Duoprism (EntityClass, 13)
From Hi.gher. Space
A duoprism is the Cartesian product of two polygons. In other words, it is the set of all combinations of points (w,x,y,z) where (w,x) is in the first polygon, and (y,z) is in the second.
Nomenclature
A duoprism made from the Cartesian product of an m-polygon and an n-polygon is referred to as an m,n-duoprism or an m-gonal n-gonal duoprism (George Olshevsky).
Geometry
An m,n-duoprism is bounded by m n-gonal prisms and n m-gonal prisms, for a total of m*n cells. The two types of cells lie on two mutually orthogonal planes. Cells within a ring are connected to each other via their m-gonal (or n-gonal) faces. Cells from different rings are joined to each other via their square faces.
The duoprism is convex if the two source polygons are convex.
The two polygons are usually assumed to be regular. If the edge lengths of the two polygons are equal, the resulting duoprism will be a uniform polychoron. (Note that the regularity of the polygons by itself is not sufficient to make the duoprism uniform.)
Limiting shapes
As m approaches infinity, an m,n-duoprism approaches an n-prismic cylinder (i. e., the Cartesian product of an n-polygon with a circle). The 4-prismic cylinder is equal to the cubinder.
If n is also allowed to diverge to infinity, the limiting shape is the duocylinder (i. e., the Cartesian product of two circles).
Due to this property, duoprisms serve as good non-quadric approximations of the prismic cylinders and the duocylinder. This is useful for implementing programs for rendering these objects, because it is much easier to implement a renderer based on polygons (using a large but finite value for m and/or n) than one based on 4D quadric surfaces.
