An Approximation of Bézier Curves by a Sequence of Circular Arcs

Taweechai Nuntawisuttiwong,Natasha Dejdumrong

doi:10.5755/j01.itc.50.2.25178

Taweechai Nuntawisuttiwong, Natasha Dejdumrong

Open Access

https://doi.org/10.5755/j01.itc.50.2.25178

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Abstract

Some researches have investigated that a Bézier curve can be treated as circular arcs. This work is to proposea new scheme for approximating an arbitrary degree Bézier curve by a sequence of circular arcs. The sequenceof circular arcs represents the shape of the given Bézier curve which cannot be expressed using any other algebraicapproximation schemes. The technique used for segmentation is to simply investigate the inner anglesand the tangent vectors along the corresponding circles. It is obvious that a Bézier curve can be subdivided intothe form of subcurves. Hence, a given Bézier curve can be expressed by a sequence of calculated points on thecurve corresponding to a parametric variable t. Although the resulting points can be used in the circular arcconstruction, some duplicate and irrelevant vertices should be removed. Then, the sequence of inner angles arecalculated and clustered from a sequence of consecutive pixels. As a result, the output dots are now appropriateto determine the optimal circular path. Finally, a sequence of circular segments of a Bézier curve can be approximatedwith the pre-defined resolution satisfaction. Furthermore, the result of the circular arc representationis not exceeding a user-specified tolerance. Examples of approximated nth-degree Bézier curves by circular arcsare shown to illustrate efficiency of the new method.

Highlights

In geometric modeling, there are different kinds of file formats used in design and production processes
Models are usually designed by Bézier curves and B-spline but such models cannot be directly used in the production process
Besides the vector graphics format used in design applications, the raster graphics format is another format applied in the production process

Summary

Introduction

There are different kinds of file formats used in design and production processes. The method to approximate Bézier curves by a sequence of arc splines with inscribed regular polygon was proposed in [13]. A Bézier curve B(t) has an appropriate number of uniform arc length sampled points, denoted by M , that is satisfied the given error of linear interpolation, called tolerance (τ ) if. Considering inner angle a1 and a2 at vertices v1 and v2 respectively, the edge e2 will be merged with circular path c0 and incidence matrix is transformed as follows: ε qi − ei.

Algorithm for Circular Arcs Approximation

Examples

Conclusions