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 / Curvilinear Coordinates Points and Vectors Vector Addition and Scalar Multiplication Scalar Product / Vector Product Basis Vectors and the Metric Tensor Orthonormal Basis Metric Tensor Matrices Scalar Multiplication and Matrix Addition Matrix Multiplication / 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 2D 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 or 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 x 2 axis x 3 axisx 1 axis x 1 = const 1 x 3 = const 3 x 2 = const 2

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 Points and Vectors

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)(x2,y2) (x0,y0)(x0,y0) (x1,y1)(x1,y1)

16. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Basis Vectors Basis (or a Set of Base Vectors) Specify the coordinate axes in any reference frame Linearly independent set of vectors  Any other vector in that space can be written as linear combination of them Vector Space Contains scalars and vectors Dimension: the number of base vectors Curvilinear coordinate- axis vectors u2u2 u1u1 u3u3

17. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Orthonormal Basis Normal Basis + Orthogonal Basis Example Orthonormal basis for 2D Cartesian reference frame Orthonormal basis for 3D Cartesian reference frame

18. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Metric Tensor Tensor Quantity having a number of components, depending on the tensor rank and the dimension of the space Vector – tensor of rank 1, scalar – tensor of rank 0 Metric Tensor for any General Coordinate System Rank 2 Elements: Symmetric:

19. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Properties of Metric Tensors The Elements of a Metric Tensor can be used to Determine Distance between two points in that space Transformation equations for conversion to another space Components of various differential vector operators (such as gradient, divergence, and curl) within that space

20. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Examples of Metric Tensors Cartesian Coordinate System Polar Coordinates

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

22. 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

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

24. 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)

25. 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