Why are there extra points in a 4-d or higher convex hull?
You may see extra points if you use options ‘i’ or ‘Ft’ without using triangulated output (‘Qt’). The extra points occur when a facet is non-simplicial (i.e., a facet with more than d vertices). For example, Qhull reports the following for one facet of the convex hull of a hypercube. Option ‘Pd0:0.5’ returns the facet along the positive-x axis: rbox c D4 | qconvex i Pd0:0.5 12 17 13 14 15 17 13 12 14 17 11 13 15 17 14 11 15 17 10 11 14 17 14 12 8 17 12 13 8 17 10 14 8 17 11 10 8 17 13 9 8 17 9 11 8 17 11 9 13 The 4-d hypercube has 16 vertices; so point “17” was added by qconvex. Qhull adds the point in order to report a simplicial decomposition of the facet. The point corresponds to the “centrum” which Qhull computes to test for convexity. Triangulate the output (‘Qt’) to avoid the extra points. Since the hypercube is 4-d, each simplicial facet is a tetrahedron. C:\qhull3.1>rbox c D4 | qconvex i Pd0:0.5 Qt 9 9 13 14 15 12 9 13 14 9 11 13 15 11 9 14 15 9 10 11 14 12 9 14 8 9 12 13 8 9 1
