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02mathematics

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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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02mathematics

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  1. 한글을 발음하는 입술 애니메이션은 다음과 같은 문제점을 갖고 있습니다 . 한글은 외국어와 차이가 있어 외국 모델을 그대로 사용할 수 없으며 연구 개요에서 보셨듯이 선행 연구는 부정확한 발음을 표현하고 동시조음도 고려하지 않아 실재와 다른 움직임을 보였습니다 . 본 연구는 음절을 자연스럽게 발음하고 음절간의 움직임도 부드럽게 표현하는 모델을 구현하는 것이 목적입니다 .
  2. 한글을 발음하는 입술 애니메이션은 다음과 같은 문제점을 갖고 있습니다 . 한글은 외국어와 차이가 있어 외국 모델을 그대로 사용할 수 없으며 연구 개요에서 보셨듯이 선행 연구는 부정확한 발음을 표현하고 동시조음도 고려하지 않아 실재와 다른 움직임을 보였습니다 . 본 연구는 음절을 자연스럽게 발음하고 음절간의 움직임도 부드럽게 표현하는 모델을 구현하는 것이 목적입니다 .
  3. 한글을 발음하는 입술 애니메이션은 다음과 같은 문제점을 갖고 있습니다 . 한글은 외국어와 차이가 있어 외국 모델을 그대로 사용할 수 없으며 연구 개요에서 보셨듯이 선행 연구는 부정확한 발음을 표현하고 동시조음도 고려하지 않아 실재와 다른 움직임을 보였습니다 . 본 연구는 음절을 자연스럽게 발음하고 음절간의 움직임도 부드럽게 표현하는 모델을 구현하는 것이 목적입니다 .
  4. 한글을 발음하는 입술 애니메이션은 다음과 같은 문제점을 갖고 있습니다 . 한글은 외국어와 차이가 있어 외국 모델을 그대로 사용할 수 없으며 연구 개요에서 보셨듯이 선행 연구는 부정확한 발음을 표현하고 동시조음도 고려하지 않아 실재와 다른 움직임을 보였습니다 . 본 연구는 음절을 자연스럽게 발음하고 음절간의 움직임도 부드럽게 표현하는 모델을 구현하는 것이 목적입니다 .
  5. 한글을 발음하는 입술 애니메이션은 다음과 같은 문제점을 갖고 있습니다 . 한글은 외국어와 차이가 있어 외국 모델을 그대로 사용할 수 없으며 연구 개요에서 보셨듯이 선행 연구는 부정확한 발음을 표현하고 동시조음도 고려하지 않아 실재와 다른 움직임을 보였습니다 . 본 연구는 음절을 자연스럽게 발음하고 음절간의 움직임도 부드럽게 표현하는 모델을 구현하는 것이 목적입니다 .
  6. 한글을 발음하는 입술 애니메이션은 다음과 같은 문제점을 갖고 있습니다 . 한글은 외국어와 차이가 있어 외국 모델을 그대로 사용할 수 없으며 연구 개요에서 보셨듯이 선행 연구는 부정확한 발음을 표현하고 동시조음도 고려하지 않아 실재와 다른 움직임을 보였습니다 . 본 연구는 음절을 자연스럽게 발음하고 음절간의 움직임도 부드럽게 표현하는 모델을 구현하는 것이 목적입니다 .