"Warp" <war### [at] tagpovrayorg> wrote in message
news:4b770ead@news.povray.org...
> clipka <ano### [at] anonymousorg> wrote:

> > Isn't such a transformation non-affine? So it wouldn't be possible to do
> > it with a simple 4x4 matrix multiplication.

>   Multiplication with a 4x4 matrix can be used for perspective projection.
> It's a 3x3 transformation matrix which is restricted to affine
transformations.
>
>   (Yes, I didn't know that either, many years ago.)

That's not correct. 3x3 matrix multiplication only gives you linear
transformations (in 3D). You need a particularly designed 4x4 matrix for
affine transformations, which includes translation on top of linear
transformations of scaling, rotation and projective ("orthogonal")
transformations. Perspective projection is not strictly achieved with matrix
multiplication applied to standard coordinates, so you include a special
equivalency clause / homogenous coordinates (a division in disguise), and
with that in mind, you can use the remaining elements of the 4x4 matrix to
that end.

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