Algebraic characterization of planar cubic and quintic Pythagorean-Hodograph B-spline curves

We provide a revised representation of planar cubic and quintic Pythagorean-Hodograph B-spline curves (PH B-splines for short) that offers the following advantages: (i) the clamped and closed cases are mostly treated together; (ii) the closed case is represented by using the minimum possible number of knots thus avoiding useless control points as well as control edges of zero length when the curve is regular. The proposed simplified representation turns out to be extremely useful to provide a unified complex algebraic characterization of clamped and closed planar PH B-splines of degree three and five. This is aimed at distinguishing regular planar cubic and quintic PH B-splines from $C^1$ cubic and $C^2$ quintic B-spline curves in general. As for planar cubic PH B-splines consisting of $m$ pieces, we obtain $m$ complex conditions that, differently from what was known so far, can be used to characterize both the clamped and the closed case. As for planar quintic PH B-splines, the complex conditions are $2m$ and, unlike what is shown for cubic PH B-splines, they also depend on the knot intervals. This is to be considered a completely new result since no complex algebraic characterization working for any arbitrarily chosen knot partition had ever been provided for either clamped or closed planar quintic PH B-splines. The proposed algebraic characterization is finally exploited to fully identify the preimage of a regular planar quintic PH B-spline resolving all the sign ambiguities that affected the existing results.