Lecture 13 Operations in Graphics and Geometric Modeling I: Projection, Rotation, and Reflection

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

2. Projection • Projection onto an axis (a,b) x axis is a vector subspace

3. Projection onto an Arbitrary Line Passing through 0 (a,b)

4. Projection on to a Plane

5. Projection on to a Line b a q p

6. Projection Matrix: on to a Line b What matrix P has the property p = Pb a q p

7. Properties of Projection on to a Line b a q p p is the points in Span(a) that is the closest to b

8. 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

9. 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]

10. 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])

11. Using Orthogonal Condition

12. 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

13. Connection to Least Square Approximation

14. Rotation q

15. Properties of The Rotation Matrix

16. Properties of The Rotation Matrix Q is an orthonormal matrix: QT Q = I

17. Rotation Matrix in High Dimensions Q is an orthonormal matrix: QT Q = I

18. Rotation Matrix in High Dimensions Q is an orthonormal matrix: QT Q = I

19. Reflection b u mirror

20. Reflection b u

21. Reflection b u mirror

22. Reflection is Symmetric and Orthonormal b u mirror

23. Orthonormal Vectors are orthonormal if

24. Orthonormal Matrices Q is orthonormal if QT Q = I The columns of Q are orthonormal vectors Theorem: For any vectors x and y,

25. Products of Orthonormal Matrices Theorem: If Q and P are both orthonormal matrices then QP is also an orthonormal matrix. Proof:

26. 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

27. 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

28. Gram-Schmidt • Suppose A = [a1 … an] • q1 = a1 /|| a1 || • For i = 2 to n What is the complexity? O(mn2)

29. Theorem: QR-Decomposition • Suppose A = [a1 … an] • There exist an upper triangular matrix R such that • A = QR

30. Using QR to Find Least Square Approximation Can be solved by back substitution