Graphics Mathematics for Computer Graphics cgvr korea ac

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

Graphics Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실 cgvr. korea. ac. kr Graphics Lab @ Korea University

Contents n CGVR Coordinate-Reference Frames 2 D Cartesian Reference Frames / Polar Coordinates n

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

Coordinate Reference Frames n CGVR Coordinate Reference Frames n Cartesian coordinate system o n

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 cgvr. korea. ac. kr 특수한 경우의 object표현에 사용. Polar, Spherical, Cylindrical 좌표계 등 Graphics Lab @ Korea University

2 D Cartesian Reference System n CGVR 2 D Cartesian Reference Frames y x

2 D Cartesian Reference System n CGVR 2 D Cartesian Reference Frames y x Coordinate origin at the lower-left screen corner cgvr. korea. ac. kr Coordinate origin in the upper-left screen corner Graphics Lab @ Korea University

Polar Coordinates n CGVR 가장 많이 쓰이는 Non-Cartesian System r n Elliptical Coordinates, Hyperbolic

Polar Coordinates n CGVR 가장 많이 쓰이는 Non-Cartesian System r n Elliptical Coordinates, Hyperbolic or Parabolic Plane Coordinates 등 원 이외에 Symmetry를 가진 다른 2차 곡선들로도 좌표계 표현 가능 cgvr. korea. ac. kr Graphics Lab @ Korea University

Why Polar Coordinates? n CGVR Circle n 2 D Cartesian : 비균등 분포 Polar

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 cgvr. korea. ac. kr 연속된 점들 사이에 일정간격유지 Polar Coordinates Graphics Lab @ Korea University

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

3 D Cartesian Reference Frames CGVR Three Dimensional Point cgvr. korea. ac. kr Graphics Lab @ Korea University

3 D Cartesian Reference Frames n 오른손 좌표계 n n CGVR 대부분의 Graphics Package에서

3 D Cartesian Reference Frames n 오른손 좌표계 n n CGVR 대부분의 Graphics Package에서 표준 왼손 좌표계 관찰자로부터 얼마만큼 떨 어져 있는지 나타내기에 편 리함 n Video Monitor의 좌표계 n cgvr. korea. ac. kr Graphics Lab @ Korea University

3 D Curvilinear Coordinate Systems n CGVR General Curvilinear Reference Frame n Orthogonal coordinate

3 D Curvilinear Coordinate Systems n CGVR General Curvilinear Reference Frame n Orthogonal coordinate system o Each coordinate surfaces intersects at right angles x 2 axis x 1 = const 1 x 3 axis x 3 = const 3 x 2 = const 2 x 1 axis A general Curvilinear coordinate reference frame cgvr. korea. ac. kr Graphics Lab @ Korea University

3 D Non-Cartesian System n Cylindrical Coordinates n Spherical Coordinates z axis z z

3 D Non-Cartesian System n Cylindrical Coordinates n Spherical Coordinates z axis z z axis P( , , z) x axis cgvr. korea. ac. kr CGVR y axis x axis P(r, , ) r y axis Graphics Lab @ Korea University

Points and Vectors n Point: 좌표계의 한 점을 차지, 위치표시 n Vector: 두 position간의

Points and Vectors 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 1 cgvr. korea. ac. kr x 2 Graphics Lab @ Korea University

Vectors n CGVR 3차원에서의 Vector z V y x n Vector Addition and Scalar

Vectors n CGVR 3차원에서의 Vector z V y x n Vector Addition and Scalar Multiplication cgvr. korea. ac. kr Graphics Lab @ Korea University

Scalar Product n CGVR Definition V 2 |V 2|cos V 1 Dot Product, Inner

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 cgvr. korea. ac. kr Graphics Lab @ Korea University

Vector Product n CGVR Definition V 1 V 2 u V 2 V 1

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 cgvr. korea. ac. kr Graphics Lab @ Korea University

Examples n CGVR Scalar Product n Vector Product (x 2, y 2) V 2

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 cgvr. korea. ac. kr Normal Vector of the Plane Graphics Lab @ Korea University

Basis Vectors n CGVR Basis (or a Set of Base Vectors) Specify the coordinate

Basis Vectors n CGVR Basis (or a Set of Base Vectors) Specify the coordinate axes in any reference frame n Linearly independent set of vectors Any other vector in that space can be written as linear combination of them n n u 2 Vector Space Contains scalars and vectors n Dimension: the number of base vectors u 1 n cgvr. korea. ac. kr u 3 Curvilinear coordinateaxis vectors Graphics Lab @ Korea University

Orthonormal Basis n Normal Basis + Orthogonal Basis n Example CGVR n Orthonormal basis

Orthonormal Basis n Normal Basis + Orthogonal Basis n Example CGVR n Orthonormal basis for 2 D Cartesian reference frame n Orthonormal basis for 3 D Cartesian reference frame cgvr. korea. ac. kr Graphics Lab @ Korea University

Metric Tensor n CGVR Tensor Quantity having a number of components, depending on the

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

Properties of Metric Tensors n CGVR The Elements of a Metric Tensor can be

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

Examples of Metric Tensors n Cartesian Coordinate System n Polar Coordinates cgvr. korea. ac.

Examples of Metric Tensors n Cartesian Coordinate System n Polar Coordinates cgvr. korea. ac. kr CGVR Graphics Lab @ Korea University

Matrices n Definition n n CGVR A rectangular array of quantities Scalar Multiplication and

Matrices n Definition n n CGVR A rectangular array of quantities Scalar Multiplication and Matrix Addition cgvr. korea. ac. kr Graphics Lab @ Korea University

Matrix Multiplication n CGVR Definition j-th column i-th row × l m n Properties

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 cgvr. korea. ac. kr m = (i, j) l n n Graphics Lab @ Korea University

Matrix Transpose n Definition n n CGVR Interchanging rows and columns Transpose of Matrix

Matrix Transpose n Definition n n CGVR Interchanging rows and columns Transpose of Matrix Product cgvr. korea. ac. kr Graphics Lab @ Korea University

Determinant of Matrix n CGVR Definition n For a square matrix, combining the matrix

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) cgvr. korea. ac. kr Graphics Lab @ Korea University

Inverse Matrix n CGVR Definition n Non-singular matrix o If and only if the

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 cgvr. korea. ac. kr Graphics Lab @ Korea University