{"title":"Optimal one-sided approximants of circular arc","authors":"Ada Šadl Praprotnik , Aleš Vavpetič , Emil Žagar","doi":"10.1016/j.cagd.2024.102401","DOIUrl":null,"url":null,"abstract":"<div><div>The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"115 ","pages":"Article 102401"},"PeriodicalIF":1.3000,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Aided Geometric Design","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167839624001353","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.
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.
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