136 research outputs found
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
Construction and evaluation of PH curves in exponential-polynomial spaces
In the past few decades polynomial curves with Pythagorean Hodograph (for
short PH curves) have received considerable attention due to their usefulness
in various CAD/CAM areas, manufacturing, numerical control machining and
robotics. This work deals with classes of PH curves built-upon
exponential-polynomial spaces (for short EPH curves). In particular, for the
two most frequently encountered exponential-polynomial spaces, we first provide
necessary and sufficient conditions to be satisfied by the control polygon of
the B\'{e}zier-like curve in order to fulfill the PH property. Then, for such
EPH curves, fundamental characteristics like parametric speed or cumulative and
total arc length are discussed to show the interesting analogies with their
well-known polynomial counterparts. Differences and advantages with respect to
ordinary PH curves become commendable when discussing the solutions to
application problems like the interpolation of first-order Hermite data.
Finally, a new evaluation algorithm for EPH curves is proposed and shown to
compare favorably with the celebrated de Casteljau-like algorithm and two
recently proposed methods: Wo\'zny and Chudy's algorithm and the dynamic
evaluation procedure by Yang and Hong
Employing Pythagorean Hodograph curves for artistic patterns
In this paper we present a novel design element creator tool for the digital artist. The purpose of our tool is to support the creation of vines, swirls, swooshes and floral components. To create visually pleasing and gentle curves we employ Pythagorean Hodograph quintic spiral curves to join a hierarchy of control circles defined by the user. The control circles are joined by spiral segments with at least G2 continuity, ensuring smooth and seamless transitions. The control circles give the user a fast and intuitive way to define the desired curve. The resulting curves can be exported as cubic Bezier curves for further use in vector graphics applications
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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
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