2.2.2 Geometric Transformations with Matrices

Introduction to 3D Transformations

In 3D geometry, transformations using matrices extend the 2D concepts to three dimensions. These transformations include reflections across planes and rotations about coordinate axes. This topic assumes knowledge of 3D vectors, providing a foundation for visualizing and calculating transformations in space.

Key 3D Transformations and Their Matrices

Reflections

A reflection in 3D mirrors points across a specified plane. The transformation matrices for reflections about the three coordinate planes are:

Reflection in the plane $x = 0$ ($yz-plane$):

$$\text{Matrix} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Reflection in the plane $y = 0$ ($xz-plane$):

$$\text{Matrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Reflection in the plane $z = 0$ ($xy-plane$):

$$\text{Matrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$$

Rotations

Rotations in 3D are performed about one of the coordinate axes:

Rotation by angle $\theta$ about the $x-axis$:

$$\text{Matrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \end{pmatrix}$$

Rotation by angle $\theta$ about the $y-axis$:

$$\text{Matrix} = \begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{pmatrix}$$

Rotation by angle $\theta$ about the $z-axis$:

$$\text{Matrix} = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

These matrices rotate vectors counterclockwise when looking along the positive axis direction.

Combined Transformations

When combining multiple transformations, the order of multiplication is critical. The resulting matrix is the product of the matrices of the individual transformations in reverse order:

If transformation $A$ is applied first, followed by $B$, the combined transformation matrix is $B \times A$

Note Summary