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lecture13

# lecture13 - Lecture 13 Operations in Graphics and Geometric...

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Lecture 13 Operations in Graphics and Geometric Modeling I: Projection, Rotation, and Reflection Shang-Hua Teng

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Projection Projection onto an axis ( a,b ) x axis is a vector subspace
Projection onto an Arbitrary Line Passing through 0 ( a,b )

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Projection on to a Plane
Projection on to a Line b a θ θ cos b a b a T = ( 29 ( 29 a a a b a a a a b a a a b p T T T = = = θ θ cos cos p

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Projection Matrix: on to a Line b a θ ( 29 ( 29 a a a b a a a a b a a a b p T T T = = = θ θ cos cos p What matrix P has the property p = P b = = = = = a a aa P b a a aa a a b a a p Pb a a a b a p T T T T T T T T
Properties of Projection on to a Line b a θ p x a b x a p a a b a x b ax x a p b a p T T ˆ minimizes which , ˆ ˆ then , of ion approximat square least the is ˆ if ) ( ) ( Span - = = = - p is the points in Span( a ) that is the closest to b

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Projection onto a Subspace Input: 1. Given a vector subspace V in R m 2. A vector b in R m Desirable Output: A vector in p in V that is closest to b The projection p of b in V A vector p in V such that ( b-p ) is orthogonal to V
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lecture13 - Lecture 13 Operations in Graphics and Geometric...

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