On Mon, 13 Jul 1998 21:48:45 +0200, Thorsten Froehlich 
<Tho### [at] csicom> wrote:
>Ron Parker wrote:
>>Right, matrices won't work.  By definition, anything that can be represented
>>as a matrix is linear.  But Fran wasn't talking about what mathematicians 
>>traditionally call a matrix; he was talking about a notational system that
>>superficially resembles the one POV now uses for linear transformations.
>
>I am not sure I fully understand what you are refering to English isn't my native
language), can you expail this a little bit more, please?

I'm not sure how to explain this.  I just meant that he wasn't talking about
what you and I might call a matrix.  He was only talking about a syntactic
way to express an array of formulas.

>To keep it simple (and possibly not covering all cases) a quadartic function (for
each component of the vector) would be enough to make an egg out of a sphere.

A perspective transformation is sufficient to transform a sphere into an
egg.  Unfortunately, a perspective transformation can also turn a sphere into 
a weird barbell-shaped thing with a singularity in the middle.  In general, 
it might not be possible to transform the ray into the space of the object 
when using perspective transformations because working with rays passing 
through the singularity would require dividing by zero.  Also the concept of 
"inside" is seriously messed up by the presence of singularities.

>Ron Parker wrote:
>>The problem of general nonlinear transformations would likely only be solved 
>>by converting the object to a mesh and then using the forward transform on 
>>its constituent elements, but unless the inverse transform is easily 
>>calculable, I don't see how you could transform the texture as well.  
>
>Splines could be used so solve this problem as well, couldn't they?

Of course.  Anything that can still be tested for intersections after 
undergoing an arbitrary transformation is fine.  The idea is to eliminate
the need for an inverse transformation.

>Yes, to generate meshes out of objects is an important feature missing. But I think
it is no (too big) problem to determine the surface of any polynomial defined object,
however I have never tried so I cannot claim this to be right.

I, too, have no experience with trying to make meshes from higher-order
polynomial surfaces.  They may indeed be easier than I think.  But they can
also be infinite, and depending on what you plan to use the triangles for,
that can be an insurmountable obstacle.

>But having a few million triangles and all their energies, etc. to calculate the
radiosity with a itterative approach (not to talk about a full matrix inversion) will
still not be easier or faster. And inorder to use raytracing with it you would have to
re-apply the light (energy) data to the objects as something like a texture. Or you
would only be able to render with radiosity and meshes. Not to talk about the
reflection problem (e.g. mirrors) which is still (???) there with radiosity based on
meshes.

True, we might not want to try to implement radiosity this way in POV, but 
it might be nice to use some of the great tools available for POV and the 
expressiveness of the POV language to generate data that could be fed into 
other renderers, of which Radiance, OpenGL, and VRML are but obvious 
examples.  

Clearly, there'd have to be extra parameters to control the mesh-generation 
process, so you don't get a few million triangles unless you want or need 
them.

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