School of Mathematics, Hefei University of Technology, Hefei 23009, China
College of Information & Network Engineering, Anhui Science and Technology University, Chuzhou 233100, China
: Received: 22 September 2016 / Approved: 23 September 2016 / Online: 23 September 2016 (08:04:57 CEST)
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.
How to cite:
Liu, Z.; Li, C.; Tan, J.; Chen, X. Analysis of Inflection and Singular Points on Parametric Curve with a Shape Factor. Preprints2016, 2016090082 (doi: 10.20944/preprints201609.0082.v1).
Liu, Z.; Li, C.; Tan, J.; Chen, X. Analysis of Inflection and Singular Points on Parametric Curve with a Shape Factor. Preprints 2016, 2016090082 (doi: 10.20944/preprints201609.0082.v1).
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.
shape factor; singular points; inflection points; local convexity; global convexity