Intersections of Polinomial Curves with Planes
Abstract
There are different approaches to determining such characteristics of a curve as osculating plane, flattening and its order. The difference in approaches is because the curves are considered in different spaces: affine and projective. Therefore, the wordings are given in terms of the relevant structures: for curves in affine space – through a linear dependence or independence of the vector systems, for curves in the projective space – through the intersection multiplicity of a curve and the planes. The interpenetration of the two points of view can be realized through the common to both spaces concept of intersection multiplicity of a curve and a surface or through a transition to affine charts for the curves in the projective space. The paper discloses the connection between the differential and algebraic definitions of these concepts. Although the equivalence of both interpretations in terms of general applicability, apparently, is meant (the author has not met the discussion on this question in literature), it is useful to provide a rigorous foundation, especially since there are partial and principal discrepancies. In this paper, an algebraic characterization of the flattening of order k, which corresponds to differential-geometric interpretation, is presented. The multiplicity of intersections of polynomial curves with the planes in n-dimensional affine space is investigated. A necessary and sufficient condition of non-degeneracy of polynomial curves is found. Bezier curves playing an important role in computer graphics are the example of such polynomial lines.
DOI 10.14258/izvasu(2017)4-26
Downloads
Metrics
References
Брус Дж., Джиблин П. Кривые и особенности: Геометрическое введение в теорию особенностей. — М., 1988.
Uribe-Vargas R. On Vertices, Focal Curvatures and Differential Geometry of Spase Curves // Bull. Braz. Math. Soc., new series. — 2005. — T. 36(3).
Лейко С.Г. Р-геодезические преобразования и их группы в касательных расслоениях, индуцированные конциркулярными преобразованиями базисного многоообразия // Известия высших учебных заведений. Математика. — 1998. — № 6 (433).
Поликанова И.В. Некоторые свойства линий с аффинно-эквивалентными дугами // Труды семинара по геометрии и математическому моделированию: сб. ст. — Вып. 2/ под ред. Е. Д. Родионова. — Барнаул, 2016.
Анисов С.С. Выпуклые кривые в RPn // Локальные и глобальные задачи теории особенностей: сб. ст. к 60-летию академика Владимира Игоревича Арнольда: труды МИАН 221. — М., 1998.
Anisov C.C. Projective convex curves // The Arnold-Gelfand mothematical seminars: geometry and singularity theory. — Boston, 1997.
Arnold V.I. On the number of flattening points on space curves // Amer. Math. Soc. Transl. (2) — 1995. — V. 171.
Седых В.Д. Теорема о четырёх вершинах пространственной кривой // Функциональный анализ и его приложения. — 1992. — Т. 26. — Вып. 1.
Седых В.Д. Теорема о четырёх вершинах плоской кривой и её обобщения // Соросовский образовательный журнал. — 2000. — Т. 6. — № 9.
Uribe-Vargas R. On 4-flattening Theorems and the Curves of Charateodory, Barner and Segre // Journal of Geometry. — 2003. — Т. 77.
Copyright (c) 2017 И.В. Поликанова
This work is licensed under a Creative Commons Attribution 4.0 International License.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
