Preprint Article Version 1 This version not peer reviewed

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

Version 1 : Received: 22 September 2016 / Approved: 23 September 2016 / Online: 23 September 2016 (08:04:57 CEST)

A peer-reviewed article of this Preprint also exists.

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. 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.

Journal reference: Math. Comput. Appl. 2017, 22, 9
DOI: 10.3390/mca22010009

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 discussed the influences of the shape factor on the shape diagram and the ability for adjusting the shape of the curve.

Subject Areas

shape factor; singular points; inflection points; local convexity; global convexity

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