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


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}$$