24 research outputs found
Interpolating sequences of 3D-data with C2 quintic PH B-spline curves
The goal of this paper is to present an effective method for interpolating sequences of 3D-data by means of C2 quintic Pythagorean-Hodograph (PH) B-spline curves. The strategy we propose works successfully with both open and closed sequences of 3D-points. It relies on calculations that are mostly explicit thanks to the fact that the interpolation conditions can explicitly be solved in dependence of the coefficients of the pre-image PH B-spline curve. In order to select a more suitable interpolant a functional is minimized in two remaining free coefficients of the pre-image PH B-spline curve and some angular parameters
Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints
In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves
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
A new class of trigonometric B-Spline Curves
We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties
Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics
We present an algorithm for C1 Hermite interpolation
using Möbius transformations of planar polynomial Pythagoreanhodograph
(PH) cubics. In general, with PH cubics, we cannot
solve C1 Hermite interpolation problems, since their lack of parameters
makes the problems overdetermined. In this paper, we
show that, for each Möbius transformation, we can introduce an
extra parameter determined by the transformation, with which we
can reduce them to the problems determining PH cubics in the
complex plane ℂ. Möbius transformations preserve the PH property
of PH curves and are biholomorphic. Thus the interpolants
obtained by this algorithm are also PH and preserve the topology
of PH cubics. We present a condition to be met by a Hermite
dataset, in order for the corresponding interpolant to be simple or
to be a loop. We demonstrate the improved stability of these new
interpolants compared with PH quintics