Mathematics for Graphics 김 창 헌

Coordinate Reference Frames Cartesian coordinate system x,y,z좌표축사용, 전형적 좌표계 Non-Cartesian coordinate system 특수한 경우의 object표현에 사용. Polar, Spherical, Cylindrical 좌표계등

2 D Coordinate System Two-dimensional Cartesian Reference Frames a) b) x y x a) b)

Polar Coordinates 가장 많이 쓰이는 non-Cartesian System Elliptical coordinates, hyperbolic, parabolic plane coordinates등 원 이외에 symmetry를 가진 다른 2차 곡선들로도 좌표계 표현 가능. r 

Cartesian Coordinates Why Polar Coordinates? 예) 원의 표현 y y d d x x dx dx 균등하게 분포되지 않은 점들 연속된 점들 사이에 일정간격유지 Cartesian Coordinates Polar Coordinates

Three Dimensional Point 3D Cartesian Frames Three Dimensional Point

- 대부분의 Graphics Package에서 표준 3D Cartesian Frames 오른손 좌표계 - 대부분의 Graphics Package에서 표준 왼손 좌표계 - 관찰자로부터 얼마만큼 떨어져 있는지 나타내기에 편리함 - Video Monitor의 좌표계

Spherical coordinates 3D NonCartesian System Cylindrical coordinates Spherical coordinates z P(,,z) x axis y axis z axis   P(r,, ) x axis y axis z axis   r

Points 가장 기본적인 Output Primitive 0차원으로 크기와 길이 측정 불가 순서쌍(x,y)나 vector형식으로 표기 Raster Scan display 의 한 Pixel차지 2D or 3D

Lines Rasterization Defined as a list of points(PolyLine) Stairstep effect(jaggies)

Matrices Scalar multiplication and Matrix Addition Definition A rectangular array of quantities Scalar multiplication and Matrix Addition

Matrix Multiplication Definition Properties Not Commutative Assotiative Distributive Scalar multiplication × = (i,j) j-th column i-th row m l n

Translation y P x (a) y P’ x (b)

Scaling x’ x y x x’ = xf + (x-xf) sx , y’ = yf + ( y- yf) sy x’ = x · Sx , y’ = y · Sy x’ x P2 (xf, yf) : fixed point x y (xf,yf) P1 P3 x’ = xf + (x-xf) sx , y’ = yf + ( y- yf) sy

Rotation P’ = R P  x = r cos  , y = r sin  x’ = r cos ( + ) = r cos  cos  - r sin  sin  y’ = r sin ( + ) = r cos  sin  + r sin  cos   x’= x cos  - y sin , y’ = x sin  + y cos  (x’,y’) r  (x,y) r  죄표중심을 회전점으로 각  만큼 회전

Homogeneous Coordinates In basic Transformations, ( Addition ) Only Translation is treated differently ( Multiplication ) ( Multiplication ) We hope to combine the multiplicative and translational terms for two-dimensional geometric transformations into a single matrix representation for enabling the composite transformations such as,

Extend the matrix We can solve this problem by extending 2 by2 matrix into 3 by 3 matrix Translation Rotation Scaling Ex) P2 (xf, yf) : fixed point x y (xf,yf) P1 P3

3D Representarion of homogeneous space Any two dimensional point can be represented by one of the points along the ray in 3D space y x z =3 z =2 z =1 P(3x1,3y1,3) P(2x1,2y1,2) P(x1,y1,1) P(hx1,hy1,h) = P(X,Y, h) 3D Representarion of homogeneous space

Point at Infinity The points with h=0 are called points at infinity, and this will not appear very often. Ex) [ X Y h ] = [ 4 3 1]

Determinant of Matrix and for a 2 by 2 matrix, Ex) For n 2, the Determinant of nn matrix A is, and for a 2 by 2 matrix, Ex) if A is a triangular matrix, det A is the product of the entries on the main diagonal of A

Solving Linear Equations where, ajk and bj are known Using Matrix Equation ※Coefficient Matrix A의 역행렬이 존재할 때만 성립

Inverse Matrix • Definition • Properties • 2  2 matrix의 경우 If ad-bc = 0, then A is not invertible.

Inverse Matrix • Algorithm for Finding A-1 : Row reduction [ I A-1] 형태의 row reduction이 존재하지 않으면, A is not invertible. Ex)

Gaussian Elimination System에서 한 행의 실수배를 다른 행에서 빼어 연립방정식의 차수를 줄여나가다, 한 변수의 해가 구해지면, 역으로 대입해 나머지 변수값을 구한다. • Elementary Row Operation 1. Multiply any row of the augmented matrix by a nonzero constant 2. Add a multiple of one row to a multiple of any other row 3. Interchange the order of any two rows

Using Row Operation

Consistency of Linear System 1. Unknown variable의 수에 비해 equation의 수가 적을 때 No Unique solution (Infinite set of solution) 2. Unknown variable의 수에 비해 equation의 수가 많을 때 1) Consistent한 경우 : 구한 해가 남은 방정식에 대해서도 성립(redundant) 2) Inconsistent한 경우 (infinity) (no solution)