On decomposition of embedded prismatoids in R3R^3 without additional points

This paper considers three-dimensional prismatoids which can be embedded in ℝ³ A subclass of this family are twisted prisms, which includes the family of non-triangulable Scönhardt polyhedra [12, 10]. We call a prismatoid decomposable if it can be cut into two smaller prismatoids (which have smaller volumes) without using additional points. Otherwise it is indecomposable. The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa. In this paper we prove two basic facts about the decomposability of embedded prismatoid in ℝ³ with convex bases. Let P be such a prismatoid, call an edge interior edge of P if its both endpoints are vertices of P and its interior lies inside P. Our first result is a condition to characterise indecomposable twisted prisms. It states that a twisted prism is indecomposable without additional points if and only if it allows no interior edge. Our second result shows that any embedded prismatoid in ℝ³ with convex base polygons can be decomposed into the union of two sets (one of them may be empty): a set of tetrahedra and a set of indecomposable twisted prisms, such that all elements in these two sets have disjoint interiors

On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph

This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega be a height function which lifts the vertices of A into R3. Every flip in triangulations of A can be associated with a direction. We first established a relatively obvious relation between monotone sequences of directed flips between triangulations of A and triangulations of the lifted point set of A in R3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case if this result in 3d has been shown by B.Joe in 1989. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra. We proved sufficient conditions for the termination of this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure

On monotone sequences of directed flips, triangulations of polyhedra, and structural properties of a directed flip graph

This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm  [21, 22]. Let A be a finite point set in R^2 and ω : A → R be a height function which lifts the vertices of A into R^3. Every flip in triangulations of A can be assigned a direction [6, Definition 6.1.1]. A sequence of directed flips is monotone if all its flips follow the same direction. We first established a relatively obvious relation between monotone sequences of directed flips on triangulations of A and triangulations of the lifted point set A^ω in R^3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case of this result in 3d has been shown in [19]. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra without using additional vertices. We prove sufficient conditions for the termination of this algorithm, and show it runs in O(n^3) time, where $n$ is the number of input vertices