A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Abstract

a b s t r a c tAlgorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G 1 Hermite data; however, one could also obtain higher order algorithms.

Skinning and blending with rational envelope surfaces

“…is an isotropic vector; see [28] for more details. Thus we can choose three arbitrary equations in each system, e.g., the first three of them.…”

Medial axis transforms yielding rational envelopes

“…In Kosinka and Lávička (2011), an interpolation method using MPH splines based on planar PH splines was introduced and thoroughly investigated. While conceptually simpler than previous methods, it still relies on the construction of a PH curve interpolating certain derived data in the plane.…”

“…In this paper, we return to this thoroughly studied problem of MPH and envelope curves (Choi et al, 1999;Kosinka and Jüttler, 2006;Kosinka and Jüttler, 2009;Kosinka and Šír, 2010;Kosinka and Lávička, 2011). As we will see shortly in Section 2, it turns out that if one allows a square-root term (Abhyankar, 1994) in the MAT's representation, a broader class of MATs corresponding to rational envelopes of the associated family of circles is obtained.…”

“…The discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation was further developed by Kosinka and Lávička (2011). The main advantage of the method presented in Kosinka and Lávička (2011) lies in the fact that it uses, only after some simple additional computations, an arbitrary algorithm for interpolation by planar PH curves also for interpolation by spatial MPH curves. Thus, compared to other MPH schemes, it does not require the complicated Clifford algebra machinery (Choi et al, 2002).…”

Applying Rational Envelope curves for skinning purposes

“…Nowadays, several Hermite interpolation methods have been proposed using MPH curves so that the resulting envelope is rational (Kim and Ahn, 2003;Jüttler, 2006, 2009;Kosinka and Šír, 2010;Kosinka and Lávička, 2011;Bizzarri et al, 2019). Bizzarri et al (2016) showed that a broader class of curves exists in R 2,1 , which yields rational boundaries: the so-called Rational Envelope (RE) curves.…”