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| A Theorem on the Convexity of Planar Bezier Curves of Degree n |
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Citation： |
Liu Dingyuan.A Theorem on the Convexity of Planar Bezier Curves of Degree n[J].Chinese Annals of Mathematics B,1982,3(1):45~56 |
| Page view： 850
Net amount： 711 |
Authors： |
Liu Dingyuan; |
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| Abstract： |
The puppose of this paper is to prove the following
Theorem. If the polygon $\[{P_0}{P_1} \cdots {P_n}{P_0}\]$ formed by the characteristic polygon $\[{P_0}{P_1} \cdots {P_n}{P_0}\]$ of a planar Bezier curve is convex,then so is the Bezier curve.
In the case that the angle of rotation from $\[\mathop {{P_0}P{}_1}\limits^ \to \]$ to $\[\mathop {{P_{n - 1}}P{}_n}\limits^ \to \]$ is not larger than \pi,we obtained the theorem by using certain properties of Bernstein polynomials.On the contrary,if the above angle of rotation is larger than \pi,then we cut the oringinal Bezier curve into two new Bezier curves,and prove that the new corresponding characteristic polygons are convex and angles of rotation betweenthe first edge and last edge of the both polygons are not larger than \pi,so that we reduce the latter case into the former discussed case.The theoremis proved.
In the present paper we also discuss the distribution of the singular points and inflection points of a planar cubic Bezier curve in detais,and thence give a classification of planar cubic Bezier curves.
This paper is prepared under the guidance of Professor Su Buchin. |
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