Planar Typical Bezier Curves Made Simple

Recently, He et al. derived several remarkable properties of the so-called typical Bezier curves, a subset of constrained Bezier 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.

Automated summary

He et al. recently derived remarkable properties of typical Bezier curves, proving that these curves have at most one curvature extremum and providing an explicit formula for the extremum parameter. Subdividing a curve at this point results in two new typical curves. Typical curves are segments of sinusoidal spirals with specific properties, which have been studied since the 18th century.