>> Firstly, has anyone looked into it? (4D geometry in POV-Ray that is.)
> 
> I have done 4D-geometry visualization long before raytracing. That was
> wireframe for red/green glasses. I haven't done it in povray.

Likewise. (I wrote a Java thingy... but Java is just painful!)

>> Anyway, I'd like to use POV-Ray to draw animations of a hypercube 
>> rotating in 4 dimensions, orthographically projected into 3 dimensions.
> 
> 
> I always prefered perspective projection.

Yeah, I know what you mean... Only trouble is working out which effects 
are due to the 4D->3D perspective, and which ones are the 3D->2D 
perspective! ;-)

> Anyway, you seem to want to use a solid 3D object. You will lose a lot of
> the 4D information that way. The 4D object has vertices, edges, faces,
> and hyperfaces as "visual" elements of its geometry. By your method you
> lose all vertices, edges, and faces that are projected to the inside of
> the 3D model. You also lose all hyperfaces alltogether.

Yeah, true.

> For povray I could think of several alternatives:
> 
> 1. Do a wireframe model using spheres and cylinders -- or one large
>    spheresweep (after all the graph of the hypercube does have a Eulerian
>    cycle). Vertcies and edges will be clearly visible, faces and hyperfaces
>    sort of.

I was planning to do this too. But working out where the hell the edges 
are seems like a harder problem then just telling POV-Ray to take the 
intersection of some simple planes.

> 2. Show the faces as semitransparent polygons. Possibly of different 
> colours.
>    Vertices, edges, and faces are visible, hyperfaces sort of. The faces 
> will
>    maybe be too many to not be confusing.

This could work - I think anyway! :-S

> 3. Show the hyperfaces by differently coloured media.

Ooo... hadn't thought of that!

> Or, maybe, use a combination.
> 
>> Question: is the intersection of the 3D projections of the 8 
>> hyperplanes equal to the 3D projection of the intersection of the 8 
>> hyperplanes?
> 
> No. and that is not a feature unique to 4D->3D. An example in 2D-1D:
> Consider the normal vectors (1,1) and (1,-1) and work it out. Another
> example: Take the sets {(0,0),(1,1)} and {(0,1),(1,0)}. Projection, then
> intersection yields {0,1}, while the other way around you obtain the empty
> set. The first collects all 1D-points that can be extended to some point
> in each of the original sets. The second collects all 1D-point that have
> one extension that pertains to both sets simultaneously.

Right...

So figuring out how to project the 8 hyperplanes into 3D and then asking 
POV-Ray for the intersection will get me nowhere fast. (I was already 
beginning to suspect this conclusion, which is why I asked! ;-)

Maybe I'll have a go at the wireframe bit for a while...

Thanks.
Andrew @ home.

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