μ-Bases for complex rational curves

Corrigendum to Example 4 in “μ-Bases for complex rational curves” [Computer Aided Geometric Design 30 (2013), 623–635]

“…

We made a mistake in Example 4 in the paper Wang and Goldman (2013). Here we correct this mistake.

In Example 4 in the paper (Wang and Goldman, 2013), we gave the following incorrect parametrization of the cornoid

Hence, using Algorithm 2 ("Determining if a real rational curve is a complex rational curve") in Wang and Goldman (2013), we claimed that the cornoid is a complex rational curve.

…”

Birational Quadratic Planar Maps with Generalized Complex Rational Representations

“…Note that the set of all syzygies of C(s, t) is also a free module of rank two over the ring R[s, t] (for more details, the interested reader is referred to the proof of Proposition 2.1 in [27]). Moreover, µ-bases of generalized complex rational curves can be calculated via these syzygies in the one-dimensional case [16,21].…”

“…Complex rational curves were introduced by J. Sánchez-Reyes by allowing complex weights in rational Bézier curves [15]. Based on the observation that complex rational curves have two special syzygies of low degrees [16], we can construct birational quadratic planar maps by exploiting their complex rational representations [17], which are complex rational maps of degree one. Complex rational planar maps of degree one represent exactly Möbius transformations.…”