Matrices and Linear Transformations - 3D Math Primer for Graphics and Game Development - page 159

Areas/
volumes
preserved

Angles
preserved

Rigid
body

Lengths
preserved Transform Linear Affine Invertible Orthogonal Determinant

Linear
transformations Y Y

Affine
transformations

Y = 0

Invertible
transformations Y

Angle-preserving
transformations Y Y Y

Orthogonal
transformations Y Y Y ±1

Rigid body
transformations

Y Y Y Y Y Y Y 1

Translation Y Y Y Y Y Y Y 1

Rotation 1

Y Y Y Y Y Y Y Y 1

Uniform scale 2 k n 3 Y Y Y Y

Non-uniform
scale Y Y Y

Orthographic
projection 4 Y Y 0

Reflection 5 Y 6 Y Y Y Y Y −1

Y 7 Shearing Y Y Y 1

1 About the origin in 2D or an axis passing through the origin in 3D.
2 About the origin in 2D or an axis passing through the origin in 3D.
3 The determinant is the square of the scale factor in 2D, and the cube of the scale factor in 3D.
4 Onto a line (2D) or plane (3D) that passes through the origin.
5 About a line (2D) or plane (3D) that passes through the origin.
6 Not considering “negative” area or volume.
7 Surprisingly!

Table 5.1. Types of transformations

5.8 Exercises

(Answers on page 763.)
1. Does the matrix below express a linear transformation? A ne?
2
4

3
5

34 1.7 π

√

2 0 18

4 −9 −1.3

2. Construct a matrix to rotate −22 o about the x-axis.
3. Construct a matrix to rotate 30 o about the y-axis.
4. Construct a matrix to rotate −15 o about the axis [0.267,−0.535, 0.802].
5. Construct a matrix that doubles the height, width, and length of an object
in 3D.