[Gnash-dev] canonical form for affine transformations

[Eric Hughes]

Sandro's question about matrices last week, and a note I saw about possibly 
implementing "skew" (really called shear) in the commit log, led me to find 
some relevant reference information.

The transformation that's implemented in matrix.cpp:
http://en.wikipedia.org/wiki/Affine_transformation

Details about what a matrix representation of the above looks like:
http://en.wikipedia.org/wiki/Transformation_matrix

As for a canonical form, I'd recommend
        rotate * x-shear * scale
which in matrix form is, in LaTeX form:
\[
\left[\begin{matrix}
        \cos \theta     &   - \sin \theta \cr
        \sin \theta     &   \cos \theta
\end{matrix}\right]
\left[\begin{matrix}
        1       &   k       \cr
        0       &   1
\end{matrix}\right]
\left[\begin{matrix}
        s_x             &   0               \cr
        0               &   s_y
\end{matrix}\right]
\]

In the above, $\theta$ is the angle of rotation, $k$ is the shear value, 
and $s_x$ and $s_y$ are the scale values.

The reason for this choice is that, since these transformations are 
naturally applied to type, with its natural baseline along the x-direction, 
that x-shear is the natural cognitive choice.  Putting it after (to the 
left of) the scaling means that the angle of shear will be independent of 
the size of the scale factors; otherwise the shear would also be scaled.

Eric