# 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 §

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}