2.2 Day 2 Reflections and
Rotations combined with Scaling
The concept of transformations inspired art by M.C. Escher
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
Problem 7
Solution to problem 7
Formula
Reflections over a vector (line)
The matrix of transformation is given by the
formula:
Note u1 and u2 are components of a unit vector pointing
in the direction of line of reflection. (will prove as next
problem)
Please note that this matrix has the following form:
Where
Note: this only works for vectors in R2 while
other formula works for in Rn
Problem 13
Solution to Problem 13
Reflections
Find the matrix of projection through
Use the matrix
Find the matrix of reflection over
For reflections in 3 D space
Reflecting a Vector over a plane
Formula for reflection over a plane:
Add this formula to your notes
Note: u is a unit vector perpendicular (normal) to the plane
Example 3
Note: we are reflecting the vector x about a plane
Formula:
Solution to example 3
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
The matrix below represents a
rotation. Find the angle of rotation
(in radians)
The matrix below represents a
rotation. Find the angle of rotation
(in radians)
Use the formula:
Answer: invcos(3/5)
Or invsin (4/5)
Rotations combined with Scaling
This is the same as the proof we did in 2.1 but now we don’t require
a2 + b 2 = 1
Why does removing this requirement
result in a rotation plus a scaling?
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
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.
Identify the following matrices
1
0
2
1
5
0
25/169 60/169
0
2
60/169 144/169
2
0
3/5
4/5
1
-2
0
0
4/5
-3/5
2
1
5/13
-12/13
12/13
5/13
Identify the following matrices
1
0
2
1
5
0
25/169 60/169
0
2
60/169 144/169
Vertical shear Combined
scaling
0
3/5
4/5
1
-2
0
0
4/5
-3/5
2
1
reflection
12/13
5/13
rotation
projection
2
Projection
Onto x-axis
with scaling
5/13
-12/13
An non-symmetrical
Projection onto y=x
Rotation with scaling
Homework
P. 65 7,11, 25,26 (d, e only) 27,28,32,34,37,38,39,
Rotations in R3
For more information on rotations visit:
http://www.songho.ca/opengl/gl_projectionmatrix.html