1. 2.2 Day 2 Reflections and Rotations combined with Scaling The concept of transformations inspired art by M.C. Escher

2. Reflections Consider a line L through the origin. We saw yesterday that and vector in R2 can be written as the sum of components perpendicular and parallel that line If we consider the parallel component minus two times the perpendicular component, The result if a resultant vector that is the a reflection of the original vector over line L You will need this formula in your notes

3. Problem 7

4. Solution to problem 7 Formula

5. Reflections over a vector (line) The matrix of transformation is given by the formula: Where Please note that this matrix has the following form: Note u 1 and u 2 are components of a unit vector pointing in the direction of line of reflection. (will prove as next problem) Note: this only works for vectors in R 2 while other formula works for in R n

6. Problem 13

7. Solution to Problem 13

8. Reflections Find the matrix of projection through

9. Use the matrix Find the matrix of reflection over

10. For reflections in 3 D space Reflecting a Vector over a plane Formula for reflection over a plane: Note: u is a unit vector perpendicular (normal) to the plane Add this formula to your notes

11. Example 3 Note: we are reflecting the vector x about a plane

12. Solution to example 3 Formula:

13. Recall: Rotations Note: We proved this in 2.1 The matrix of counterclockwise rotation in real 2 dimensional space through angle theta is Note this is a matrix of the form

14. The matrix below represents a rotation. Find the angle of rotation (in radians)

15. Answer: invcos(3/5) Or invsin (4/5) Use the formula:

16. Rotations combined with Scaling This is the same as the proof we did in 2.1 but now we don’t require a 2 + b 2 = 1 Why does removing this requirement result in a rotation plus a scaling?

17. What matrices should we have in our library of basic matrices? Identity Matrix Projection Matrices Projection onto x-axis Projection onto y-axis Rotation Matrix Reflection Matrix Rotation with Scaling One directional Scaling Mixed Scaling Horizontal Shear Vertical Shear

18. How do you identify an unknown matrix? 1)Check your library of basic matrices. 2) Use your knowledge of matrix multiplication. 3) Plug in values. To be efficient use the elementary matrices.

19. Identify the following matrices 50 0 2 5/13 12/13 -12/13 5/13 20 0 3/5 4/5 4/5 -3/5 1 0 2 1 1 -2 2 1 25/169 60/169 60/169 144/169

20. Identify the following matrices An non-symmetrical Projection onto y=x 50 0 2 5/13 12/13 -12/13 5/13 rotation Combined scaling 20 0 Projection Onto x-axis with scaling 3/5 4/5 4/5 -3/5 reflection 1 0 2 1 Vertical shear 1 -2 2 1 Rotation with scaling 25/169 60/169 60/169 144/169 projection

21. Homework P. 65 7,11, 25,26 (d, e only) 27,28,32,34,37,38,39,

22. Rotations in R 3 For more information on rotations visit: http://www.songho.ca/opengl/gl_projectionmatrix.html