On the uniqueness of the multirational blossom

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Computer Aided Geometric Design Pub Date : 2023-10-12 DOI:10.1016/j.cagd.2023.102252

O. Oğulcan Tuncer , Plamen Simeonov , Ron Goldman

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引用次数: 0

Abstract

The multirational blossom of order k and degree −n of a k-times differentiable function $f (t)$ is defined as a multivariate function $f (u_{1}, \dots, u_{k} / v_{1}, \dots, v_{k + n})$ characterized by four axioms: bisymmetry in the u and v parameters, multiaffine in the u parameters, satisfies a cancellation property and reduces to $f (t)$ along the diagonal. The existence of a multirational blossom was established in Goldman (1999a) by providing an explicit formula for this blossom in terms of divided differences. Here we show that these four axioms uniquely characterize the multirational blossom. We go on to introduce a homogeneous version of the multirational blossom. We then show that for differentiable functions derivatives can be computed in terms of this homogeneous multirational blossom. We also use the homogeneous multirational blossom to convert between the Taylor bases and the negative degree Bernstein bases.

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论多有理开花的唯一性

k次可微函数f（t）的k阶和次数−n的多域开花被定义为一个多元函数f（u1，…，uk/v1，…，vk+n），其特征在于四个公理：u和v参数中的双对称性，u参数中的多仿射，满足消去性质并沿对角线减少到f（t）。在Goldman（1999a）中，通过根据分裂的差异为多民族开花提供明确的公式，确立了多民族开花的存在。在这里，我们证明了这四个公理是多民族开花的唯一特征。我们接着介绍多民族开花的同质版本。然后，我们证明了对于可微函数，导数可以根据这个齐次多域开花来计算。我们还使用齐次多域开花来转换Taylor基和负度Bernstein基。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.