math&com graphics lab vector hyoungseok b. kim dept. of multimedia eng. dongeui university

Math&Com Graphics Lab

Vector

Hyoungseok B. Kim

Dept. of Multimedia Eng.Dongeui University

Math&Com Graphics Lab.

Dongeui University 2

What is Computer Game ? To make us fun by using computer Computer Game

Sense of Sight (Computer Graphics) Sense of Hearing (Sound) Sense of Touch (Interaction) Interesting Story

What is Computer Graphics Technologies of creating virtual space and displaying it on

computer monitor by using computer Computer Graphics

Modeling Rendering Animation

Game and Graphics



Computer Graphics

3D Virtual Space = Model + Light

2D Virtual Space

Camera (Clipping, Projection, Hidden-Surface Removal)

Rasterization

Screen Space

3 차원 공간3 차원 공간

필름 필름

사진 현상 사진 현상



transformation 좌표 변환을 위한 4×4 matrix multiplication

clipping projection plane 상에서 불필요한 부분 제거

projection 3D object 2D image mapping

rasterization image 를 frame buffer 에 저장하는 과정

OpenGL Pipeline Architecture



Where is Mathematics in Computer Graphics ? Creation of Objects

Vertices, edges, faces, box, sphere, cylinder, torus, … Handle of Objects

Transformation : translation, scaling, rotation Handle of Camera

Position, Orientation, Lens, Projection Handle of Light

Shadow Handle of Motion

Character motion, animation of all kinds of objects Handle of Rendering

Image based rendering …..

Game Mathematics



Standard Language in Mathematics Quantity

Scalar : Magnitude( 크기 ) Vector : Magnitude( 크기 ), Orientation( 방향 )

Representation of Quantities Scalar : real number

1 , 2, 0.72534, 3 / 7

Vector : Arrow 시점과 종점

Scalar vs Vector

시점

종점

inefficient

Your answer ?



Equality and Efficient Representation Find out the samesame vectors as the given vector A ?

same magnitude, same color Blue arrow

Efficient Representation Assume that the start point of all vectors is the Origin in the space.

Vector

A



Vector

Representation of Vector

Position of the End Point :

),( yx

시점

종점

종점

종점종점

32 ),,(,),( RzyxRyx



Operators of Vectors

Same if and only if

Magnitude

Addition

Inner Product

Cross Product

VU xx vu yy vu zz vu

222zyx uuu U

),,( zzyyxx vuvuvu VU

zzyyxx vuvuvu VU

),,( xyyxzxxzyzzy

zyx

zyx vuvuvuvuvuvu

vvv

uuu kji

VU



Inner Product

Properties

1. Why ?

2. Why ?

코사인 제 2 법칙

2UUU

cosVUVU

U

V

VU



Inner Product

1.

2.

What is the inner product used for ? To confirm whether the angle is right or not…….

2)()( VUVUVU

cosVUVU

222)()( VVUUVUVU

cos2222

VUVUVU



Inner Product’s Applications

1. ?

2. Compute a plane P passing through a given point

with the normal vector

)0,0,1(UVU (0,1,0)V

),,( cbaU),,( 111 zyxA

UAP ),,(),,( 111 zyxzyx0),,(),,( 111 cbazzyyxx

111 czbyaxczbyax



Inner Product’s Applications

3. Compute the distance of a point

from a line L passing through

with a unit directional vector

4. Compute the distance of a point

from a Plane P passing through

with a unit Normal vector

distance = ?

5. Up side or Down side ?

),,( 111 zyxA

),,( 222 zyxB),,( cbaU

UUBABA )( d

),,( 111 zyxA

),,( 222 zyxB),,( cbaN

NBA



Back Face Removal

Which are back faces ?

vx

vy

vz

front faceback face

How to compute ?

What is a counter-example ?



Back Face Removal

Counter-example



Cross Product

Properties : A new vector orthogonal to both two vectors

Area of a Triangle with edges and

VUsinVUVU

2222)()()( xyyxzxxzyzzy vuvuvuvuvuvu VU

22222222)())(( zzyyxxzyxzyx vuvuvuvvvuuu VU

222222cosVUVUVU

2222222sin)cos1( VUVUVU

U V

U

V

VU2

1Area Why ?



Cross Product’s Applications

Normal Vector Computation Parametric Surface

vu

v

zv

yv

x

v

u

zu

yu

x

u

vuz

vuy

vux

vu

ppn

pp

p

,

),(

),(

),(

),(



Cross Product’s Applications

Normal Vector Computation Polygonal Mesh

cba

)()( 0201 ppppn

),,( 1111 zyxp

),,( 2222 zyxp

),,( 0000 zyxp

0 dczbyax



Vector Space

The set of vectors satisfying 9 properties addition, scalar multiplication, identity, additive inverse, commutative law, distributive law

Examples)

Properties) Linearly dependence

linearly independent

linearly dependent

Is the set linearly dependent ?

32 ,RR

2R)2,1(1 U )1,2(2 U

)4,2(3 U

21 UU

31 UU

},{ 21 UU

},{ 21 UU

)5,1(4 U

},,,{ 4321 UUUU



Vector Space

1 차 독립 (Linearly Independent) 만약 다음을 만족한다면

은 “ 1 차 독립”라고 함

만약 그렇지 않다면 , 은 “ 1 차 종속”이라 함

즉 , 이라면

0321 n 0332211 nnUUUU

},,,{ 21 nUUU

},,,{ 21 nUUU

0m

)(1

332211 nnm

m UUUUU



Vector Space

Linearly dependence in

linearly dependent linearly independent linearly dependent

3R

)0,1,2(1 U

)0,2,1(2 U

)0,3,3(3 U

)1,1,1(4 U

},,{ 321 UUU

},,{ 421 UUU

},,,{ 4321 UUUU



Vector Space

Basis 이 벡터공간 에서 1 차 독립이며 그 공간에 있는 모든 벡터들을 다음과 같이 1 차 선형조합으로 표현가능 하면

이 벡터공간 의 기저 (basis)

예제

의 basis 에는 어떤 것이 있는가 ?

},,,{ 21 nUUU V

nnP UUUU 332211

},,,{ 21 nUUU V

2RV

)}1,2(),2,1{(

)}4,2(),2,1{(

)}3,2(),3,1(),2,1{(



Vector Space

Question 1. 서로 수직인 basis 는 무엇인가 ?

Orthogonal basis 가 왜 필요하나 ?

일반 basis 를 orthogonal basis 로 바꾸는 방법은 있을까 ? Gram-schmit Orthogonalization

2RV )}1,0(),0,1{(

'1

12'

''

k

i

k k

kiii e

e

eeee



Frame

Frame Point + Orthogonal basis : three orthogonal vectors

(0, 0, 0) +

Frame Transformation

kj,i,

kji cbaP

CBA fedP

i

j

k

A

B

C

P

d = ?

e = ?

f = ?



Coordinate Transform : Point

original frame v1, v2, v3, P0

new frame u1, u2, u3, Q0

mapping

frame newin

frame originalin

0332211

0332211

Q

PP

uuu

vvv

03432421410

3332321313

3232221212

3132121111

PQ

vvv

vvvu

vvvu

vvvu

1

0

0

0

434241

333231

232221

131211

M

0

3

2

1

321

0

3

2

1

321

0

3

2

1

321 111

PPQ

v

v

v

v

v

v

Mu

u

u

w

math&com graphics lab vector hyoungseok b. kim dept. of multimedia eng. dongeui university

Documents