1. Mathematics for Computer Graphics
Graphics Laboratory
Korea University

2. Contents Coordinate Systems Points and Vectors Matrices
Parametric vs. Nonparametric Representations

3. Coordinate Systems Rectangular (Cartesian) Polar Cylindrical Spherical
x, y, z axes
Typical coordinate system
Right/left-hand system
Polar
Cylindrical
Spherical

4. 2D Rectangular Coordinate Systemx y x
Coordinate origin at the
lower-left screen corner
Coordinate origin at the
upper-left screen corner
<Window Coordinate System>
<Screen Coordinate System>

5. 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. Polar Coordinate System
p(r,) r s  x

7. Why Polar Coordinates in Circles?
In rectangular system
Irregular distribution of continuous points y y d d x x dx dx
Constant Distance
among the Adjacent Points
Irregularly Distributed Adjacent Points
Rectangular Coordinate System
Polar Coordinate System

8. Cylindrical / Spherical Systemx z   r
p(r,, ) z y x r 
p(r,,z)
Cylindrical Coordinate System
Spherical Coordinate System

9. Points and Vectors Point: location, position
Vector: direction from one point to another
Represented by using magnitude and unit direction y P2 y2 V  y1 P1 x x1 x2

10. Vectors 3D Vector Vector addition and scalar multiplication z V   y   V x z y
3D Vector
Vector addition and scalar multiplication

11. Dot Product / Inner Product
Scalar Product
Definition
Properties
Commutative
Distributive
|V2|cos  V2 V1
Dot Product / Inner Product

12. Careful for its direction!!!
Vector Product
Definition
Properties
Anti-commutative
Not associative
Distributive V1 V2
V1  V2  u
Cross Product / Outer Product
Careful for its direction!!!

13. Plane Normal Calculation
Frequently used in
Back face detection
Shading function
Vector product between
Two edges of the target polygon
N = V1 × V2 P4 V2 P3 P0 V1 P2 P1

14. Back Face Detection Not drawing the back faces to be culled
Can make the drawing speed faster
Scalar product between
Eye direction Deye and face normal vector Ni
If DeyeNi > 0
Fi is back face

15. Back Face Example Eye N4 N3 F4 F3 N2 N5 F2 F5 Eye Direction Deye(1,0)
(-0.9, -0.1) N2
(-0.8, 0.2) N3
(-0.2, 0.8) N4
(0.3, 0.7) N5
(0.8, 0.2) F1 F2 F3 F4 F5

16. Back Face Calculation F1 F2 F3 F4 F5
DeyeN1 = (1,0)(-0.9, -0.1) =  F1 is a front face F2
DeyeN2 = (1,0)(-0.8, 0.2) =  F2 is a front face F3
DeyeN3 = (1,0)(-0.2, 0.8) =  F3 is a front face F4
DeyeN4 = (1,0)(0.3, 0.7) =  F4 is a back face F5
DeyeN5 = (1,0)(0.8, 0.2) =  F5 is a back face

17. Back Face Culled ResultN4
(0.3, 0.7) N3
(-0.2, 0.8) F4 F3 N2
(-0.8, 0.2) N5
(0.8, 0.2) F2
Eye Direction Deye(1,0) F5
Eye F1 N1
(-0.9, -0.1)

18. <Simple Shading Function>
The amount of illumination depends on cos
If the incoming light Iin is perpendicular to the surface
Isurf is maximum, so the surface is fully illuminated
 = 0, cos = 1  N L
Isurf: intensity of the surface
Iin: intensity of the incident light
k: surface reflection coefficient
L: direction from the surface to a light source
<Simple Shading Function>

19. Matrices Definition Scalar multiplication and matrix addition
A rectangular array of quantities
Scalar multiplication and matrix addition

20. Matrix Multiplication
Definition
Properties
Not commutative
Associative
Distributive
Scalar multiplication
j-th column
i-th row × m = l l
(i,j) m n n

21. Matrix Transpose Definition Transpose of matrix product
Interchanging rows and columns
Transpose of matrix product

22. Determinant of Matrix Definition 2  2 matrix
For a square matrix
Combining the matrix elements to product a single number
2  2 matrix
Determinant of nn matrix A (n 2)
Determinant of N x N Matrix

23. Inverse Matrix Definition 2  2 matrix Properties Non-singular matrix
If and only if the determinant of the matrix is non-zero
2  2 matrix
Properties

24. 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: y y
Which one is balanced? -2 2 x -2 -1 1 2 x
Parametric Expression
Interval of : /4
Nonparametric Expression
Interval of x: 1

25. 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