300 likes | 321 Views
Lecture 13 Operations in Graphics and Geometric Modeling I: Projection, Rotation, and Reflection. Shang-Hua Teng. Projection. Projection onto an axis. ( a,b ). x axis is a vector subspace. Projection onto an Arbitrary Line Passing through 0. ( a,b ). Projection on to a Plane.
E N D
Lecture 13Operations in Graphics and Geometric Modeling I:Projection, Rotation, and Reflection Shang-Hua Teng
Projection • Projection onto an axis (a,b) x axis is a vector subspace
Projection on to a Line b a q p
Projection Matrix: on to a Line b What matrix P has the property p = Pb a q p
Properties of Projection on to a Line b a q p p is the points in Span(a) that is the closest to b
Projection onto a Subspace • Input: • Given a vector subspace V in Rm • A vector b in Rm… • 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
How to Describe a Vector Subspace V in Rm • If dim(V) = n, then V has n basis vectors • a1, a2, …, an • They are independent • V = C(A) where A = [a1, a2, …, an]
Projection onto a Subspace • Input: • Given n independent vectors a1, a2, …, an in Rm • A vector b in Rm… • Desirable Output: • A vector in p in C([a1, a2, …, an]) that is closest to b • The projection p of b in C([a1, a2, …, an]) • A vector p in V such that (b-p) is orthogonal to C([a1, a2, …, an])
Think about this Picture dim r dim r C(A) xr C(AT) p b Rn Rm xn b-p dim n- r N(A) N(AT) dim m- r
Rotation q
Properties of The Rotation Matrix Q is an orthonormal matrix: QT Q = I
Rotation Matrix in High Dimensions Q is an orthonormal matrix: QT Q = I
Rotation Matrix in High Dimensions Q is an orthonormal matrix: QT Q = I
Reflection b u mirror
Reflection b u
Reflection b u mirror
Reflection is Symmetric and Orthonormal b u mirror
Orthonormal Vectors are orthonormal if
Orthonormal Matrices Q is orthonormal if QT Q = I The columns of Q are orthonormal vectors Theorem: For any vectors x and y,
Products of Orthonormal Matrices Theorem: If Q and P are both orthonormal matrices then QP is also an orthonormal matrix. Proof:
Orthonormal Basis and Gram-Schmidt • Input: an m by n matrix A • Desirable output: Q such that • C(A) = C(Q), and • Q is orthonormal
Basic Idea • Suppose A = [a1 … an] • If n = 1, then Q = [a1 /|| a1 ||] • If n = 2, • q1 = a1 /|| a1 || • Start with a2 and subtract its projection along a1 • Normalize
Gram-Schmidt • Suppose A = [a1 … an] • q1 = a1 /|| a1 || • For i = 2 to n What is the complexity? O(mn2)
Theorem: QR-Decomposition • Suppose A = [a1 … an] • There exist an upper triangular matrix R such that • A = QR
Using QR to Find Least Square Approximation Can be solved by back substitution