>>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/)
> 
> 
> Well, so much for polyhedra - you can surely create any polyhedron you
> want with meshes.  
> 

That is a possible way to describe the surface of a "convex polytope".

To attain a solid, it is possible to take an intersection of planes 
(http://www.f-lohmueller.de/pov_tut/shapes/plane1e.htm) plane { n, d } 
where n runs through the "outer normal vectors" of the "facets" (2 dim. 
faces) of the polytope. The values n and d can be evaluated from the 
vertices of a polytope with "polymake" for example.

Nevertheless, I mentioned polytopes only as an example of possible sets 
to be combined via a "(vector) sum operation".

My main idea was to visualize the mathematical defintion of the vector 
sum (also known as Minkowski sum, see 
http://mathworld.wolfram.com/MinkowskiSum.html) of two arbitrary sets.



>>>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)".
> 
> 
> So the vector sum of an arbitrary surface and a sphere at <0,0,0> with
> radius 1 is the same as with any other sphere with the radius 1?  Sounds
> strange.  Furthermore this means that this sum is not commutative.  
> 

The sum of a set with any sphere of the same radius is the same, up to a 
translation. Moreover, the operation is commutative.

Achill

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