Graphics Mathematics for Computer Graphics Graphics Lab Korea
Graphics Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실 Graphics Lab @ Korea University
Contents n CGVR Coordinate-reference Frames 2 D Cartesian Reference Frames n Polar Coordinates n 3 D Cartesian Reference Frames n 3 D Curvilinear Coordinate Systems n n Points and Vector Addition and Scalar Multiplication n Scalar Product n Vector Product n n Matrices n n n Scalar Multiplication and Matrix Addition Matrix Multiplication Matrix Transpose Determinant of a Matrix Inverse Graphics Lab @ Korea University
Coordinate Reference Frames n CGVR Coordinate Reference Frames n Cartesian coordinate system o n x, y, z 좌표축사용, 전형적 좌표계 Non-Cartesian coordinate system o o 특수한 경우의 object표현에 사용. Polar, Spherical, Cylindrical 좌표계 등 Graphics Lab @ Korea University
2 D Cartesian Reference System n CGVR Two-dimensional Cartesian Reference Frames y x Coordinate origin at the lower-left screen corner Coordinate origin in the upper-left screen corner Graphics Lab @ Korea University
Polar Coordinates n 가장 많이 쓰이는 CGVR Non-Cartesian System r n Elliptical coordinates, hyperbolic, parabolic plane coordinates등 원 이외에 symmetry를 가진 다른 2차 곡선 들로도 좌표계 표현 가능. Graphics Lab @ Korea University
Why Polar Coordinates? n CGVR Circle n 2 D Cartesian : 비균등 분포 Polar Coordinate y y d d x x dx dx 균등하게 분포되지 않은 점들 Cartesian Coordinates 연속된 점들 사이에 일정간격유지 Polar Coordinates Graphics Lab @ Korea University
3 D Cartesian Reference Frames CGVR Three Dimensional Point Graphics Lab @ Korea University
3 D Cartesian Reference Frames n CGVR 오른손 좌표계 n 대부분의 Graphics Package에서 표준 n 왼손 좌표계 n 관찰자로부터 얼마만큼 떨어져 있는지 나타내기에 편리함 n Video Monitor의 좌표계 Graphics Lab @ Korea University
3 D Curvilinear Coordinate Systems n CGVR General curvilinear reference frame n Orthogonal coordinate system o Each coordinate surfaces intersects at right angles A general Curvilinear coordinate reference frame Graphics Lab @ Korea University
3 D Non-Cartesian System n Cylindrical coordinates n Spherical coordinates z axis z z axis P( , , z) x axis CGVR y axis x axis P(r, , ) r y axis Graphics Lab @ Korea University
Point and Vector n Point: 좌표계의 한 점을 차지, 위치표시 n Vector: 두 position간의 차로 정의 n CGVR Magnitude와 Direction으로도 표기 P 2 y 1 V P 1 x 2 Graphics Lab @ Korea University
Vectors n CGVR 3차원에서의 Vector z V y x n Vector Addition and Scalar Multiplication Graphics Lab @ Korea University
Scalar Product n CGVR Definition V 2 |V 2|cos V 1 Dot Product, Inner Product라고도 함 n For Cartesian Reference Frame n Properties n Commutative n Distributive Graphics Lab @ Korea University
Vector Product n CGVR Definition V 1 V 2 u V 2 V 1 Cross Product, Outer Product라고도 함 n For Cartesian Reference Frame n Properties Anti. Commutative n Not Associative n Distributive n Graphics Lab @ Korea University
Examples n CGVR Scalar Product n Vector Product (x 2, y 2) V 2 (x 0, y 0) V 1 (x 1, y 1) Angle between Two Edges Normal Vector of the Plane Graphics Lab @ Korea University
Matrices n Definition n n CGVR A rectangular array of quantities Scalar multiplication and Matrix Addition Graphics Lab @ Korea University
Matrix Multiplication n CGVR Definition j-th column i-th row × l m n Properties n Not Commutative n Associative n Distributive n Scalar Multiplication m = (i, j) l n n Graphics Lab @ Korea University
Matrix Transpose n Definition n n CGVR Interchanging rows and columns Transpose of Matrix Product Graphics Lab @ Korea University
Determinant of Matrix n CGVR Definition n For a square matrix, combining the matrix elements to product a single number n 2 2 matrix n Determinant of n n matrix A (n 2) Graphics Lab @ Korea University
Inverse Matrix n CGVR Definition n Non-singular matrix o If and only if the determinant of the matrix is non-zero n 2 2 matrix n Properties Graphics Lab @ Korea University
Homework (1/3) CGVR n Cylindrical 좌표 (2, π/6, 1), (2, π/6, -1), (-2, π/6, 1), ( -2, π/6, -1), Spherical 좌표 (4, π/6, π/3), (-4, π/6, - π/3)를 Cartesian 좌표 로 변환하여라. n 세 점 A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)으로 구성된 삼각형 T(A, B, C)의 Face Normal을 구하시오. ( 반드시 Normalizing(정규화) 시킬 것) Graphics Lab @ Korea University
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