른 행에서 빼어 연립방정식의 차수를 줄여나가다 , 한 변수의 해가 구해지면 , 역으로 대입해 나머지 변수값을 구한다 .

• Elementary Row Operation 1. Multiply any row of the augmented matrix by a nonzero constant 2. Add a multiple of one row to a multiple of any other row 3. Interchange the order of any two rows

222

1132

1223

321

321

321

xxx

xxx

xxx

1874

2177

1223

32

32

321

xx

xx

xxx

4221

2177

1223

3

32

321

x

xx

xxx

312 123 x,x,x


Using Row Operation

2

11

12

122

321

213

3

2

1

x

x

x

2

11

12

122

321

213

|

|

|

b|A

13

12

23

13

2

11

12

122

321

213

R)(R

R)(R

23 4718

21

12

740

770

213

RR

42

21

12

2100

770

213

42

21

168

2100

0210

02163

42

21

189

2100

0210

0063

213

2

1

3

2

1

3

100

010

001

321

x,x,x,x,

222

1132

1223

321

321

321

xxx

xxx

xxx


Consistency of Linear System

1. Unknown variable 의 수에 비해 equation 의 수가 적을 때 No Unique solution (Infinite set of solution)

2. Unknown variable 의 수에 비해 equation 의 수가 많을 때 1) Consistent한 경우

: 구한 해가 남은 방정식에 대해서도 성립 (redundant)

2) Inconsistent 한 경우

(infinity)

(no solution)

1132

1223

321

321

xxx

xxx

6223 yx,yx

7223 yx,yx


Singular Matrix

• nn system 이 unique solution 을 갖지 않을 때 , 그 system의 coefficient matrix 를 singular 하다고 한다 .

• 일반적으로 coefficient matrix 를 triangulization 했을 때 ,diagonal 에 0 이 있으면 singular 하다 .

~

211141

7142

5321

1000

3780

5321

( No solution,Inconsistent )

( Infinity,redundant )

0000

3780

5321


Cramer’s Rule

For invertible nn matrix A and b, let Ai(b) be the matrix obtained from A by replacing column i by the vector b,

x of Ax=b is,

niA

bAx i

i ,...,2,1,det

)(det

],...,,...,[)( 1 ni ababA

column i

Ex)845

623

21

21

xx

xx

85

63)(,

48

26)(,

45

2321 bAbAA

detA=2 272

3024

det

)(det,20

2

1624

det

)(det 22

11

A

bAx

A

bAx


Iterative Method

,...

11

131321211 a

xaxaxabx nn ......,

...

12

121211122 a

xaxaxabx nn

- Gaussian Elimination 의 roundoff error 를 해결하기 위함- 처음에는 임의의 기대값을 해로 준 다음 error 가 최소가 될 때까지 다음 식에 반복 대입

427

1292

88

321

321

321

xxx

xxx

xxx

213

312

321

111022203331

286014305710

125012501

x.x..x

x.x..x

x.x.x

First Second Third Fourth Fifth Sixth Seventh Eighthx1

x 2

x 3

0 1.000 1.095 0.995 0.993 1.002 1.001 1.000 0 0.571 1.095 1.026 0.990 0.998 1.001 1.000 0 1.333 1.048 0.969 1.000 1.004 1.001 1.000

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Mathmatics for Computer Graphics