es

This section is divided into a number of subsections, links to which are:

Rotations

2D Rotations

3D Rotations

Euler Theorem

Quaternions

 

This section shows how to take a sequence of transformation matrices and combine (or concatenate or catenate) them into one single transformation matrix. This new matrix represents the cumulative result of applying all of the original transformations in order. It's actually quite easy. The transformation that results from applying the transformation with matrix A followed by the transformation with matrix B has matrix B A. That is, matrix multiplication is how we compose transformations represented as matrices.

Compositions

Reflections are the elements of the orthogonal group O(n) whose canonical form is

\[ \begin{bmatrix} -1 & {\bf 0} \\ \phantom{-}0 & {\bf I} \end{bmatrix} , \]
where I is the (n − 1) × (n − 1) identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.

2D Compositions

In dimension two, every rotation can be decomposed into a product of two reflections. More precisely, a rotation of angle θ is the product of two reflections whose axes form an angle of θ / 2.

For rotations and reflections, we have the following relations:

\begin{align*} \mbox{Rot}(\theta )\,\mbox{Rot}(\phi ) &= \mbox{Rot}(\theta + \phi ) = \mbox{Rot}(\phi + \theta ) = \mbox{Rot}(\phi )\,\mbox{Rot}(\theta ) , \\ \mbox{Ref}(\theta )\,\mbox{Ref}(\phi ) &= \mbox{Rot}(2\theta - 2\phi ), \\ \mbox{Rot}(\theta )\,\mbox{Ref}(\phi ) &= \mbox{Ref}\left( \phi + \frac{1}{2}\,\theta \right) , \\ \mbox{Ref}(\theta )\,\mbox{Rot}(\phi ) &= \mbox{Ref}\left( \theta - \frac{1}{2}\,\phi \right) . \end{align*}

3D Compositions



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