In this paper we consider geodesic triangulations of the surface of the regular dodecahedron. We are especially interested in triangulations with angles not larger than π/2, with as few triangles as possible. The obvious triangulation obtained by taking the centres of all faces consists of 20 acute triangles. We show that there exists a geodesic triangulation with only 10 non-obtuse triangles, ...

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...

Let be a set of points in the plane and consider a family of (nondegenerate) pairwise congruent triangles whose vertices belong to . While the number of such triangles can grow superlinearly in — as it happens in lattice sections of the integer grid — it has been conjectured by Brass that the number of pairwise congruent empty triangles is only at most linear. We disprove this conjecture by con...

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...