Related papers: Building geometrically continuous splines
We give a thoroughful explanation of the general properties of different, general scales, corresponding to different (all possible) mathematical functions f(x), we mention and analyse many examples. These observations and statements might…
A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.
A procedure for interpolating between specified points of a curve or surface is described. The method guarantees slope continuity at all junctions. A surface panel divided into p x q contiguous patches is completely specified by the…
An integer generalized spline is a set of vertex labels on an edge-labeled graph that satisfy the condition that if two vertices are joined by an edge, the vertex labels are congruent modulo the edge label. Foundational work on these…
Given a graph $G$ whose edges are labeled by ideals of a commutative ring $R$ with identity, a generalized spline is a vertex labeling of $G$ by the elements of $R$ so that the difference of labels on adjacent vertices is an element of the…
Isogeometric analysis allows to define shape functions of global $C^{1}$ continuity (or of higher continuity) over multi-patch geometries. The construction of such $C^{1}$-smooth isogeometric functions is a non-trivial task and requires…
A 2-dimensional point-line framework is a collection of points and lines in the plane which are linked by pairwise constraints that fix some angles between pairs of lines and also some point-line and point-point distances. It is rigid if…
The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end…
Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with…
The main purpose of the present paper is to introduce the notion of squeezing functions of bounded domains and study some properties of them. The relation to geometric and analytic structures of bounded domains will be investigated.…
In this paper we present a method for direct evaluation of generalized B-splines (GB-splines) via the local representation of these curves as piecewise functions. To accomplish this we introduce a local structure that makes GB-spline curves…
A $\textit{regular polygon surface}$ $M$ is a surface graph $(\Sigma, \Gamma)$ together with a continuous map $\psi$ from $\Sigma$ into Euclidean 3-space which maps faces to regular Euclidean polygons. When $\Sigma$ is homeomorphic to the…
When considering regularity of surfaces, it is its geometry that is of interest. Thus, the concept of geometric regularity or geometric continuity of a specific order is a relevant concept. In this paper we discuss necessary and sufficient…
In this paper, we address the problem of constructing $C^2$ cubic spline functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The $C^2$ cubic…
We review the construction and a few properties of the S-basis, a simplex spline basis for the $C^1$ quadratic splines on the Powell-Sabin 12-split.
The space of $C^1$ cubic Clough-Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a…
Using a recent description of the geometric stability manifold, we show the geometric stability manifold associated to any smooth projective complex surface is contractible. We then use this result to demonstrate infinitely many new…
Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If…
This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on…
Concepts and techniques from the theory of G-structures of higher order are applied to the study of certain structures (volume forms, conformal structures, linear connections and projective structures) defined on a pseudo-Riemanniann…