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02mathematics

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02mathematics

  1. 1. Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실
  2. 2. Contents <ul><li>Coordinate-Reference Frames </li></ul><ul><ul><li>2D Cartesian Reference Frames / Polar Coordinates </li></ul></ul><ul><ul><li>3D Cartesian Reference Frames / Curvilinear Coordinates </li></ul></ul><ul><li>Points and Vectors </li></ul><ul><ul><li>Vector Addition and Scalar Multiplication </li></ul></ul><ul><ul><li>Scalar Product / Vector Product </li></ul></ul><ul><li>Basis Vectors and the Metric Tensor </li></ul><ul><ul><li>Orthonormal Basis </li></ul></ul><ul><ul><li>Metric Tensor </li></ul></ul><ul><li>M a trices </li></ul><ul><ul><li>Scalar Multiplication and Matrix Addition </li></ul></ul><ul><ul><li>Matrix Multiplication / Transpose </li></ul></ul><ul><ul><li>Determinant of a Matrix / Matrix Inverse </li></ul></ul>
  3. 3. Coordinate Reference Frames <ul><li>Coordinate Reference Frames </li></ul><ul><ul><li>Cartesian coordinate system </li></ul></ul><ul><ul><ul><li>x, y, z 좌표축사용 , 전형적 좌표계 </li></ul></ul></ul><ul><ul><li>Non-Cartesian coordinate system </li></ul></ul><ul><ul><ul><li>특수한 경우의 object 표현에 사용 . </li></ul></ul></ul><ul><ul><ul><li>Polar, Spherical, Cylindrical 좌표계 등 </li></ul></ul></ul>
  4. 4. 2D Cartesian Reference System <ul><li>2D Cartesian Reference Frames </li></ul>Coordinate origin at the lower-left screen corner y x y x Coordinate origin in the upper-left screen corner
  5. 5. Polar Coordinates <ul><li>가장 많이 쓰이는 Non-Cartesian System </li></ul><ul><li>Elliptical Coordinates, Hyperbolic or Parabolic Plane Coordinates 등 원 이외에 Symmetry 를 가진 다른 2 차 곡선들로도 좌표계 표현 가능 </li></ul> r
  6. 6. Why Polar Coordinates? <ul><li>Circle </li></ul><ul><ul><li>2D Cartesian : 비균등 분포 </li></ul></ul><ul><ul><li> Polar Coordinate </li></ul></ul>x x y y dx dx d  d  균등하게 분포되지 않은 점들 연속된 점들 사이에 일정간격유지 Polar Coordinates Cartesian Coordinates
  7. 7. 3D Cartesian Reference Frames Three Dimensional Point
  8. 8. 3D Cartesian Reference Frames <ul><li>오른손 좌표계 </li></ul><ul><ul><li>대부분의 Graphics Package 에서 표준 </li></ul></ul><ul><li>왼손 좌표계 </li></ul><ul><ul><li>관찰자로부터 얼마만큼 떨어져 있는지 나타내기에 편리함 </li></ul></ul><ul><ul><li>Video Monitor 의 좌표계 </li></ul></ul>
  9. 9. 3D Curvilinear Coordinate Systems <ul><li>General Curvilinear Reference Frame </li></ul><ul><ul><li>Orthogonal coordinate system </li></ul></ul><ul><ul><ul><li>Each coordinate surfaces intersects at right angles </li></ul></ul></ul>A general Curvilinear coordinate reference frame x 2 axis x 3 axis x 1 axis x 1 = const 1 x 3 = const 3 x 2 = const 2
  10. 10. 3D Non-Cartesian System <ul><li>Cylindrical Coordinates </li></ul><ul><li>Spherical Coordinates </li></ul>z P(  ,  ,z) x axis y axis z axis   P(r,  ,  ) x axis y axis z axis   r
  11. 11. <ul><li>Point : 좌표계의 한 점을 차지 , 위치표시 </li></ul><ul><li>Vector : 두 position 간의 차로 정의 </li></ul><ul><ul><li>Magnitude 와 Direction 으로도 표기 </li></ul></ul>Points and Vectors V P 2 P 1 x 1 x 2 y 1 y 2
  12. 12. Vectors <ul><li>3 차원에서의 Vector </li></ul><ul><li>Vector Addition and Scalar Multiplication </li></ul>   V x z y
  13. 13. Scalar Product <ul><li>Definition </li></ul><ul><li>For Cartesian Reference Frame </li></ul><ul><li>Properties </li></ul><ul><ul><li>Commutative </li></ul></ul><ul><ul><li>Distributive </li></ul></ul>Dot Product, Inner Product 라고도 함 |V 2 |cos   V 2 V 1
