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

AbstractAlgorithms 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 G1 Hermite data; however, one could also obtain higher order algorithms

Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Given a polynomial space curve r(ξ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r˜(ξ) with the same rotation–minimizing frame as r(ξ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ξ)=r′(ξ)×r″(ξ) and distances from the origin specified in terms of a rational function f(ξ) as f(ξ)/‖u(ξ)‖. An explicit characterization of the rational curves r˜(ξ) generated by a given RRMF curve r(ξ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ξ), obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ξ). Criteria for identifying low–degree instances of the curves r˜(ξ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples

On Normals and Control Nets

This paper characterizes when the normals of a spline curve or spline surface lie in the more easily computed cone of the normals of the segments of the spline control net

Generalized plane offsets and rational parameterizations

In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown