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:

Where

Please note that this matrix has the following form:

Note u1 and u2 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 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:

Note: u is a unit vector perpendicular (normal) to the plane

Add this formula to your notes

Example 3

Note: we are reflecting the vector x about a plane

Solution to example 3Formula:

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)

Answer: invcos(3/5)Or invsin (4/5)

Use the formula:

Rotations combined with Scaling

This is the same as the proof we did in 2.1 but now we don’t requirea2 + b2 = 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 MatrixReflection Matrix

Rotation with ScalingOne directional ScalingMixed ScalingHorizontal ShearVertical 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

5 00 2

5/13 12/13-12/13 5/13

2 0 0 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

Identify the following matrices

An non-symmetricalProjection onto y=x

5 00 2

5/13 12/13-12/13 5/13

rotationCombined scaling

2 0 0 0

ProjectionOnto 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

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