Storing the subdivision of a polyhedral surface

On the refinements of a polyhedral subdivision

Let π:P→Q be an affine projection map between two polytopesP and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by π (or π-induced) and the fiber polytope of the projection: a polytope Σ(P,π) of dimension dim(P )−dim(Q) whose faces are in correspondence with the coherent π-induced subdivisions (or π-coherent subdivisions). In this paper we invest...

On the Reenements of a Polyhedral Subdivision

Let : P ! Q be an aane projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by (or-induced) and the ber polytope of the projection: a polytope (P;) of dimension dim(P) ? dim(Q) whose faces are in correspondence with the coherent-induced subdivisions (or-coherent subdivisions). In this paper we investigate the ...

Vertex-Rounding a Three-Dimensional Polyhedral Subdivision

Polyhedral Subdivision for Free-Form Algebraic Surfaces

We present a robust algorithm to construct an “inner” simplicial hull Σ as a single step of subdivision of an input polyhedron P in three dimensional space. Similar to traditional subdivision schemes P becomes the ‘control net’ for free-form modeling while an inner surface triangulation T of Σ is a second level mesh. Free-Form C1 cubic A-patches and C2 quintic A-patches can then be constructed ...

Reversing a Polyhedral Surface

We introduce a new variety of flexatube, a rhomboflexatube. It is obtained from a cardboard rhombohedron by removing a pair of opposite faces (rhombi), and then subdividing the remaining four faces by pairs of diagonals. It is reversible, that is, it can be turned inside out by a series of folds, using edges and diagonals of the rhombi. To turn a rhomboflexatube inside out is quite a challengin...