C2 Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation

doi:10.1016/j.cagd.2010.04.005

Computer Aided Geometric Design

Volume 27, Issue 8, November 2010, Pages 631-643

https://doi.org/10.1016/j.cagd.2010.04.005 Get rights and content

Abstract

We describe and fully analyze an algorithm for $C^{2}$ Hermite interpolation by Pythagorean hodograph curves of degree 9 in Minkowski space $R^{2, 1}$ . We show that for any data there exists a four-parameter system of interpolants and we identify the one which preserves symmetry and planarity of the input data and which has the optimal approximation degree. The new algorithm is applied to an efficient approximation of segments of the medial axis transform of a planar domain leading to rational parameterizations of the offsets of the domain boundaries with a high order of approximation.

References (21)

H.I. Choi et al.
Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves
Comput. Aided Design
(1999)
W.L.F. Degen
Exploiting curvatures to compute the medial axis for domains with smooth boundary
Comput. Aided Geom. Design
(2004)
R.T. Farouki
Pythagorean hodograph curves
R.T. Farouki et al.
Characterization and construction of helical polynomial space curves
J. Comput. Appl. Math.
(2004)
B. Jüttler et al.
Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling
Comput. Aided Design
(1999)
G.-I. Kim et al.
$C^{1}$ Hermite interpolation using MPH quartic
Comput. Aided Geom. Design
(2003)
J. Kosinka et al.
$G^{1}$ Hermite interpolation by Minkowski Pythagorean hodograph cubics
Comput. Aided Geom. Design
(2006)
H.P. Moon
Minkowski Pythagorean hodographs
Comput. Aided Geom. Design
(1999)
F. Aurenhammer et al.
Structural and computational advantages of circular boundary representations
F. Chazal et al.
Stability and finiteness properties of medial axis and skeleton
J. Dyn. Control Syst.
(2004)

There are more references available in the full text version of this article.

Cited by (15)

Interpolation of Hermite data by clamped Minkowski Pythagorean hodograph B-spline curves
2021, Journal of Computational and Applied Mathematics
In this paper, the problem of Hermite interpolation by clamped Minkowski Pythagorean hodograph (MPH) B-spline curves is considered. Using the properties of B-splines, our intention is to use the MPH curves of degrees lower than in algorithms designed before. Special attention is devoted to $C^{1} ∕ C^{2}$ Hermite interpolation by MPH B-spline cubics/quintics. The resulting interpolants are obtained by exploiting properties of B-spline basis functions and via solving special quadratic and linear equations in Clifford algebra $C ℓ_{2, 1}$ . All the presented algorithms are purely symbolic. The results are confirmed by several applications, in particular we use them to generate an approximate conversion of a given analytic curve to MPH B-spline curve with a high order of approximation, then to an efficient approximation of the medial axis transform of a planar domain leading to NURBS representation of the (trimmed) offsets of the domain boundaries, and to skinning of systems of circles in plane.
Linear computational approach to interpolations with polynomial Minkowski Pythagorean hodograph curves
2019, Journal of Computational and Applied Mathematics
Methods using Pythagorean hodographs both in Euclidean plane and Minkowski space are often used in geometric modelling when necessary to solve the problem of rationality of offsets of planar domains. A main justification for studying and formulating approximation and interpolation algorithms based on the called Minkowski Pythagorean hodograph (MPH) curves is the fact that they make the trimming procedure of inner offsets considerably simpler. This is why one can find many existing techniques in literature. In this paper a simple computational approach to parametric/geometric Hermite interpolation problem by polynomial MPH curves in $R^{2, 1}$ is presented and an algorithm to construct such interpolants is described. The main idea is to construct first not a tangent but a normal vector space satisfying the prescribed MPH property that matches the given first order conditions, and then to compute a curve possessing this constructed normal vector space and satisfying all the remaining interpolation conditions. Compared to other methods using special formalisms (e.g. Clifford algebra), the presented approach is based only on solving systems of linear equations. The results are confirmed by number of examples.
Minkowski Pythagorean-hodograph preserving mappings
2016, Journal of Computational and Applied Mathematics
We state and prove the sufficient and necessary condition for a mapping to be a scaled MPH-preserving mapping which preserves the MPH property of a curve with rescaling the speed by a rational function in $R^{2, 1}$ , and show how to produce polynomial scaled MPH-preserving mappings from given generating polynomials. We introduce s-cubic MPH-preserving mappings of the first kind, and their corresponding surfaces. We show that these mappings can be used to solve interpolation problems for $C^{1}$ Hermite data-sets with admissible velocity vectors on their corresponding surfaces.
Medial axis transforms yielding rational envelopes
2016, Computer Aided Geometric Design
Citation Excerpt :
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).
Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes.
To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves.
Parameterizing rational offset canal surfaces via rational contour curves
2013, CAD Computer Aided Design
Citation Excerpt :
Nevertheless, we need some well-known testing procedure to present a functionality of the presented method. For the sake of illustration, we have chosen a variant of the algorithm introduced in [37] however any arbitrary alternative method can be used — we recall e.g. [30,38–40]. In this paper, we studied a condition guaranteeing the rationality of contour curves on canal surfaces.
A canal surface is the envelope of a 1-parameter set of spheres centered at the spine curve $m (t)$ and with the radii described by the function $r (t)$ . Any canal surface given by rational $m (t)$ and $r (t)$ possesses a rational parameterization. However, an arbitrary rational canal surface does not have to fulfill the PN (Pythagorean normals) condition. Most (exact or approximate) parameterization methods are based on a construction of a rational unit normal vector field guaranteeing rational offsets. In this paper, we will study a condition which guarantees that a given canal surface has rational contour curves, which are later used for a straightforward computation of rational parameterizations of canal surfaces providing rational offsets. Using the contour curves in the parameterization algorithm brings another extra feature; the parameter lines do not unnecessarily wind around the canal surface. Our approach follows a construction of rational spatial MPH curves from the associated planar PH curves introduced in Kosinka and Lávička (2010) [28] and gives it to the relation with the contour curves of canal surfaces given by their medial axis transforms. We also present simple methods for computing approximate PN parameterizations of given canal surfaces and rational offset blends between two canal surfaces.
A unified Pythagorean hodograph approach to the medial axis transform and offset approximation
2011, Journal of Computational and Applied Mathematics
Citation Excerpt :
Indeed, if a part of the medial axis transform of a planar domain is an MPH curve, then the corresponding domain boundary segments and all their offsets possess rational parametrisations. Interpolation and approximation methods based on MPH curves were thoroughly investigated in e.g. [22–27]. Although algorithms based on Pythagorean hodographs in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains, there exist many efficient but separate techniques for Hermite interpolation based on PH and MPH curves.
Algorithms 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.

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C2 Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation

Abstract

Comput. Aided Design

Comput. Aided Geom. Design

J. Comput. Appl. Math.

Comput. Aided Design

Comput. Aided Geom. Design

Comput. Aided Geom. Design

Comput. Aided Geom. Design

Structural and computational advantages of circular boundary representations

Stability and finiteness properties of medial axis and skeleton

J. Dyn. Control Syst.

$C^{2}$ Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation