Ok, I see what you're talking about now.  Yes, as we're talking about 
meshes, the non linear transformations occur on the verticies and 
normals.  I agree that there is indeed no way to apply a non linear 
transformation to edges and faces in conventional meshes as they are 
represented as a relationship between verticies, not as independent 
entities.  When you think about it, even subdividing meshes isn't 
applying a non linear transformation to edges and faces, as what is 
actually happening is a linear transformation plus one or more 
additional edges and faces per original edge and face.  The only way to 
do this would be to change the format of the mesh such that the edges 
were represented as an algorithm rather than a relationship, and then it 
would not be a mesh anymore, more like a collection of Bezier patches.

Peter D.

Warp wrote:
> Peter Duthie <pd_### [at] warlordsofbeercom> wrote:
> 
>>I mean to the mesh.  In that the mesh consists of verticies, edges, 
>>faces, uv_vectors, and normals.  Transforming the verticies will 
>>generally transform the edges and faces with which they are associated 
>>unless my understanding of meshes is fundamentally flawed.
> 
> 
>   We are talking about non-linear transformation here.
> 
>   Moving the vertices around will only apply a linear transformation
> to the edges and faces.
>   Applying a true non-linear transformation to the mesh would bend the
> edges of the triangles (and thus their surfaces). However, no renderer
> I know of can do this (some renderers subdivide the triangles to get
> more "bending", but it's still just an approximation, not a true
> non-linear transformation).
> 
>   Usually a "non-linear" transformation of a mesh is performed by just
> moving the vertices (and normals). The edges and faces keep straight
> regardless (which means that only linear transformations are performed
> to them in practice).
> 
> 
>>I'd be interested in 
>>knowing specifically what problems you can foresee with this sort of 
>>mesh transformation so that I can try to come up with a solution.
> 
> 
>   The problem is that the "non-linear" transformation of a mesh is only
> as good as the size of its triangles.
>   If you have for example a box consisting of two triangles per side,
> twisting the box is basically impossible (without subdividing the
> triangles).
>

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