Improvement of the modifiable Bézier curves

Yan Lanlan,Han Xuli(College of Science, East China Institute of Technology, Nanchang 330013, China;School of Mathematics and Statistics, Central South University, Changsha 410083, China)

Abstract
Objective When a Bézier curve is used to describe complex shapes, the problem of joining curve segments smoothly has to be solved. To maintain the continuity of the whole curve, adjacent curve segments must meet strict continuity conditions. A higher the requirement for continuity usually causes conditions to become more complex and involves a larger number of control points. This study improves the modifiable Bézier curve in the literature to achieve smooth connection between curves automatically and to construct a piecewise composite curve with numerous merits. Method We first present a sufficient condition of Gl continuity for two curves with continuous position.On the basis of the sufficient condition Gl, we prove that the modifiable Bézier curve can achieve a smooth connection under conditions that usually guarantee Gl continuity only for the usual Bézier curve and most Bézier-like curves in the literature. We then use a transition matrix to convert the modifiable Bézier basis to a new set of basis functions. We employ this set of basis functions to define a new kind of curve according to the definition mode of the standard Bézier curve. We then analyze the smooth connection conditions of the new curve.Considering theses smooth connection conditions and by using a special definition mode, we construct a kind of piecewise composite curve. The connection of the control points between adjacent curve segmentsis apparently similar to that of the classical B-spline curve. However, the connections are actually different. The B-spline curve only has one edge between the control polygons of two adjacent curve segments, that is, only one control point is different. Nevertheless, the composite curve defined in this study only has one edge. Thus, only two control points are the same. Result The new curve defined by the new basis function has relatively simple and special smooth connection conditions. Two neighboring curves can be smoothly joined automatically as long as the last control edge of the former coincides with the first control edge of the latter. Furthermore, the degree of smoothness at the meeting point can be freely adjusted by simply changing the value of the parameter. The piecewise composite curve of the new curve possesses numerous desirable properties, such as geometric invariance, symmetry, automatic smoothness property, and local control capability similar to that of the classical B-spline curve. However, the main difference between the newly defined piecewise curve and the standard B-spline curve is that in the new composite curve, each segment can be defined by different numbers of control points. By contrast, the number of segments must be equal forthe usual B-spline curve.This difference is the main reason that the new piecewise curve has a stronger local control capability than the B-spline curve. Furthermore, a suitable parameter can enable the new composite curve to achieve the expected smoothness at each connection point. In addition, changing one control point can alter the shape of two curve segments. When the parameter of one curve segment is changed, only the shape of the current curve and the degree of smoothness at two connection points will change. Conclusion This study presents the general Method to construct curves. This Method can easily achieve a smooth connection. We further present the general Method to construct composite curves. This Method can achieve a smooth connection automatically.
Keywords
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