Transformation of 3D Shapes

Transformation of 3D Shapes
An Introduction

Computer Science Department

bernstdh@jmu.edu

Motivation

What We Have:
- Methods for representing lines, triangles, and convex polygons in 2D and 3D
- The ability to transform 2D shapes
What We Need:
- The ability to transform 3D shapes in various ways

Coordinate Systems

In 2D:
- We considered scaling, reflection, shear, rotation and translation in both Cartesian and homogeneous coordinates
In 3D:
- We could consider both Cartesian and homogeneous coordinates but we will only consider homogeneous coordinates

Remember

This discussion assumes that we are using column vectors
If you implement using row vectors then you will need to reverse the order of the operations

Visualization

As we have seen, it can be useful to visualize these kinds of operations
Unfortunately, we can't draw figures in three dimensions (we have to project onto two)
This can cause misconceptions so it is better to use physical models and not drawings

Translation

Translation Matrix:
- \( \bs{T} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & \tau_{1} \\ 0 & 1 & 0 & \tau_{2} \\ 0 & 0 & 1 & \tau_{3} \\ 0 & 0 & 0 & 1 \end{array} \right] \)
Inverse Translation Matrix:
- \( \bs{T}^{-1} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & -\tau_{1} \\ 0 & 1 & 0 & -\tau_{2} \\ 0 & 0 & 1 & -\tau_{3} \\ 0 & 0 & 0 & 1 \end{array} \right] \)

Scaling

Scaling Matrix:
- \( \bs{S} = \left[ \begin{array}{c c c c} \sigma_{1} & 0 & 0 & 0 \\ 0 & \sigma_{2} & 0 & 0 \\ 0 & 0 & \sigma_{3} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \)
Inverse Scaling Matrix:
- \( \bs{S}^{-1} = \left[ \begin{array}{c c c c} 1/\sigma_{1} & 0 & 0 & 0 \\ 0 & 1/\sigma_{2} & 0 & 0 \\ 0 & 0 & 1/\sigma_{3} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \)

Rotation Around Axes

Rotation Around the \(x\)-Axis:
- \( \bs{R_{x}} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & \cos \phi & - \sin \phi & 0 \\ 0 & \sin \phi & \cos \phi & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \)
Rotation Around the \(y\)-Axis:
- \( \bs{R_{y}} = \left[ \begin{array}{c c c c} \cos \theta & 0 & \sin \theta & 0 \\ 0 & 1 & 0 & 0 \\ - \sin \theta & 0 & \cos \theta & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \)
Rotation Around the \(z\)-Axis:
- \( \bs{R_{z}} = \left[ \begin{array}{c c c c} \cos \psi & - \sin \psi & 0 & 0 \\ \sin \psi & \cos \psi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array} \right] \)

Rotation Around Axes (cont.)

Inverse Rotation Matrices:
- Just rotate an equal amount in the opposite direction (e.g.,. replace \(\theta\) with \(- \theta\))
An Additional Result:
- All of the \(\cos\) terms are on the diagonal and all of the \(\sin\) terms are off the diagonal
- \(\cos(-\theta) = \cos \theta\)
- \(\sin(-\theta) = - \sin \theta\)
- So, \(\bs{R}^{-1} = \bs{R}^{T}\)

Shear

Defined:
- A shear transforms the point \(\left[ \begin{array}{c c c c}x \\ y \\ z \\ 1\end{array} \right]\)
- to the point \(\left[ \begin{array}{c c c c}x+ay+bz \\ cx+y+dz \\ ex+fy+z \\ 1\end{array} \right]\)
Shear Matrix:
- \( \left[ \begin{array}{c c c c} 1 & a & b & 0 \\ c & 1 & d & 0 \\ e & f & 1 & 0 \\ 0 & 0 & 0 & 1\end{array} \right] \)

Concatenating Transforms

Sequential Transformations:
Being Careful About Dimensionality:
- \(\bs{q} = \bs{A} \bs{p}\)
- \(\bs{r} = \bs{B} \bs{q} = \bs{B} (\bs{A} \bs{p})\)
- \(\bs{s} = \bs{C} \bs{r} = \bs{C} (\bs{B} \bs{q}) = \bs{C} (\bs{B} (\bs{A} \bs{p}))\)

Concatenating Transforms (cont.)

Associativity of Matrix Multiplication:
- \(\bs{C} (\bs{B} (\bs{A} \bs{p})) = ((\bs{C} \bs{B}) \bs{A}) \bs{p}\)
The Concatenated Transform:

Complicated Rotations

Around the point \(\bs{p}\):
- Translate by \(-\bs{p}\)
- Rotate
- Translate by \(\bs{p}\)
Rotation Around Multiple Axes:
- \(\bs{R} = \bs{R_{x}} \bs{R_{y}} \bs{R_{z}}\)
Rotation Around a Line:
- Combine the two above ideas

There's Always More to Learn