1. Lecture 13 Operations 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 Lineb
What matrix P has the property
p = Pb a q p

7. Properties of Projection on to a Lineb 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 xr
C(A)
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 Orthonormalb 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 ||]
Start with a2 and subtract its projection along a1
Normalize

28. Gram-Schmidt Suppose A = [a1 … an] What is the complexity? O(mn2)
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