1. Introduction. It is often said that when the conditions are right, discoveries are made by different people at the same time. The discovery of the non-Euclidean geometry, and the invention of calculus are the most frequently mentioned examples. Here is another example (with an incomparably less significant subject) which I noticed recently. My note [4] described a starshaped polyhedron which...

In this paper we present a new framework, based on subdivision surface fitting, for high rate compression and coding of 3D models. Our algorithm fits the input 3D model, represented by a polygonal mesh, with a piecewise smooth subdivision surface represented by a coarse control polyhedron. Our fitting scheme, particularly suited for meshes issued from mechanical or CAD parts, aims at getting cl...

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-unfold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron...

We present efficient algorithms to construct both C1 and C 2 smooth meshes of cubic and quintic A-patches to approximate a given polyhedron P in three dimensions. The A~patch is a smooth and single-sheeted zero-contour patch of a trivariate polynomial in Bernstein-Bezier (BB) form defined within a tetrahedron. The smooth mesh constructions rely on a novel scheme to build an inner simplicial hul...

To determine the relative position of any two surfaces in a system, one approach is to use operations (Minkowski sum and intersection) on sets of constraints. These constraints are made compliant with half-spaces of n  where each set of half-spaces defines an operand polyhedron. These operands are generally unbounded due to the inclusion of degrees of invariance for surfaces and degrees of fre...

For P ⊂ H3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/mij (mij ≥ 2 an integer), there is a naturally associated Coxeter group ΓP . Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O+(3, 1) = Isom(H3), with fundamental domain the original polyhedron P . In this paper, we provide a procedure...

The 3-D reconstruction of visible polyhedral faces from a pair of general perspective views with the aid of a calibration plate is addressed. A polyhedron is placed on a planar calibration plate and two side views of both the polyhedron and the calibration plate are taken. Through proper arrangements we may assume that in the two views a number of polyhedral edges lying on the calibration plate...

Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to minimally-dependent subsystems of the presentation of the polyhedron. These can be understood as the set of all possible edge directions, including edges that may ari...

It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresp...

Recently, Todd got a new bound on the diameter of a polyhedron using an analysis due to Kalai and Kleitman in 1992. In this short note, we prove that the bound by Todd can further be improved. Although our bound is not valid when the dimension is 1 or 2, it fits better for a high-dimensional polyhedron with a large number of facets. Keyword: Polytopes, Diameter, Kalai and Kleitman bound.