An Approximation of Bézier Curves by a Sequence of Circular Arcs

Abstract

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