Generalized plane offsets and rational parameterizations

Geometric Hermite interpolation by rational curves of constant width

“…Remark 2. As noticed in [22], a rationally parameterized curve by a support function h as in ( 2) is convex and regular if and only if 4 h(t)…”

“…In particular, the authors proved that any hedgehog hypersurface (envelope of a family of supporting hyperplanes [11]) parameterized by a rational support function can be rationally parameterized. A detailed description of the planar case can be found in [22], which leads to the following definition. Definition 2.…”

“…In this section, we will study the conditions that the support function of a rationally parameterized curve of constant width must satisfy. First, we characterize the property of having constant width (see also [22]). Proposition 2.…”

“…The authors of [9] showed elegantly that odd rational support functions correspond to those rational surfaces which can be equipped with a linear field of normal vectors (LN surfaces), and then that their offsets are rational constant width surfaces. The planar version can also be deduced from [22], where it is noted that any rationally parameterized curve is Pythagorean-hodograph, and thus, it has a rational offset. For this reason and by Remark 1, we can focus without loss of generality on discussing what a support function of a rational projective hedgehog must be.…”

“…With this, curves of constant width can also be integrated into the scheme of rational Bézier curves. There are some previous works in this direction, such as [1] or [22]. Other related works are [14], [9] and [24].…”