1. Graphics Graphics Lab @ Korea University cgvr.korea.ac.kr Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실

2. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Contents Coordinate-reference Frames 2D Cartesian Reference Frames Polar Coordinates 3D Cartesian Reference Frames 3D Curvilinear Coordinate Systems Points and Vector Vector Addition and Scalar Multiplication Scalar Product Vector Product Matrices Scalar Multiplication and Matrix Addition Matrix Multiplication Matrix Transpose Determinant of a Matrix Matrix Inverse

3. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Coordinate Reference Frames Cartesian coordinate system  x, y, z 좌표축사용, 전형적 좌표계 Non-Cartesian coordinate system  특수한 경우의 object 표현에 사용.  Polar, Spherical, Cylindrical 좌표계 등

4. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 2D Cartesian Reference System Two-dimensional Cartesian Reference Frames Coordinate origin at the lower-left screen corner y x y x Coordinate origin in the upper-left screen corner

5. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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.  r

6. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Why Polar Coordinates? x x y y dx dd dd 균등하게 분포되지 않은 점들 연속된 점들 사이에 일정간격유지 Polar Coordinates Cartesian Coordinates Circle 2D Cartesian : 비균등 분포  Polar Coordinate

7. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Cartesian Reference Frames Three Dimensional Point

8. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Cartesian Reference Frames 오른손 좌표계 대부분의 Graphics Package 에서 표준 왼손 좌표계 관찰자로부터 얼마만큼 떨어져 있는지 나타내기에 편리함 Video Monitor 의 좌표계

9. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Curvilinear Coordinate Systems General curvilinear reference frame Orthogonal coordinate system  Each coordinate surfaces intersects at right angles A general Curvilinear coordinate reference frame

10. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Non-Cartesian System Cylindrical coordinates Spherical coordinates z P( , ,z) x axis y axis z axis   P(r, ,  ) x axis y axis z axis   r

11. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Point : 좌표계의 한 점을 차지, 위치표시 Vector : 두 position 간의 차로 정의 Magnitude 와 Direction 으로도 표기 V P2P2 P1P1 x1x1 x2x2 y1y1 y2y2 Point and Vector

12. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Vectors 3 차원에서의 Vector Vector Addition and Scalar Multiplication    V x z y

13. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Scalar Product Definition For Cartesian Reference Frame Properties Commutative Distributive |V 2 |cos   V2V2 V1V1 Dot Product, Inner Product 라고도 함

14. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Vector Product Definition For Cartesian Reference Frame Properties AntiCommutative Not Associative Distributive Cross Product, Outer Product 라고도 함 V1V1 V2V2 V1  V2V1  V2  u

15. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Examples Scalar Product Vector Product Normal Vector of the Plane  V2V2 V1V1 Angle between Two Edges (x2,y2) (x0,y0) (x1,y1)

16. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Matrices Definition A rectangular array of quantities Scalar multiplication and Matrix Addition

17. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Matrix Multiplication Definition Properties Not Commutative Associative Distributive Scalar Multiplication ×= (i,j) j-th column i-th row m l n n m l

18. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Matrix Transpose Definition Interchanging rows and columns Transpose of Matrix Product

19. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Determinant of Matrix Definition For a square matrix, combining the matrix elements to product a single number 2  2 matrix Determinant of n  n matrix A (n  2)

20. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Inverse Matrix Definition Non-singular matrix  If and only if the determinant of the matrix is non-zero 2  2 matrix Properties

21. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Homework (1/3) Cylindrical 좌표 (2, π/6, 1), (2, π/6, -1), (-2, π/6, 1), (-2, π/6, -1), Spherical 좌표 (4, π/6, π/3), (4, π/6, - π/3), (-4, π/6, π/3), (-4, π/6, - π/3) 를 Cartesian 좌표로 변환하여라. 세 점 A(1, 0, 0), B(0, 1, 0), C(0, 0, 1) 으로 구성된 삼각형 T(A, B, C) 의 Face Normal 을 구하시오. ( 반드시 Normalizing( 정규화 ) 시킬 것 )

22. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Homework (2/3) View Vector V(0, 0, -1) 에 대해의 위의 삼각형 T 의 앞면이 보이는지 뒷면이 보이는지 기술하시오. ( 반드시 이유도 기술 ) 다음의 식을 행렬로 표현하시오. (1) (2) (3)

23. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Homework (3/3) 삼각형 T 의 세 꼭지점 A, B, C 에 대해 위의 행렬 연산을 모두 적용시킨 후, 새로운 꼭지점의 좌표를 계산하여라. ( 꼭지점에 대한 연산 적용 순서는 (1)  (2)  (3) 이고, 가능한 간단한 행렬 식으로 표현할 것 )