{"title":"带有形状参数的多项式基础,用于曲线和曲面建模","authors":"Bahareh Nouri , Imre Juhász , Jamshid Saeidian","doi":"10.1016/j.matcom.2024.10.029","DOIUrl":null,"url":null,"abstract":"<div><div>Based on Bernstein polynomials, a system of functions with a free parameter is proposed in the space of polynomials of degree at most <span><math><mi>n</mi></math></span>. The system inherits several properties of Bernstein polynomials, such as linear independence, non-negativity, partition of unity and symmetry. This new family of functions are employed to construct control point based parametric curves. The free parameter serves as a shape adjustment parameter, by means of which a one-parameter family of polynomial curves is obtained. The new family of curves is in common with Bézier curves in most of the geometric properties, providing a smooth transition between the Bézier curve and the straight line segment joining the first and last control points. Shape preserving properties, such as monotonicity preservation, as well as length, hodograph and variation diminishing are studied. The proposed basis can also be used to create tensor product surfaces. The extent to which the suggested basis generation method can be applied to other (non-polynomial) function spaces is also being investigated.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"229 ","pages":"Pages 690-705"},"PeriodicalIF":4.4000,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A polynomial basis with a shape parameter for curve and surface modeling\",\"authors\":\"Bahareh Nouri , Imre Juhász , Jamshid Saeidian\",\"doi\":\"10.1016/j.matcom.2024.10.029\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Based on Bernstein polynomials, a system of functions with a free parameter is proposed in the space of polynomials of degree at most <span><math><mi>n</mi></math></span>. The system inherits several properties of Bernstein polynomials, such as linear independence, non-negativity, partition of unity and symmetry. This new family of functions are employed to construct control point based parametric curves. The free parameter serves as a shape adjustment parameter, by means of which a one-parameter family of polynomial curves is obtained. The new family of curves is in common with Bézier curves in most of the geometric properties, providing a smooth transition between the Bézier curve and the straight line segment joining the first and last control points. Shape preserving properties, such as monotonicity preservation, as well as length, hodograph and variation diminishing are studied. The proposed basis can also be used to create tensor product surfaces. The extent to which the suggested basis generation method can be applied to other (non-polynomial) function spaces is also being investigated.</div></div>\",\"PeriodicalId\":49856,\"journal\":{\"name\":\"Mathematics and Computers in Simulation\",\"volume\":\"229 \",\"pages\":\"Pages 690-705\"},\"PeriodicalIF\":4.4000,\"publicationDate\":\"2024-10-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematics and Computers in Simulation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0378475424004270\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics and Computers in Simulation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0378475424004270","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
A polynomial basis with a shape parameter for curve and surface modeling
Based on Bernstein polynomials, a system of functions with a free parameter is proposed in the space of polynomials of degree at most . The system inherits several properties of Bernstein polynomials, such as linear independence, non-negativity, partition of unity and symmetry. This new family of functions are employed to construct control point based parametric curves. The free parameter serves as a shape adjustment parameter, by means of which a one-parameter family of polynomial curves is obtained. The new family of curves is in common with Bézier curves in most of the geometric properties, providing a smooth transition between the Bézier curve and the straight line segment joining the first and last control points. Shape preserving properties, such as monotonicity preservation, as well as length, hodograph and variation diminishing are studied. The proposed basis can also be used to create tensor product surfaces. The extent to which the suggested basis generation method can be applied to other (non-polynomial) function spaces is also being investigated.
期刊介绍:
The aim of the journal is to provide an international forum for the dissemination of up-to-date information in the fields of the mathematics and computers, in particular (but not exclusively) as they apply to the dynamics of systems, their simulation and scientific computation in general. Published material ranges from short, concise research papers to more general tutorial articles.
Mathematics and Computers in Simulation, published monthly, is the official organ of IMACS, the International Association for Mathematics and Computers in Simulation (Formerly AICA). This Association, founded in 1955 and legally incorporated in 1956 is a member of FIACC (the Five International Associations Coordinating Committee), together with IFIP, IFAV, IFORS and IMEKO.
Topics covered by the journal include mathematical tools in:
•The foundations of systems modelling
•Numerical analysis and the development of algorithms for simulation
They also include considerations about computer hardware for simulation and about special software and compilers.
The journal also publishes articles concerned with specific applications of modelling and simulation in science and engineering, with relevant applied mathematics, the general philosophy of systems simulation, and their impact on disciplinary and interdisciplinary research.
The journal includes a Book Review section -- and a "News on IMACS" section that contains a Calendar of future Conferences/Events and other information about the Association.