Minkowski Pythagorean Hodograph Research Articles

On null Cartan normal isophotic and normal silhouette curves on a timelike surface in Minkowski 3‐space

We introduce generalized Darboux frames along a null Cartan curve lying on a timelike surface in Minkowski space and define null Cartan normal isophotic and normal silhouette curves in terms of the vector field that lies in the normal plane of the curve and belongs to its generalized Darboux frame of the first kind. We investigate null Cartan normal isophotic and normal silhouette curves with constant geodesic curvature and constant geodesic torsion . We obtain the parameter equations of their axes and prove that such curves are the null Cartan helices or the null Cartan cubics. In particular, we show that null Cartan normal isophotic curves with a non‐zero constant curvatures and have a remarkable property that they are general helices, relatively normal‐slant helices and isophotic curves with respect to the same axis. We prove that null Cartan cubics lying on a timelike surface are normal isophotic curves with a spacelike axis and normal silhouette curves with a lightlike axis. We obtain the relation between Minkowski Pythagorean hodograph cubic curves and null Cartan normal isophotic and normal silhouette curves. Finally, we give numerical examples of null Cartan normal isophotic and normal silhouette curves obtained by integrating the system of two the first order differential equations under the initial conditions.

Double and Type (3,0) Minkowski Pythagorean Hodograph Curves

In present paper, Double Minkowski Pythagorean Hodograph (DMPH) curves and type (3,0) Minkowski Pythagorean Hodograph (MPH) curves. Firstly, we obtained the conditions for a MPH curve to be a DMPH curve. Then, we examined these conditions in split quaternion form. Finally, a special class of seventh degree MPH curves is characterized and illustrative examples are given.

Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the R2,1 Minkowski space. Extending the class of MPH curves, a new class of Rational Envelope (RE) curve has been introduced. These are special curves in R2,1 that define rational boundaries for the corresponding domain. A method to use RE and MPH curves for skinning purposes, i.e., for circle-based modeling, has been developed recently. In this paper, we continue this study by proposing a new, more flexible way how these curves can be used for skinning a discrete set of circles. We give a thorough overview of our algorithm, and we show a significant advantage of using RE and MPH curves for skinning purposes: as opposed to traditional skinning methods, unintended intersections can be detected and eliminated efficiently.

Interpolation of Hermite data by clamped Minkowski Pythagorean hodograph B-spline curves

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 C1∕C2 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 Cℓ2,1. 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.

Applying Rational Envelope curves for skinning purposes

Special curves in the Minkowski space such as Minkowski Pythagorean hodograph curves play an important role in computer-aided geometric design, and their usages are thoroughly studied in recent years. Bizzarri et al. (2016) introduced the class of Rational Envelope (RE) curves, and an interpolation method for G1 Hermite data was presented, where the resulting RE curve yielded a rational boundary for the represented domain. We now propose a new application area for RE curves: skinning of a discrete set of input circles. We show that if we do not choose the Hermite data correctly for interpolation, then the resulting RE curves are not suitable for skinning. We introduce a novel approach so that the obtained envelope curves touch each circle at previously defined points of contact. Thus, we overcome those problematic scenarios in which the location of touching points would not be appropriate for skinning purposes. A significant advantage of our proposed method lies in the efficiency of trimming offsets of boundaries, which is highly beneficial in computer numerical control machining.

Construction of Minkowski Pythagorean hodograph B-spline curves

Following and extending the recent results of Albrecht et al. (2017) for planar Pythagorean hodograph (PH) B-spline curves to the Minkowski 3-space, we introduce a class of Minkowski Pythagorean hodograph (MPH) B-spline curves. The distinguished property of these curves is that the Minkowski norm of their hodograph is a B-spline function. We focus mainly on the clamped case and using Clifford algebra representation we present formulas for their construction. The closed case is also mentioned. Then we solve two practical problems – construction of MPH B-spline curves with control polygon close to a given control polygon, and construction of MPH B-spline curves going through given points. We emphasize symbolic solutions wherever it is possible. The results and approaches are illustrated on several examples.

Linear computational approach to interpolations with polynomial Minkowski Pythagorean hodograph curves

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

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 R2,1, 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 C1 Hermite data-sets with admissible velocity vectors on their corresponding surfaces.

Medial axis transforms yielding rational envelopes

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.

On rational Minkowski Pythagorean hodograph curves

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorean hodograph curve can be obtained in terms of its associated planar rational Pythagorean hodograph curve and an additional rational function. Moreover, both in the original polynomial and new rational case, we investigate the close relationship between these associated curves in Euclidean plane and Minkowski space.

Formula omitted] Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation

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.

