New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves

IF 1.3 4区计算机科学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Computer Aided Geometric Design Pub Date : 2023-12-06 DOI:10.1016/j.cagd.2023.102256

Kai Hormann , Lucia Romani , Alberto Viscardi

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Abstract

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

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平面五次毕达哥拉斯曲线的新代数和几何特征

这项工作的目的是为平面五角勾股定理曲线提供新的特征。前两个是代数方程，分别由贝塞尔控制多边形边缘的两个和三个复数方程组成。这些方程相对于边缘指数是对称的，并且涵盖了具有一般控制多边形和退化控制多边形的曲线。最后两个特征是几何特征，都只依赖于控制多边形外的两个辅助点。一种方法要求两个（可能是退化的）四边形相似，另一种方法强调三个相似三角形的两个族。所有特征都是在现有技术基础上向前迈出的一步，它们可以与平面立方勾股定理曲线的成熟对等特征联系起来。证明上述结果的关键要素是曲线勾股定理的新颖一般表达式。

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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.

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