In this paper, we expand circular arc triangulations with parabolic arcs, which are triangulations whose edges are circular or parabolic arcs. Creating triangulations this way bears several advantages if angles are to be optimized, especially for finite element method. A typical reason for the occurrence of small angles in a straight line triangulation is that the geometry of the vertex set for...

We examine the eigenstructure of generalized isosceles triangles and explore the possibilities of analytic solutions to the general eigenvalue problem in other triangles. Starting with work based off of Brian McCartin’s paper on equilateral triangles, we first explore the existence of analytic solutions within the space of all isosceles triangles. We find that this method only leads to consiste...

A thermodynamically stable vortex-antivortex pattern has been revealed in equilateral mesoscopic type I superconducting triangles, contrary to type II superconductors where similar patterns are unstable. The stable vortex-antivortex "molecule" appears due to the interplay between two factors: a repulsive vortex-antivortex interaction in type I superconductors and the vortex confinement in the m...

In this paper we study geometrical properties of the iterative 4triangles longest-side partition of triangles (and of a 3-triangles partition), as well as practical algorithms based on these partitions, used both directly for the triangulation refinement problem, and as a basis for point insertion strategies in Delaunay refinement algorithms. The 4-triangles partition is obtained by joining the...

Simplicial complexes consist of a set of vertices together with designated subsets. They can be thought of as embedded in Rn with the induced metric and topology, where n is large enough that the points can all be geometrically independent. The sets of points spanned by these designated subsets then form triangles or their higher or lower dimensional analogues according to some restrictions. By...

We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is Õ(n + nt), and the running time of the algorithm for sparse graphs is Õ(m + mt), where n is the number of vertices, m is the number of edges, t is the number of triangles to be listed, and ω < 2.373 is the exponent of fast matrix multiplication. With the current boun...

In this paper we consider the problem of finding pairs triangles whose sides are perfect squares integers, and which have a common perimeter area. We find two such triangles, prove that there exist infinitely many with specified properties.

We give a complete set of orthogonal invariants for triangles in G2 ( n ). As a consequence we characterize regular triangles and isoclinic triangles and we exhibit the existence regions of these objects in comparison with the angular invariants associated to them.

We investigate the lines tangent to four triangles in R3. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even.