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Constructive Solid Geometry is a way of describing/building up solid objects from simpler primitive objects. Let's assume you can perform affine transformations on objects, along with the CSG set operations. Has there been any investigation into what solids you can exactly reproduce with a finite number of CSG operations, and a given set of primitives? Where would I look to read about it?

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Well, if the primitives are, say, quadratic surfaces, then the combinations will be piecewise quadratic. Similarly, if the primitives are linear surfaces (i.e. planes) then the combinations will be piecewise linear (i.e. polyhedra).

More generally, the surfaces of objects formed by a finite number of CSG operations will consist of a finite number of pieces of the primitive surfaces, with the edge of each piece consisting of a finite number pieces of pairwise intersections of the primitive surfaces.

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...and if you allow (rational) Béziers as primitives, I suppose one can do pretty much any nice curved shape with this CSG business... –  J. M. Aug 30 '11 at 22:19

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