A C1 simplex-spline basis for the Alfeld split in Rs

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Computer Aided Geometric Design Pub Date : 2025-01-13 DOI:10.1016/j.cagd.2025.102412

Tom Lyche , Jean-Louis Merrien , Hendrik Speleers

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引用次数: 0

Abstract

The Alfeld split is obtained by subdividing a simplex in

R^{s}

into

s + 1

subsimplices with the barycenter as one of their vertices. On this split, we consider the space of

C^{1}

splines of degree d (

d \geq s + 1

), for which we construct a basis of simplex-splines with knots at the barycenter and the vertices of the simplex. The basis consists of two types of simplex-splines: firstly Bernstein polynomials with domain points on the facets of the simplex and secondly certain simplex-splines with at least one knot at the barycenter. Partition of unity, Marsden-like identities, and domain points are shown. We also provide

C^{1}

smoothness conditions across a facet between two simplices.

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来源期刊

Computer Aided Geometric Design 工程技术-计算机：软件工程

CiteScore

3.50

自引率

13.30%

发文量

审稿时长

60 days

期刊介绍： The journal Computer Aided Geometric Design is for researchers, scholars, and software developers dealing with mathematical and computational methods for the description of geometric objects as they arise in areas ranging from CAD/CAM to robotics and scientific visualization. The journal publishes original research papers, survey papers and with quick editorial decisions short communications of at most 3 pages. The primary objects of interest are curves, surfaces, and volumes such as splines (NURBS), meshes, subdivision surfaces as well as algorithms to generate, analyze, and manipulate them. This journal will report on new developments in CAGD and its applications, including but not restricted to the following: -Mathematical and Geometric Foundations- Curve, Surface, and Volume generation- CAGD applications in Numerical Analysis, Computational Geometry, Computer Graphics, or Computer Vision- Industrial, medical, and scientific applications. The aim is to collect and disseminate information on computer aided design in one journal. To provide the user community with methods and algorithms for representing curves and surfaces. To illustrate computer aided geometric design by means of interesting applications. To combine curve and surface methods with computer graphics. To explain scientific phenomena by means of computer graphics. To concentrate on the interaction between theory and application. To expose unsolved problems of the practice. To develop new methods in computer aided geometry.