1 / 25

Mathematics for Computer Graphics

Mathematics for Computer Graphics. 고려대학교 컴퓨터 그래픽스 연구실. Contents. Coordinate-Reference Frames 2D Cartesian Reference Frames / Polar Coordinates 3D Cartesian Reference Frames / Curvilinear Coordinates Points and Vectors Vector Addition and Scalar Multiplication

indra
Download Presentation

Mathematics for Computer Graphics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실 cgvr.korea.ac.kr

  2. Contents • Coordinate-Reference Frames • 2D Cartesian Reference Frames / Polar Coordinates • 3D Cartesian Reference Frames / Curvilinear Coordinates • Points and Vectors • Vector Addition and Scalar Multiplication • Scalar Product / Vector Product • Basis Vectors and the Metric Tensor • Orthonormal Basis • Metric Tensor • Matrices • Scalar Multiplication and Matrix Addition • Matrix Multiplication / Transpose • Determinant of a Matrix / Matrix Inverse cgvr.korea.ac.kr

  3. Coordinate Reference Frames • Coordinate Reference Frames • Cartesian coordinate system • x, y, z 좌표축사용, 전형적 좌표계 • Non-Cartesian coordinate system • 특수한 경우의 object표현에 사용. • Polar, Spherical, Cylindrical 좌표계 등 cgvr.korea.ac.kr

  4. 2D Cartesian Reference System • 2D Cartesian Reference Frames y x y x Coordinate origin at the lower-left screen corner Coordinate origin in the upper-left screen corner cgvr.korea.ac.kr

  5. Polar Coordinates • 가장 많이 쓰이는 Non-Cartesian System • Elliptical Coordinates, Hyperbolic or Parabolic Plane Coordinates 등 원 이외에 Symmetry를 가진 다른 2차 곡선들로도 좌표계 표현 가능 r  cgvr.korea.ac.kr

  6. Why Polar Coordinates? • Circle • 2D Cartesian : 비균등 분포  Polar Coordinate y y d d x x dx dx 균등하게 분포되지 않은 점들 연속된 점들 사이에 일정간격유지 Polar Coordinates Cartesian Coordinates cgvr.korea.ac.kr

  7. 3D Cartesian Reference Frames Three Dimensional Point cgvr.korea.ac.kr

  8. 3D Cartesian Reference Frames • 오른손 좌표계 • 대부분의 Graphics Package에서 표준 • 왼손 좌표계 • 관찰자로부터 얼마만큼 떨어져 있는지 나타내기에 편리함 • Video Monitor의 좌표계 cgvr.korea.ac.kr

  9. 3D Curvilinear Coordinate Systems • General Curvilinear Reference Frame • Orthogonal coordinate system • Each coordinate surfaces intersects at right angles x2 axis x3 = const3 x1 = const1 x3 axis x2 = const2 x1 axis A general Curvilinear coordinate reference frame cgvr.korea.ac.kr

  10. Cylindrical Coordinates Spherical Coordinates z axis z z axis P(,,z) P(r,, )  r y axis  y axis   x axis x axis 3D Non-Cartesian System cgvr.korea.ac.kr

  11. P2 y2 V y1 P1 x1 x2 Points and Vectors • Point:좌표계의 한 점을 차지, 위치표시 • Vector:두 position간의 차로 정의 • Magnitude와 Direction으로도 표기 cgvr.korea.ac.kr

  12. z  V   y x Vectors • 3차원에서의Vector • Vector Addition and Scalar Multiplication cgvr.korea.ac.kr

  13. V2  V1 |V2|cos Scalar Product • Definition • For Cartesian Reference Frame • Properties • Commutative • Distributive Dot Product, Inner Product라고도 함 cgvr.korea.ac.kr

  14. V1 V2 V2 u  V1 Vector Product • Definition • For Cartesian Reference Frame • Properties • AntiCommutative • Not Associative • Distributive Cross Product, Outer Product라고도 함 cgvr.korea.ac.kr

  15. Scalar Product Vector Product Examples (x2,y2) V2  (x1,y1) V1 (x0,y0) Angle between Two Edges Normal Vector of the Plane cgvr.korea.ac.kr

  16. u2 u1 u3 Basis Vectors • Basis (or a Set of Base Vectors) • Specify the coordinate axes in any reference frame • Linearly independent set of vectors  Any other vector in that space can be written as linear combination of them • Vector Space • Contains scalars and vectors • Dimension: the number of base vectors Curvilinear coordinate-axis vectors cgvr.korea.ac.kr

  17. Orthonormal Basis • Normal Basis + Orthogonal Basis • Example • Orthonormal basis for 2D Cartesian reference frame • Orthonormal basis for 3D Cartesian reference frame cgvr.korea.ac.kr

  18. Metric Tensor • Tensor • Quantity having a number of components, depending on the tensor rank and the dimension of the space • Vector – tensor of rank 1, scalar – tensor of rank 0 • Metric Tensor for any General Coordinate System • Rank 2 • Elements: • Symmetric: cgvr.korea.ac.kr

  19. Properties of Metric Tensors • The Elements of a Metric Tensor can be used to Determine • Distance between two points in that space • Transformation equations for conversion to another space • Components of various differential vector operators (such as gradient, divergence, and curl) within that space cgvr.korea.ac.kr

  20. Examples of Metric Tensors • Cartesian Coordinate System • Polar Coordinates cgvr.korea.ac.kr

  21. Matrices • Definition • A rectangular array of quantities • Scalar Multiplication and Matrix Addition cgvr.korea.ac.kr

  22. j-th column i-th row m × = (i,j) l l n m n Matrix Multiplication • Definition • Properties • Not Commutative • Associative • Distributive • Scalar Multiplication cgvr.korea.ac.kr

  23. Matrix Transpose • Definition • Interchanging rows and columns • Transpose of Matrix Product cgvr.korea.ac.kr

  24. 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) cgvr.korea.ac.kr

  25. Inverse Matrix • Definition • Non-singular matrix • If and only if the determinant of the matrix is non-zero • 2  2 matrix • Properties cgvr.korea.ac.kr

More Related