[cairo] [patch] enable projective transformations

[Andrea Canciani]

On Tue, Aug 17, 2010 at 1:24 PM, Arjen Nienhuis <a.g.nienhuis at gmail.com> wrote:
> On Tue, Aug 17, 2010 at 12:56 PM, Andrea Canciani <ranma42 at gmail.com> wrote:
>> On Tue, Aug 17, 2010 at 12:52 PM, Arjen Nienhuis <a.g.nienhuis at gmail.com> wrote:
>>>> To use it for 3D transformations you would need 4x3 (and 4x4 if you
>>>> want projections, too).
>>>> The question was: what would 3D transformations be used for?
>>>> (Remember: cairo uses 2D surfaces, thus the third dimension input
>>>> would be constant and its output would get discarded "soon")
>>>>
>>>
>>> Use case: 2 windows with buttons that 'stick out'.
>>>
>>> def draw_window(ctx, w):
>>>  draw_background(ctx, w.background)
>>>  ctx.save()
>>>  ctx.translate_z(-10)
>>>  draw_buttons(ctx, w.buttons)
>>>  ctx.restore()
>>>
>>> def main():
>>>  draw_window(ctx, w1)
>>>  ctx.rotate_y(0.5)
>>>  ctx.translate(300, 0)
>>>  draw_window(ctx, w2)
>>>
>>> I think this needs a 4x3 matrix. You need the value of the rotation
>>> around the y axis to make translate_z have the right effect.
>> Are you saying that if you want to use 4x3 transforms (rotate_y), you
>> need 4x3 transforms?
>>
>
> I'm saying:
>
> I have this use case. I think it needs rotate_y and translate_z. I
> think it needs 4x3 transforms.
Oh, if this is the case, then I can correct you easily.
Your use case doesn't need 3D transforms, you can obtain the same
effect with a 2D transforms:
let
A = [ [ a b c d ] [ e f g h ] [ i j k l ] [ m n o p ] ]
be your 4x4 transform (if you just want 4x3, you will have d==h==l==0 && p==1)
when you transform a point P [x y z w] you get:
P' = P * A = [ (ax+by+cz+dw), (ex+fy+gz+hw), (ix+jy+kz+lw), (mx+ny+oz+pw) ]
but if P is constrained to be in 2D (i.e. z==0) you get:
P' = P*A = [ (ax+by+dw), (ex+fy+hw), (ix+jy+lw), (mx+ny+pw) ]
and (assuming you won't use the output z, you will just get the same
as P*B, with
B = [ [ a b d ] [ e f g ] [ m n p ]]
Notice that you are able to use projective transforms (since w is not
necessarily ==1).
Even more interesting, you can do concatenate all your 3D
transformations (4x4), throw away the z row and column and get a 3x3
matrix that does exactly what you wanted.
NB: concatenating, then throwing away the elements is *NOT* the same
as throwing away those elements, then concatenating.
>
> Do you have better way to deal with this use case? Do you have
> different use cases? Is my use case useful/clear/typical?
The only use cases I can see for 3D transformation matrices involve 3D
sources/destinations (for example a path whose points are not (x,y)
but (x,y,z)), but I think this is beyond cairo scope.