UVA CS 445 - Mathematical Foundations (10 pages)

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Mathematical Foundations



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Mathematical Foundations

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Pages:
10
School:
University Of Virginia
Course:
Cs 445 - Introduction to Computer Graphics
Introduction to Computer Graphics Documents

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Introduction Mathematical Foundations CS 445 645 Introduction to Computer Graphics David Luebke Spring 2003 Admin Introductions Dave Nate Call roll Go over syllabus Fill out course action forms I ll collect sign and turn in after class David Luebke 01 14 19 2 Mathematical Foundations I ll give a brief informal review of some of the mathematical tools we ll employ Geometry 2D 3D Trigonometry Vector and affine spaces Points vectors and coordinates Dot and cross products Linear transforms and matrices Bear with me David Luebke 01 14 19 3 2D Geometry Know your high school geometry Total angle around a circle is 360 or 2 radians When two lines cross Opposite angles are equivalent Angles along line sum to 180 Similar triangles All corresponding angles are equivalent Corresponding pairs of sides have the same length ratio and are separated by equivalent angles Any corresponding pairs of sides have same length ratio David Luebke 01 14 19 4 Trigonometry Sine opposite over hypotenuse Cosine adjacent over hypotenuse Tangent opposite over adjacent Unit circle definitions sin y cos x tan y x Etc David Luebke 01 14 19 x y 5 3D Geometry To model animate and render 3D scenes we must specify Location Displacement from arbitrary locations Orientation We ll look at two types of spaces Vector spaces Affine spaces We will often be sloppy about the distinction David Luebke 01 14 19 6 Vector Spaces Two types of elements Scalars real numbers Vectors n tuples u v w Supports two operations Addition operation u v with v 0 v Inverse v v 0 Scalar multiplication Distributive rule u v u v u u u Identity David Luebke 01 14 19 0 7 Vector Spaces A linear combination of vectors results in a new vector v 1v1 2v2 nvn If the only set of scalars such that 1v1 2v2 nvn 0 is 1 2 3 0 then we say the vectors are linearly independent The dimension of a space is the greatest number of linearly independent vectors possible in a vector set For a vector space of dimension n any set of n linearly independent



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