靠近脐带的平面剖面

Q3 Mathematics
Peter Giblin, Aleksandr Pukhlikov
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引用次数: 0

摘要

研究了带脐点的光滑代数曲面与在脐点处平行且接近切平面的平面的交。这个问题起源于对二维图像中等光照曲线的研究。特别地,我们研究了与这条相交曲线有特殊相切的圆:一点相切,另一点相切;三点的常切线;和一个顶点的四点相切。两点常相切的圆心画出一条曲线,该曲线的闭包是交点曲线的对称集,上述例外圆分别给出了交点曲线的顶点、三交点和端点。我们分析了由接触点和异常圆的中心跟踪的曲线,当平面接近切平面在脐带。并通过一个典型的例子简要地讨论了对称集的整体结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Planar Sections of a Surface Close to an Umbilic

We study the intersection between a smooth algebraic surface with an umbilic point and a plane parallel and close to the tangent plane at the umbilic. The problem has its origin in the study of isophote (equal illumination) curves in a 2-dimensional image. In particular, we study the circles which have exceptional tangency to this intersection curve: ordinary tangency at one point and osculating at another; ordinary tangency at three points; and 4-point tangency at a vertex. The centres of circles having ordinary tangency at two points trace out a curve whose closure is the symmetry set of the intersection curve, and the exceptional circles above give respectively cusps, triple crossings and endpoints of this set. We analyse the curves traced out by the contact points and centres of the exceptional circles as the plane approaches the tangent plane at the umbilic. We also briefly discuss the global structure of the symmetry set by means of a typical example.

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来源期刊
Arnold Mathematical Journal
Arnold Mathematical Journal Mathematics-Mathematics (all)
CiteScore
1.50
自引率
0.00%
发文量
28
期刊介绍: The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis.  Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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