Ear-clipping Based Algorithms of Generating High-quality Polygon Triangulation

A basic and an improved ear clipping based algorithm for triangulating simple polygons and polygons with holes are presented. In the basic version, the ear with smallest interior angle is always selected to be cut in order to create fewer sliver triangles. To reduce sliver triangles in further, a bound of angle is set to determine whether a newly formed triangle has sharp angles, and edge swapping is accepted when the triangle is sharp. To apply the two algorithms on polygons with holes, "Bridge" edges are created to transform a polygon with holes to a degenerate polygon which can be triangulated by the two algorithms. Applications show that the basic algorithm can avoid creating sliver triangles and obtain better triangulations than the traditional ear clipping algorithm, and the improved algorithm can in further reduce sliver triangles effectively. Both of the algorithms run in O(n2) time and O(n) space.Comment: Proceedings of the 2012 International Conference on Information Technology and Software Engineering Lecture Notes in Electrical Engineering Volume 212, 2013, pp 979-98

Polyomorphisms Conjugate to Dilations

Consider polynomial maps f: Cn → Cn and their dilations sf(x) by complex scalars s. That is, maps f whose components fi are polynomials with complex coefficients in the n variables (x1, x2,..., xn) = x ∈ Cn. The question, first raised by O.-H. Keller in 1939 [10] for polynomials over the integers but now also raised for complex polynomials and, as such, known as The Jacobian Conjecture (JC), asks whether a polynomial map f with nonzero constant Jacobian determinant det f ′(x) need be a polyomorphism: I.e., bijective with polynomial inverse. It suffices to prove injectivity because in 1960–62 it was proved, first in dimension 2 by Newman [19] and then in all dimensions by Bia lynicki-Birula and Rosenlicht [4], that, for polynomial maps, surjectivity follows from injectivity; and furthermore, under Keller’s hypothesis, the inverse f−1(x) will be polynomial, at least in the complex case, if the polynomial map is bijective. The group of all polyomorphisms of Cn is denoted GAn(C). It is isomorphic to the group AutC[x] of automorphisms σ of the polynomial ring C[x] by means of the correspondence φ(f) = σ where σ(xi) = fi(x). Polynomial maps f(x) satisfying det f ′(x) = const 6 = 0 are called Keller maps. We can and do assume that f(0) = 0 and f ′(0) = I. Five main problems arise: Problem#1. Classify all Keller maps f: Cn → Cn. Open for n ≥ 2

Degree Aware Triangulation of Annular Regions

Generating constrained triangulation of point sites distributed in the plane is an important problem in computational geometry. We present theoretical and experimental investigation results for generating triangulations for polygons and point sites that address node degree constraints. We characterize point sites that have almost all vertices of odd degree. We present experimental results on the node degree distribution of Delaunay triangulation of point sites generated randomly. Additionally, we present a heuristic algorithm for triangulating a given normal annular region with increased number of even degree vertices