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

Contents Coordinate Systems Points and Vectors Matrices Parametric vs. Nonparametric Representations http://kucg.korea.ac.kr

Coordinate Systems Rectangular (Cartesian) Polar Cylindrical Spherical x, y, z axes Typical coordinate system Right/left-hand system Polar Cylindrical Spherical http://kucg.korea.ac.kr

2D Rectangular Coordinate System x y x Coordinate origin at the lower-left screen corner Coordinate origin at the upper-left screen corner <Window Coordinate System> <Screen Coordinate System> 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 http://kucg.korea.ac.kr

Polar Coordinate System p(r,) r s  x http://kucg.korea.ac.kr

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 http://kucg.korea.ac.kr

Cylindrical / Spherical System x z   r p(r,, ) z y x r  p(r,,z) Cylindrical Coordinate System Spherical Coordinate System http://kucg.korea.ac.kr

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 http://kucg.korea.ac.kr

Vectors 3D Vector Vector addition and scalar multiplication z V   y    V x z y 3D Vector Vector addition and scalar multiplication http://kucg.korea.ac.kr

Dot Product / Inner Product Scalar Product Definition Properties Commutative Distributive |V2|cos  V2 V1 Dot Product / Inner Product http://kucg.korea.ac.kr

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!!! http://kucg.korea.ac.kr

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 http://kucg.korea.ac.kr

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 http://kucg.korea.ac.kr

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 http://kucg.korea.ac.kr

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

Back Face Culled Result N4 (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) http://kucg.korea.ac.kr

<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> http://kucg.korea.ac.kr

Matrices Definition Scalar multiplication and matrix addition A rectangular array of quantities Scalar multiplication and matrix addition http://kucg.korea.ac.kr

Matrix Multiplication Definition Properties Not commutative Associative Distributive Scalar multiplication j-th column i-th row × m = l l (i,j) m n n http://kucg.korea.ac.kr

Matrix Transpose Definition Transpose of matrix product Interchanging rows and columns Transpose of matrix product http://kucg.korea.ac.kr

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 http://kucg.korea.ac.kr

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 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: 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 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 http://kucg.korea.ac.kr