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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

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Mathematics for Computer Graphics

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  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

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