The fact that the Darboux frame is rotation-minimizing along lines of curvature of a smooth surface is invoked to construct rational surface patches whose boundary curves are lines of curvature. For ...given patch corner points and associated frames defining the surface normals and principal directions, the patch boundaries are constructed as quintic RRMF curves, i.e., spatial Pythagorean-hodograph (PH) curves that possess rational rotation-minimizing frames. The interior of the patch is then defined as a Coons interpolant, matching the boundary curves and their associated rotation-minimizing frames as surface Darboux frames. The surface patches are compatible with the standard rational Bézier/B-spline representations, and
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continuity between adjacent patches is easily achieved. Such patches are advantageous in surface design with more precise control over the surface curvature properties.
Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational ...curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.
We investigate the properties of a special kind of frame, which we call the Euler–Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. It is a frame that can be naturally ...constructed from the PH condition. It turns out that this ERF enjoys some nice properties. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a characterization of PH curves whose ERF achieves rotation-minimizing property. This computation leads into a new fact that this ERF is equivalent to the Frenet frame on cubic PH curves. Furthermore, we prove that the minimum degree of non-planar PH curves whose ERF is an rotation-minimizing frame is seven, and provide a parameterization of the coefficients of those curves.
Pythagorean Element on UFD and PH Curve Shou, Huahao; Jiang, Yu; Song, Congwei ...
Theoretical and Mathematical Foundations of Computer Science
Book Chapter
Recenzirano
Necessary and sufficient conditions are given for Pythagorean elements on unique factorization domains. Sets of easy to use polynomial coefficient based Pythagorean discriminant equations are derived ...for degree 2 polynomials on polynomial rings. Finally, obtained results are applied to degree 3 Pythagorean Hodograph curves in geometric modeling.
Aiming at the problem of online cooperative flight trajectory planning for multiple UAVs, a curvilinear trajectory planning method based on velocity obstacle method is proposed. The three-dimensional ...model of mission planning battlefield is established by using satellite elevation data. An improved Pythagorean Hodograph (PH) curve trajectory planning method is proposed for the problem of curvature constraint in UAV track planning. By introducing the adaptive chaos optimization strategy into the estimation distribution algorithm, an estimation distribution algorithm based on adaptive chaos optimization (AC-EDA) is presented to avoid blind trial and error of curve programming parameters. The improved particle sedimentation method is used to extend the two-dimensional trajectory to three-dimensional space, and the principle of the velocity obstacle method is combined with the PH curve planning method to find a suitable insertion point when facing the threat of a dynamic target, and realize the combination of curved trajectory planning and obstacle avoidance behavior.
This paper aims to solve the problem that when smoothing the UAV trajectory, the common smoothing optimization algorithm will be limited by the UAV flight performance, and it is difficult to ...guarantee the smoothness and curvature continuity of the whole trajectory. In order to make the UAV flight trajectory meet the actual flight requirements, this paper proposes a UAV 3D trajectory smoothing optimization algorithm based on improved PH curves. To solve the problem of curvature discontinuity between a shaped PH curve, this paper utilizes the properties of the first and second order derivatives of two adjacent PH curves. By subjecting the smoothed segment of the PH curve to the overall smoothing constraint, the entire trajectory can satisfy both the performance constraint and the smooth flyability of the trajectory. From the simulation experiment results, it can be seen that using the improved PH curve smoothing to optimize the UAV 3D flight trajectory can effectively reduce the number of turns during the UAV flight, and smooth the UAV flight trajectory.
Les problèmes d'interpolation ont été largement étudiés dans la Conception Géométrique Assistée par Ordinateur. Ces problèmes consistent en la construction de courbes et de surfaces qui passent ...exactement par un ensemble de données. Dans ce cadre, l'objectif principal de cette thèse est de présenter des méthodes d'interpolation de données 2D et 3D au moyen de courbes Algébriques Trigonométriques à Hodographe Pythagorien (ATPH). Celles-ci sont utilisables pour la conception de modèles géométriques dans de nombreuses applications. En particulier, nous nous intéressons à la modélisation géométrique d'objets odontologiques. À cette fin, nous utilisons les courbes spatiales ATPH pour la construction de surfaces développables dans des volumes odontologiques. Initialement, nous considérons la construction de courbes planes ATPH avec continuité C² qui interpolent une séquence ordonnée de points. Nous employons deux méthodes pour résoudre ce problème et trouver la « bonne » solution. Nous étendons les courbes ATPH planes à l'espace tridimensionnel. Cette caractérisation 3D est utilisée pour résoudre le problème d'interpolation Hermite de premier ordre. Nous utilisons ces splines ATPH spatiales C¹ continues pour guider des facettes développables, qui sont déployées à l'intérieur de volumes tomodensitométriques odontologiques, afin de visualiser des informations d'intérêt pour le professionnel de santé. Cette information peut être utile dans l'évaluation clinique, diagnostic et/ou plan de traitement.
Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.
Pythagorean-hodograph (PH) curves admit closed-form expressions for the integral of the square of the curvature with respect to arc length (the “energy” integral) involving only rational functions, ...arctangents, and natural logarithms. In particular, the
complex formulation of PH curves greatly facilitates the derivation of these expressions, yielding compact and efficient implementations in any high-level language that provides complex arithmetic. Explicit formulae are presented for the case of Tschirnhausen's cubic and the regular PH quintics, and in the latter case the use of the energy integral in optimizing the “fairness” of geometric Hermite interpolants is discussed. Compelling empirical evidence indicates that, for “reasonable” derivative data, first-order PH quintic Hermite interpolants are systematically of lower energy than their ordinary cubic counterparts.
We show that Möbius transformations preserve the rotation-minimizing frames which are associated with space curves. In addition, these transformations are known to preserve the class of rational ...Pythagorean-hodograph curves and rational frames. Based on these observations we derive an algorithm for G1 Hermite interpolation by rational Pythagorean-hodograph curves with rational rotation-minimizing frames.
In this paper, a new method for blending two canal surfaces is proposed. The blending surface is itself a generalized canal surface, the spine curve of which is a PH (Pythagorean-Hodograph) curve. ...The blending surface possesses an attractive property—its representation is rational. The method is extensible to blend general surfaces as long as the blending boundaries are well-defined.
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