Abstract
Recently, He et al. derived several remarkable properties of the so-called typical Bézier curves, a subset of constrained Bézier curves introduced by Mineur et al. In particular, He et al. proved that such curves display at most one curvature extremum, give an explicit formula of the parameter at the extremum, and show that subdividing a curve at this point furnishes two new typical curves. We recall that typical curves amount to segments of a special family of sinusoidal spirals, curves already studied by Maclaurin in the early 18th century and whose properties are well-known. These sinusoidal spirals display only one curvature extremum (i.e., vertex), whose parameter is simply that corresponding to the axis of symmetry. Subdividing a segment at an arbitrary point, not necessarily the vertex, always yields two segments of the same spiral, hence two typical curves.
Highlights
We recall that typical curves coincide with segments of a family of offset-rational sinusoidal spirals first introduced in Bézier form by Ueda [21,22] via a pedal-point construction
(Section 3), to make the article self-contained, we briefly review their construction by raising a straight line to the nth power in the complex plane, concluding that spiral segments coincide with typical curves
Degree-n typical Bézier curves amount to segments of a classical family of sinusoidal spirals, of negative index −1/n, an elucidating relationship overlooked in the literature
Summary
We recall that typical curves coincide with segments of a family of offset-rational sinusoidal spirals first introduced in Bézier form by Ueda [21,22] via a pedal-point construction. Since the term spiral usually implies monotonic curvature, each curve in the family (n ≥ 2) is composed of two semi-infinite spiral segments: that corresponding to θ ∈ [0, n π2 ), partially plotted, and its mirror image θ ∈ Their curvature κ (θ ) admits a simple expression [30,34]:.
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