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Math. Comput. Appl. 2017, 22(1), 9; doi:10.3390/mca22010009

Analysis of Inflection and Singular Points on a Parametric Curve with a Shape Factor

1
School of Mathematics, Hefei University of Technology, Hefei 23009, China
2
College of Information &Network Engineering, Anhui Science and Technology University, Chuzhou 233100, China
*
Author to whom correspondence should be addressed.
Academic Editor: Chungang Zhu
Received: 19 September 2016 / Revised: 25 December 2016 / Accepted: 11 January 2017 / Published: 19 January 2017
(This article belongs to the Special Issue Information and Computational Science)
View Full-Text   |   Download PDF [1377 KB, uploaded 19 January 2017]   |  

Abstract

The features of a class of cubic curves with a shape factor are analyzed by means of the theory of envelope and topological mapping. The effects of the shape factor on the cubic curves are made clear. Necessary and sufficient conditions are derived for the curve to have one or two inflection points, a loop or a cusp, or to be locally or globally convex. Those conditions are completely characterized by the relative position of the edge vectors of the control polygon and the shape factor. The results are summarized in a shape diagram, which is useful when the cubic parametric curves are used for geometric modeling. Furthermore, we discuss the influences of the shape factor on the shape diagram and the ability for adjusting the shape of the curve. View Full-Text
Keywords: shape factor; singular points; inflection points; local convexity; global convexity shape factor; singular points; inflection points; local convexity; global convexity
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Liu, Z.; Li, C.; Tan, J.; Chen, X. Analysis of Inflection and Singular Points on a Parametric Curve with a Shape Factor. Math. Comput. Appl. 2017, 22, 9.

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