>>Mathematically, the (vector) sum A+B of two sets A and B is defined as
>>the set of all points x=a+b with a in A and b in B.
>>A+B = { x : x=a+b, a in A and b in B }.
> 
> 
> I have no idea what you are talking about but we are not dealing with sets
> of points but with surfaces.  Your definition of a vector sum does not
> seem to be applicable for surfaces.  

The defintion is applyable to arbitrary pointsets --- also too surfaces.
Besides, I thought povray is using solids also.



> 
>>A "simple" example is the sum of two convex polyhedra (cube,
>>tetrahedron, octahedron, etc.) which is again a polyhedron.
> 
> That's not an example.  

Well, ok.
A concrete example would be the sum of the tetrahedron T with vertices
<0,0,0>, <1,0,0>, <0,1,0> and <0,0,1> with "its negative" -T with 
vertices <0,0,0>, <-1,0,0>, <0,-1,0> and <0,0,-1>.
The outcome is a polyhedron (polytope) with vertices
<-1, 0, 0>, <0, -1, 0>, <0, 0, -1>, <1, 0, 0>, <1, -1, 0>, <1, 0, -1>, 
<0, 1, 0>, <-1, 1, 0>, <0, 1, -1>, <0, 0, 1>, <-1, 0, 1>, <0, -1, 1>.

Generally, such vector sums of polytopes can be evaluated for example 
with the program "polymake" (www.math.tu-berlin.de/polymake/)


> Let's say we have a unit cube between <0,0,0> and <1,1,1> and a sphere at
> <0,0,0> with radius 1.  What is the 'vector sum' of these objects?

As far as I understand the macro
"Round_Box_Union(PtA, PtB, EdgeRadius)",
it would be "Round_Box_Union(<0,0,0>, <1,1,1>, 1)".


Achill

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