Rational Pythagorean-hodograph space curves

https://doi.org/10.1016/j.cagd.2011.01.002Get rights and content

Abstract

A method for constructing rational Pythagorean-hodograph (PH) curves in R3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve osculating planes. Augmenting this orientation information with a rational support function, that specifies the distance of each osculating plane from the origin, then completely defines a one-parameter family of osculating planes, whose envelope is a developable ruled surface. The rational PH space curve is identified as the edge of regression (or cuspidal edge) of this developable surface. Such curves have rational parametric speed, and also rational adapted frames that satisfy the same conditions as polynomial PH curves in order to be rotation-minimizing with respect to the tangent. The key properties of such rational PH space curves are derived and illustrated by examples, and simple algorithms for their practical construction by geometric Hermite interpolation are also proposed.

Highlights

► Rational PH space curves defined by vector field and scalar function. ► Curve is specified as edge of regression of tangent developable. ► Geometric Hermite interpolation by solution of linear equations. ► Conditions for existence of rational rotation-minimizing frames.

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      Citation Excerpt :

      Albrecht et al. (2017, 2020) introduced PH B spline curves, whose offsets are NURBS curves. The theoretical results of PH curves are also extended to spatial PH curves (Sakkalis and Farouki, 2012; Farouki et al., 2002; Albrecht et al., 2020), Minkowski PH curves (Moon, 2008) and rational PH curves (Pottmann, 1995; Farouki and Šír, 2011). More details can be found in comprehensive references (Farouki et al., 2008; Kosinka and Lavicka, 2014).

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    This paper has been recommended for acceptance by G.E. Farin.

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