1. Graphics Graphics Lab @ Korea University http://kucg.korea.ac.kr Mathematics for Computer Graphics Graphics Laboratory Korea University

2. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Contents Coordinate Systems Points and Vectors Matrices Parametric vs. Nonparametric Representations

3. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Coordinate Systems Rectangular x, y, z axes Typical coordinate system Right/left-hand system Polar Cylindrical Spherical

4. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr 2D Rectangular Coordinate System Coordinate origin at the lower-left screen corner y x y x Coordinate origin at the upper-left screen corner

5. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr 3D Rectangular Coordinate System Right-hand system Standard in most graphics packages Left-hand system Easy to know the distance from the viewer Video monitor coordinate system

6. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Polar Coordinate System  r s x y p(r,  )

7. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Why Polar Coordinates in Circles? x y dddd dddd Irregularly Distributed Adjacent Points Constant Distance among the Adjacent Points Polar Coordinate System Rectangular Coordinate System In rectangular system Irregular distribution of continuous points x y dxdx

8. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Cylindrical / Spherical Systemzy x r z  p(r, ,z) y xz  r p(r, ,  ) Spherical Coordinate System Cylindrical Coordinate System

9. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Point: location, position Vector: direction from one point to another Represented by using magnitude and unit direction V P2P2P2P2 P1P1P1P1 x1x1x1x1 x2x2x2x2 y1y1y1y1 y2y2y2y2 Points and Vectors  x y

10. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Vectors 3D Vector Vector addition and scalar multiplication    V xzy

11. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Scalar Product Definition Properties Commutative Distributive Dot Product / Inner Product |V 2 |cos   V2V2V2V2 V1V1V1V1

12. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Vector Product Definition Properties Anti-commutative Not associative Distributive Cross Product / Outer Product V1V1V1V1 V2V2V2V2 V1  V2V1  V2V1  V2V1  V2 u Careful for its direction!!!

13. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Plane Normal Calculation Frequently used in Back face detection Shading function Vector product Vector product between Two edges of the target polygon V1V1V1V1 V2V2V2V2 N = V 1 × V 2 P0P0P0P0 P1P1P1P1 P2P2P2P2 P3P3P3P3 P4P4P4P4

14. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Back Face Detection Not drawing the back faces to be culled Can make the drawing speed faster Scalar product Scalar product between D eye N i  Eye direction D eye and face normal vector N i D eye  N i If D eye  N i > 0  F i  F i is back face

15. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Back Face Example Eye Eye Direction D eye (1,0) N 1 (-0.9, -0.1) N 2 (-0.8, 0.2) N 3 (-0.2, 0.8) N 4 (0.3, 0.7) N 5 (0.8, 0.2) F1F1F1F1 F2F2F2F2 F3F3F3F3 F4F4F4F4 F5F5F5F5

16. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Back Face Calculation F 1 D eye  N 1 = (1,0)  (-0.9, -0.1) = -0.9  F 1 is a front face F 2 D eye  N 2 = (1,0)  (-0.8, 0.2) = -0.8  F 2 is a front face F 3 D eye  N 3 = (1,0)  (-0.2, 0.8) = -0.2  F 3 is a front face F 4 D eye  N 4 = (1,0)  (0.3, 0.7) = 0.3  F 4 is a back face F 5 D eye  N 5 = (1,0)  (0.8, 0.2) = 0.8  F 5 is a back face

17. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Back Face Culled Result Eye Eye Direction D eye (1,0) N 1 (-0.9, -0.1) N 2 (-0.8, 0.2) N 3 (-0.2, 0.8) N 4 (0.3, 0.7) N 5 (0.8, 0.2) F1F1F1F1 F2F2F2F2 F3F3F3F3 F4F4F4F4 F5F5F5F5

18. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Shading Function cos  The amount of illumination depends on cos  I in If the incoming light I in is perpendicular to the surface  I surf  I surf is maximum, so the surface is fully illuminated   cos    = 0, cos  = 1  NL I surf : intensity of the surface I in : intensity of the incident light k : surface reflection coefficient L : direction from the surface to a light source

19. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Matrices Definition A rectangular array of quantities Scalar multiplication and matrix addition

20. KUCG Graphics Lab @ Korea University http://kucg.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

21. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Matrix Transpose Definition Interchanging rows and columns Transpose of matrix product

22. KUCG Graphics Lab @ Korea University http://kucg.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)

23. KUCG Graphics Lab @ Korea University http://kucg.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

24. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Parametric vs. Nonparametric Representations Circle example in computer graphics  radius 2, centered at the origin Parametric expression: x = 2cos , y = 2sin  Nonparametric expression  Implicit:, explicit: x y x y 0 12-2 Parametric Expression Interval of  :  /4 Nonparametric Expression Interval of x: 1 Which one is balanced? 2-2

25. KUCG Graphics Lab @ Korea University http://kucg.korea.ac.kr Parametric Representation Easy to draw the shape of an object smoothly Just increase one parameter ex)   The other parameters are automatically calculated by  Especially for symmetric objects  Circle, sphere, ellipsoid, etc. Preferred in computer graphics Nonparametric representation is used mainly in numerical analysis