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Math&Com Graphics Lab Vector Hyoungseok B. Kim Dept. of Multimedia Eng. Dongeui University
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Math&Com Graphics Lab. Dongeui University2 What is Computer Game ? To make us fun by using computer Computer Game Sense of Sight (Computer Graphics) Sense of Hearing (Sound) Sense of Touch (Interaction) Interesting Story What is Computer Graphics Technologies of creating virtual space and displaying it on computer monitor by using computer Computer Graphics Modeling Rendering Animation Game and Graphics
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Math&Com Graphics Lab. Dongeui University3 Computer Graphics 3D Virtual Space = Model + Light 2D Virtual Space Camera (Clipping, Projection, Hidden-Surface Removal) Rasterization Screen Space 3 차원 공간 필름 사진 현상
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Math&Com Graphics Lab. Dongeui University4 transformation 좌표 변환을 위한 4×4 matrix multiplication clipping projection plane 상에서 불필요한 부분 제거 projection 3D object 2D image mapping rasterization image 를 frame buffer 에 저장하는 과정 OpenGL Pipeline Architecture
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Math&Com Graphics Lab. Dongeui University5 Where is Mathematics in Computer Graphics ? Creation of Objects Vertices, edges, faces, box, sphere, cylinder, torus, … Handle of Objects Transformation : translation, scaling, rotation Handle of Camera Position, Orientation, Lens, Projection Handle of Light Shadow Handle of Motion Character motion, animation of all kinds of objects Handle of Rendering Image based rendering ….. Game Mathematics
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Math&Com Graphics Lab. Dongeui University6 Standard Language in Mathematics Quantity Scalar : Magnitude( 크기 ) Vector : Magnitude( 크기 ), Orientation( 방향 ) Representation of Quantities Scalar : real number 1, 2, 0.72534, 3 / 7 Vector : Arrow 시점과 종점 Scalar vs Vector 시점 종점 inefficient Your answer ?
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Math&Com Graphics Lab. Dongeui University7 Equality and Efficient Representation same Find out the same vectors as the given vector A ? same magnitude, same color Blue arrow Efficient Representation Assume that the start point of all vectors is the Origin in the space. Vector A
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Math&Com Graphics Lab. Dongeui University8 Vector Representation of Vector Position of the End Point : 시점 종점
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Math&Com Graphics Lab. Dongeui University9 Operators of Vectors Same if and only if Magnitude Addition Inner Product Cross Product
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Math&Com Graphics Lab. Dongeui University10 Inner Product Properties 1. Why ? 2. Why ? 코사인 제 2 법칙
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Math&Com Graphics Lab. Dongeui University11 Inner Product 1. 2. What is the inner product used for ? To confirm whether the angle is right or not…….
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Math&Com Graphics Lab. Dongeui University12 Inner Product’s Applications 1. ? 2. Compute a plane P passing through a given point with the normal vector
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Math&Com Graphics Lab. Dongeui University13 Inner Product’s Applications 3. Compute the distance of a point from a line L passing through with a unit directional vector 4. Compute the distance of a point from a Plane P passing through with a unit Normal vector distance = ? 5. Up side or Down side ?
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Math&Com Graphics Lab. Dongeui University14 Back Face Removal Which are back faces ? front faceback face How to compute ? What is a counter-example ?
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Math&Com Graphics Lab. Dongeui University15 Back Face Removal Counter-example
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Math&Com Graphics Lab. Dongeui University16 Cross Product Properties : A new vector orthogonal to both two vectors Area of a Triangle with edges and Why ?
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Math&Com Graphics Lab. Dongeui University17 Cross Product’s Applications Normal Vector Computation Parametric Surface
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Math&Com Graphics Lab. Dongeui University18 Cross Product’s Applications Normal Vector Computation Polygonal Mesh
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Math&Com Graphics Lab. Dongeui University19 Vector Space The set of vectors satisfying 9 properties addition, scalar multiplication, identity, additive inverse, commutative law, distributive law Examples) Properties) Linearly dependence linearly independent linearly dependent Is the set linearly dependent ?
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Math&Com Graphics Lab. Dongeui University20 Vector Space 1 차 독립 (Linearly Independent) 만약 다음을 만족한다면 은 “1 차 독립 ” 라고 함 만약 그렇지 않다면, 은 “1 차 종속 ” 이라 함 즉, 이라면
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Math&Com Graphics Lab. Dongeui University21 Vector Space Linearly dependence in linearly dependent linearly independent linearly dependent
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Math&Com Graphics Lab. Dongeui University22 Vector Space Basis 이 벡터공간 에서 1 차 독립이며 그 공간에 있는 모든 벡터들을 다음과 같이 1 차 선형조합으로 표현가능 하면 이 벡터공간 의 기저 (basis) 예제 의 basis 에는 어떤 것이 있는가 ?
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Math&Com Graphics Lab. Dongeui University23 Vector Space Question 1. 서로 수직인 basis 는 무엇인가 ? Orthogonal basis 가 왜 필요하나 ? 일반 basis 를 orthogonal basis 로 바꾸는 방법은 있을까 ? Gram-schmit Orthogonalization
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Math&Com Graphics Lab. Dongeui University24 Frame Point + Orthogonal basis : three orthogonal vectors (0, 0, 0) + Frame Transformation d = ? e = ? f = ?
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Math&Com Graphics Lab. Dongeui University25 Coordinate Transform : Point original frame v 1, v 2, v 3, P 0 new frame u 1, u 2, u 3, Q 0 mapping
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