Cubic Helices in Minkowski Space

Inhalt: W. LARCHER: Altitudinal Variation in Flowering Time of Lilac (Syringa vulgaris L.) in the Alps in Relation to Temperatures; F. SCHWEIGER: A Note on the Eigenvalues for Periodic Three-Dimensional Jacobi-Perron Algorithms; J. KOSINKA – B. JÜTTLER: Cubic Helices in Minkowski Space; E. HLAWKA: Pythagoräische Tripel höherer Ordnung; B.J. GRUBER: Mathematical-Physical Properties of Musical Tone Systems II: Applications; A. HOFINGER: The Metrics of Prokhorov and Ky Fan for Assessing Uncertainty in Inverse Problems; W. JabŁoŃski – L. REICH: Stability of the Translation Equation in Rings of Formal Power Series and Partial Extensibility of One-Parameter Groups of Truncated Formal Power Series; J. CIGLER: Remarks on Some Sequences of Binomial Sums; B. ODEHNAL: Die Linienelemente des P3; G. LARCHER – F. PILLICHSHAMMER: An Extremum Problem for Convex Polygons; C. HEUBERGER – A. PETHŐ – R.F. TICHY: Thomas' Family of Thue Equations over Imaginary Quadratic Fields, II; S.J. BAUER: Electron Density Distribution in the Topside Ionosphere of Mars and Associated Atmospheric Parameters

C 1 Hermite interpolation by Pythagorean hodograph quintics in Minkowski space

Curves in the Minkowski space $\mathbb{R}^{2,1} $ are very well suited to describe the medial axis transform (MAT) of planar domains. Among them, Minkowski Pythagorean hodograph (MPH) curves correspond to domains where both the boundaries and their offsets admit rational parameterizations (Choi et al., Comput Aided Design 31:59–72, 1999; Moon, Comput Aided Geom Design 16:739–753; 1999). We construct MPH quintics which interpolate two points with associated first derivative vectors and analyze the properties of the system of solutions, including the approximation order of the ‘best’ interpolant.

G1 Hermite interpolation by Minkowski Pythagorean hodograph cubics

As observed by [Choi, H.I., Han, Ch.Y., Moon, H.P., Roh, K.H., Wee, N.S., 1999. Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves. Computer-Aided Design 31, 59-72], ...

Formula omitted] Hermite interpolation by Minkowski Pythagorean hodograph cubics

As observed by [Choi, H.I., Han, Ch.Y., Moon, H.P., Roh, K.H., Wee, N.S., 1999. Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves. Computer-Aided Design 31, 59–72], curves in Minkowski space R 2 , 1 are very well suited to describe the medial axis transform (MAT) of a planar domain, and Minkowski Pythagorean hodograph (MPH) curves correspond to domains, where both the boundaries and their offsets are rational curves [Moon, H.P., 1999. Minkowski Pythagorean hodographs. Computer Aided Geometric Design 16, 739–753]. Based on these earlier results, we give a thorough discussion of G 1 Hermite interpolation by MPH cubics, focusing on solvability and approximation order. Among other results, it is shown that any analytic space-like curve without isolated inflections can be approximately converted into a G 1 spline curve composed of MPH cubics with the approximation order being equal to four. The theoretical results are illustrated by several examples. In addition, we show how the curvature of a curve in Minkowski space is related to the boundaries of the associated planar domain.

Euclidean and Minkowski Pythagorean hodograph curves over planar cubics

Starting with a given planar cubic curve [ x ( t ) , y ( t ) ] T , we construct Pythagorean hodograph (PH) space curves of the form [ x ( t ) , y ( t ) , z ( t ) ] T in Euclidean and in Minkowski space, which interpolate the tangent vector at a given point. We prove the existence of these curves for any regular planar cubic and we express all solutions explicitly. It is shown that the constructed curves provide upper and lower polynomial bounds on the parametrical speed and the arc-length function of the given cubic. We analyze the approximation order and derive an explicit formula for the gap between the bounds. In addition, we discuss the approximation of the offset curves. Finally we define an invariant which measures the deviation of a given planar cubic from being a PH curve.

C1 Hermite interpolation using MPH quartic

In this paper, we study the C 1 Hermite interpolation problem using Minkowski Pythagorean Hodograph (MPH) quartics in R 2,1 . As a preliminary step, we characterize MPH curves in R 2,1 by the roots of the hodographs of their complexified spine curves. We present two schemes for this interpolation problem: one is a subdivision scheme using direct C 1 interpolation and the other is a two step scheme using a new concept, C 1/2 interpolation.

Minkowski Pythagorean hodographs

We introduce the Minkowski Pythagorean hodograph (MPH) curve as a polynomial curve whose speed measured under the Minkowski metric is polynomial. It is a generalization of the Pythagorean hodograph (PH) curve. The MPH curve is well adapted to the representation of the medial axis transform of a planar domain. In fact, if the smooth curve segment of the medial axis transform is written in the MPH form, the boundaries of the corresponding domain are easily computed as rational curves of the MPH curve parameter. Furthermore, just subtracting a constant value from the radius, the offset curves can be obtained as rational curves. We also give the characterization of MPH curves which is invariant under the Lorentz transform.

Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves

We present a new approach to medial axis transform and offset curve computation. Our algorithm is based on the domain decomposition scheme which reduces a complicated domain into a union of simple subdomains each of which is very easy to handle. This domain decomposition approach gives rise to the decomposition of the corresponding medial axis transform which is regarded as a geometric graph in the three dimensional Minkowski space R 2,1. Each simple piece of the domain, called the fundamental domain, corresponds to a space-like curve in R 2,1. Then using the new spline, called the Minkowski Pythagorean hodograph curve which was recently introduced, we approximate within the desired degree of accuracy the curve part of the medial axis transform with a G 1 cubic spline of Minkowski Pythagorean hodograph. This curve has the property of enabling us to write all offset curves as rational curves. Further, this Minkowski Pythagorean hodograph curve representation together with the domain decomposition lemma makes the trimming process essentially trivial. We give a simple procedure to obtain the trimmed offset curves in terms of the radius function of the MPH curve representing the medial axis transform.