Matrix Transformation
The matrix transformation for a matrix is defined by the function Note:
- For , we have that
- Notice how the matrix is , but the function is a mapping from to , because think how Matrix multiplication works
Example
Let . Then and . We can compute the matrix transformation to vector by the following:
Linear Transformation are usually written in Matrix form, hence doing Matrix Transformation.
Some Transformations
- Euclidean Transformation using a matrix from the Special Euclidean Group
- Similarity Transformation
- Affine Transformation
- Perspective Transformation
Some transformations are cool because they maintain some property of the vectors that are applied on them. Rotation is a transformation. Translation can be written as a matrix transformation (use the Euclidean Transformation with identity matrix for rotation)!
\hline
\text{Transform Name} & \text{Matrix Form} & \text{Degrees of Freedom} & \text{Invariance} \\
\hline
\text{Euclidean} \rule{0pt}{20 pt} & \left[ {\begin{array}{*{20}{c}}
\mathbf{R} & \mathbf{t}\\
{{\mathbf{0}^T}}&1
\end{array}} \right] & 6 & \text{Length, angle, volume} \\
\text{Similarity} \rule{0pt}{20 pt}& \left[ {\begin{array}{*{20}{c}}
{s \mathbf{R}}& \mathbf{t}\\
{{ \mathbf{0}^T}}&1
\end{array}} \right] & 7 & \text{volume ratio} \\
\text{Affine} \rule{0pt}{20 pt}& \left[ {\begin{array}{*{20}{c}}
\mathbf{A} & \mathbf{t}\\
{{\mathbf{0}^T}} & 1
\end{array}} \right] & 12 & \text{Parallelism, volume ratio} \\
\text{Perspective} \rule{0pt}{20 pt} & \left[ {\begin{array}{*{20}{c}}
\mathbf{A} & \mathbf{t}\\
{{\mathbf{a}^T}} & v
\end{array}} \right] & 15 & \text{Plane intersection and tangency} \rule{0pt}{20 pt}\\
\hline
\end{array}$$