At one of their meetings, Klein challenged the group to solve a complex problem in planar geometry. Klein proposed that the group consider five points on a flat surface, where no three of the five formed a straight line. It was obvious to all that when four of the points were joined, they formed a quadrilateral, but Klein also noticed that given five points, four of them always appeared to defi...

We introduce a new algorithm to convert triangular meshes of polygonal regions, with or without holes, into strictly convex quadrilateral meshes of small bounded size. Our algorithm includes all vertices of the triangular mesh in the quadrilateral mesh, but may add extra vertices (called Steiner points). We show that if the input triangular mesh has t triangles, our algorithm produces a mesh wi...

We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled hyperbolic triangle or quadrilateral with algebraic side lengths. We conjecture that there does not exist a rational angled hyperbolic polygon with algebraic side ...

We give a new proof of a recent inequality for bicentric quadrilaterals that is an extension of the Euler-like inequality R ≥ √ 2r. A bicentric quadrilateral ABCD is a convex quadrilateral that has both an incircle and a circumcircle. In [6], Zhang Yun called these “double circle quadrilaterals” and proved that

It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour-type reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Ribaucour transformations is presented in the vectorial framework. Finally, the quadratic reduction a...

We present a simple and practical technique for seamlessly texturing quadrilateral meshes. Using our technique any image can be converted to an isotropic texture that can be mapped to any quadrilateral mesh without any discontinuity or singularity. Using our technique, we can make any abstract painter like Miro to seamlessly paint any smooth manifold surface. The surface can have any number of ...

A quadrilateral is a bordered surface homeomorphic to the closed disc, with 4 marked boundary points called vertices, and equipped with a Riemannian metric of constant curvature K ∈ {0, 1,−1}, such that the sides (the arcs between the vertices) are geodesic. So the metric has the so-called conic singularities at the vertices. Let παj > 0 be the interior angles at the vertices. When K = 0, we kn...