In a recent work, the combinatorial interpretation of the polynomial α(n; k1, k2, . . . , kn) counting the number of Monotone Triangles with bottom row k1 < k2 < · · · < kn was extended to weakly decreasing sequences k1 ≥ k2 ≥ · · · ≥ kn. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Genera...

Screen space rendering statistics were gathered from 150 3D objects, each modeled by between 2K and 40K triangles. While there is wide variance by individual object, the overall trend is that the distribution of triangles by screen size is roughly exponential in the direction of small triangles. From a subjective esthetics point of view, tessellations required 10K visible triangles per quarter ...

Proposed in this paper is an efficient algorithm to remove self-intersections from the raw offset triangular mesh. The resulting regular mesh can be used in shape inflation, tool path generation, and process planning to name a few. Objective is to find the valid region set of triangles defining the outer boundary of the offset volume from the raw offset triangular mesh. Starting with a seed tri...

We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides \shape" and \size" guarantees: All triangles have a bounded aspect ratio. The number of triangles is within a constant factor of optimal. Such \quality" triangulations are desirable as meshes for the nite element method, in which the running time generally increases with the number of triang...

In grade school, students learn a standard set of Euclidean triangles. Among this set, the usual 45–45–90 and 30–60–90 triangles are the only right triangles with rational angles and side lengths each containing at most one square root. Are there any other such right triangles? We answer this question completely and present an elegant complement called Ailles’ rectangle that deserves to be in e...

Abstract. Monotone triangles are certain triangular arrays of integers, which correspond to n × n alternating sign matrices when prescribing (1, 2, . . . , n) as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved mo...