Related papers: Analytical Methods for Squaring the Disc

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution,…

An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers…

We describe three identification spaces of the square, interesting choices of immersion into $\mathbb{R}^3$, and a process to construct 3d-printable models of their parametrizations.

In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider…

We investigate an idealized model for the size reduction and smoothing of a polygonal rock due to repeated chipping at corners. Each chip is sufficiently small so that only a single corner and a fraction of its two adjacent sides are cut…

Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their…

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal…

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…

We study the spherical slice transform which assigns to a function on the $n$-dimensional unit sphere the integrals of that function over cross-sections of the sphere by $k$-dimensional affine planes passing through the north pole. These…

Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space.

In this paper we define and construct a new class of algebraic surfaces in three-dimensional Euclidean space generated by a curve and a congruence of circles. We study their properties and visualize them with the program Mathematica.

See http://www.youtube.com/watch?v=izbGXdjvK_I for a YouTube video showing part of the results in this paper.We will consider surfaces whose mean curvature at a point is a linear function of the square of the distance from that point to the…

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

We investigate discrete spin transformations, a geometric framework to manipulate surface meshes by controlling mean curvature. Applications include surface fairing -- flowing a mesh onto say, a reference sphere -- and mesh extrusion --…

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in…

We employ the framework of operational calculus to derive the operators associated with the spherical mean and a class of related averaging means of a function in $n$-dimensional space. Beginning with the classical definition of the…

In the article we consider the composite conformal map which maps annulus to infinite region with symmetric hole and nearly circular hole. It is shown that such transformation is good if the distance between centers of holes are large or…

In this paper, we define the notion of a mapping on soft classes and study several properties of images and inverse images of soft sets supported by examples and counterexamples. Finally, these notions have been applied to the problem of…

We find all analytic surfaces in space $\mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of…