  14. 14. Vector Product <ul><li>Definition </li></ul><ul><li>For Cartesian Reference Frame </li></ul><ul><li>Properties </li></ul><ul><ul><li>AntiCommutative </li></ul></ul><ul><ul><li>Not Associative </li></ul></ul><ul><ul><li>Distributive </li></ul></ul>Cross Product, Outer Product 라고도 함 V 1 V 2 V 1  V 2  u
  15. 15. Examples <ul><li>Scalar Product </li></ul><ul><li>Vector Product </li></ul>Normal Vector of the Plane  V 2 V 1 Angle between Two Edges ( x 2 , y 2 ) ( x 0 , y 0 ) ( x 1 , y 1 )
  16. 16. Basis Vectors <ul><li>Basis (or a Set of Base Vectors) </li></ul><ul><ul><li>Specify the coordinate axes in any reference frame </li></ul></ul><ul><ul><li>Linearly independent set of vectors </li></ul></ul><ul><ul><li> Any other vector in that space can be written as linear combination of them </li></ul></ul><ul><li>Vector Space </li></ul><ul><ul><li>Contains scalars and vectors </li></ul></ul><ul><ul><li>Dimension : the number of </li></ul></ul><ul><ul><li>base vectors </li></ul></ul>Curvilinear coordinate-axis vectors u 2 u 1 u 3
  17. 17. Orthonormal Basis <ul><li>Normal Basis + Orthogonal Basis </li></ul><ul><li>Example </li></ul><ul><ul><li>Orthonormal basis for 2D Cartesian reference frame </li></ul></ul><ul><ul><li>Orthonormal basis for 3D Cartesian reference frame </li></ul></ul>
  18. 18. Metric Tensor <ul><li>Tensor </li></ul><ul><ul><li>Quantity having a number of components, depending on the tensor rank and the dimension of the space </li></ul></ul><ul><ul><li>Vector – tensor of rank 1, scalar – tensor of rank 0 </li></ul></ul><ul><li>Metric Tensor for any General Coordinate System </li></ul><ul><ul><li>Rank 2 </li></ul></ul><ul><ul><li>Elements: </li></ul></ul><ul><ul><li>Symmetric: </li></ul></ul>
  19. 19. Properties of Metric Tensors <ul><li>The Elements of a Metric Tensor can be used to Determine </li></ul><ul><ul><li>Distance between two points in that space </li></ul></ul><ul><ul><li>Transformation equations for conversion to another space </li></ul></ul><ul><ul><li>Components of various differential vector operators (such as gradient, divergence, and curl) within that space </li></ul></ul>
  20. 20. Examples of Metric Tensors <ul><li>Cartesian Coordinate System </li></ul><ul><li>Polar Coordinates </li></ul>
  21. 21. Matrices <ul><li>Definition </li></ul><ul><ul><li>A rectangular array of quantities </li></ul></ul><ul><li>Scalar Multiplication and Matrix Addition </li></ul>
  22. 22. Matrix Multiplication <ul><li>Definition </li></ul><ul><li>Properties </li></ul><ul><ul><li>Not Commutative </li></ul></ul><ul><ul><li>Associative </li></ul></ul><ul><ul><li>Distributive </li></ul></ul><ul><ul><li>Scalar Multiplication </li></ul></ul>× = ( i,j ) j -th column i -th row m l n n m l
  23. 23. Matrix Transpose <ul><li>Definition </li></ul><ul><ul><li>Interchanging rows and columns </li></ul></ul><ul><li>Transpose of Matrix Product </li></ul>
  24. 24. Determinant of Matrix <ul><li>Definition </li></ul><ul><ul><li>For a square matrix, combining the matrix elements to product a single number </li></ul></ul><ul><li>2  2 matrix </li></ul><ul><li>Determinant of n  n Matrix A (n  2) </li></ul>
  25. 25. Inverse Matrix <ul><li>Definition </li></ul><ul><ul><li>Non-singular matrix </li></ul></ul><ul><ul><ul><li>If and only if the determinant of the matrix is non-zero </li></ul></ul></ul><ul><li>2  2 matrix </li></ul><ul><li>Properties </li></ul>

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