02mathematics

Mathematics for Computer Graphics 고려대학교 컴퓨터 그래픽스 연구실

Contents ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Coordinate Reference Frames ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

2D Cartesian Reference System ,[object Object],Coordinate origin at the lower-left screen corner y x y x Coordinate origin in the upper-left screen corner

Polar Coordinates ,[object Object],[object Object], r

Why Polar Coordinates? ,[object Object],[object Object],[object Object],x x y y dx dx d  d  균등하게 분포되지 않은 점들 연속된 점들 사이에 일정간격유지 Polar Coordinates Cartesian Coordinates

3D Cartesian Reference Frames Three Dimensional Point

3D Cartesian Reference Frames ,[object Object],[object Object],[object Object],[object Object],[object Object]

3D Curvilinear Coordinate Systems ,[object Object],[object Object],[object Object],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

3D Non-Cartesian System ,[object Object],[object Object],z P(  ,  ,z) x axis y axis z axis   P(r,  ,  ) x axis y axis z axis   r

[object Object],[object Object],[object Object],Points and Vectors V P 2 P 1 x 1 x 2 y 1 y 2

Vectors ,[object Object],[object Object],   V x z y

Scalar Product ,[object Object],[object Object],[object Object],[object Object],[object Object],Dot Product, Inner Product 라고도 함 |V 2 |cos   V 2 V 1

Vector Product ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Cross Product, Outer Product 라고도 함 V 1 V 2 V 1  V 2  u

Examples ,[object Object],[object Object],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 )

Basis Vectors ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Curvilinear coordinate-axis vectors u 2 u 1 u 3

Orthonormal Basis ,[object Object],[object Object],[object Object],[object Object]

Metric Tensor ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Properties of Metric Tensors ,[object Object],[object Object],[object Object],[object Object]

Examples of Metric Tensors ,[object Object],[object Object]

Matrices ,[object Object],[object Object],[object Object]

Matrix Multiplication ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],× = ( i,j ) j -th column i -th row m l n n m l

Matrix Transpose ,[object Object],[object Object],[object Object]

Determinant of Matrix ,[object Object],[object Object],[object Object],[object Object]

Inverse Matrix ,[object Object],[object Object],[object Object],[object Object],[object Object]

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Editor's Notes