1. Math&Com Graphics Lab Vector Hyoungseok B. Kim Dept. of Multimedia Eng. Dongeui University

2. Math&Com Graphics Lab. Dongeui University2 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

3. Math&Com Graphics Lab. Dongeui University3 Computer Graphics 3D Virtual Space = Model + Light 2D Virtual Space Camera (Clipping, Projection, Hidden-Surface Removal) Rasterization Screen Space 3 차원 공간 필름 사진 현상

4. Math&Com Graphics Lab. Dongeui University4 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

5. Math&Com Graphics Lab. Dongeui University5 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

6. Math&Com Graphics Lab. Dongeui University6 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 ?

7. Math&Com Graphics Lab. Dongeui University7 Equality and Efficient Representation same Find out the same 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

8. Math&Com Graphics Lab. Dongeui University8 Vector Representation of Vector Position of the End Point : 시점 종점

9. Math&Com Graphics Lab. Dongeui University9 Operators of Vectors Same if and only if Magnitude Addition Inner Product Cross Product

10. Math&Com Graphics Lab. Dongeui University10 Inner Product Properties 1. Why ? 2. Why ? 코사인 제 2 법칙

11. Math&Com Graphics Lab. Dongeui University11 Inner Product 1. 2.  What is the inner product used for ?   To confirm whether the angle is right or not…….

12. Math&Com Graphics Lab. Dongeui University12 Inner Product’s Applications 1. ? 2. Compute a plane P passing through a given point with the normal vector

13. Math&Com Graphics Lab. Dongeui University13 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 ?

14. Math&Com Graphics Lab. Dongeui University14 Back Face Removal Which are back faces ? front faceback face How to compute ? What is a counter-example ?

15. Math&Com Graphics Lab. Dongeui University15 Back Face Removal Counter-example

16. Math&Com Graphics Lab. Dongeui University16 Cross Product Properties : A new vector orthogonal to both two vectors Area of a Triangle with edges and Why ?

17. Math&Com Graphics Lab. Dongeui University17 Cross Product’s Applications Normal Vector Computation Parametric Surface

18. Math&Com Graphics Lab. Dongeui University18 Cross Product’s Applications Normal Vector Computation Polygonal Mesh

19. Math&Com Graphics Lab. Dongeui University19 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 ?

20. Math&Com Graphics Lab. Dongeui University20 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

21. Math&Com Graphics Lab. Dongeui University21 Vector Space Linearly dependence in linearly dependent linearly independent linearly dependent

22. Math&Com Graphics Lab. Dongeui University22 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 ?

23. Math&Com Graphics Lab. Dongeui University23 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 

24. Math&Com Graphics Lab. Dongeui University24 Frame Point + Orthogonal basis : three orthogonal vectors  (0, 0, 0) + Frame Transformation d = ? e = ? f = ?

25. Math&Com Graphics Lab. Dongeui University25 Coordinate Transform : Point original frame v 1, v 2, v 3, P 0 new frame u 1, u 2, u 3, Q 0 mapping