The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has not been studied yet in the ...literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G0, cubic G1, quartic G2 and quintic G3 interpolation. Several numerical examples are presented which confirm theoretical results.
•The optimal geometric interpolation of circular arcs is studied.•The Hausdorff distance is used as an error measure.•A detailed analysis of some special cases is done.•A robust algorithm for the construction is provided.
In this paper, the problem of interpolation of two points, two corresponding tangent directions and curvatures, and the arc length sampled from a circular arc (circular arc data) is considered. ...Planar Pythagorean–hodograph (PH) curves of degree seven are used since they possess enough free parameters and are capable of interpolating the arc length in an easy way. A general approach using the complex representation of PH curves is presented first and the strong dependence of the solution on the general data is demonstrated. For circular arc data, a complicated system of nonlinear equations is reduced to a numerical solution of only one algebraic equation of degree 6 and a detailed analysis of the existence of admissible solutions is provided. In the case of several solutions, some criteria for selecting the most appropriate one are described and an asymptotic analysis is given. Numerical examples are included which confirm theoretical results.
•The interpolation of G2 data and an arc lengths is stated in general.•A detailed analysis for circular arc data is provided.•Four real solutions of the problem are confirmed.•An asymptotic analysis together with asymptotic approximation order is given.•A selection of appropriate solution is suggested and everal numerical examples are shown.
Quasi-elastic cubic splines in Rd Johnson, Hakim S.; Johnson, Michael J.
Computer aided geometric design,
August 2020, Volume:
81
Journal Article
Peer reviewed
•We construct a fair parametric cubic interpolating curve through points in Rd (d≥2).•When stencil angles do not exceed arccos13, the resultant spline is G2 and unique.•The angle arccos13 above is ...sharp.•When chord lengths satisfy 12<LjLj−1<2 and stencil angles do not exceed π2, the resultant spline is G2 and unique.
Given points P1,P2,…,Pn in Rd (d≥2), we consider the problem of constructing a fair interpolating curve. For d=2, we proposed and analyzed, in Johnson and Johnson (2016), a method which first generates a family of G1 interpolating curves, where each piece is a parametric cubic. An energy functional, that loosely approximates bending energy, is defined on this family and then one seeks a curve with minimal energy. Such optimal curves, called quasi-elastic cubic splines, always exist and are always G1, but often they are both G2 and unique. In the present article we extend the construction and analysis to d≥2 and prove sufficient a priori conditions (on the interpolation points) for G2 regularity and uniqueness of the quasi-elastic cubic spline. These sufficient conditions constitute significant improvements over those obtained in Johnson and Johnson (2016). For example, we show that if the exterior angles of the data polygon do not exceed the threshold anglearccos(1/3)≈70.5∘, then the quasi-elastic cubic spline is G2 and unique. In contrast, the threshold angle obtained for d=2 in Johnson and Johnson (2016) is only ≈30.5∘. As in Johnson and Johnson (2016), we first develop a framework and then apply it to the particular example of quasi-elastic cubic splines. This framework potentially applies to other minimal energy interpolation methods.
Geometric iterative methods (GIM), including the progressive–iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves ...and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, present the local properties and accelerating techniques, and show their convergence. Moreover, because it is easy to integrate geometric constraints in the iterative procedure, GIM has been widely applied in geometric design and related areas. We survey the successful applications of geometric iterative methods, including applications in geometric design, data fitting, reverse engineering, mesh and NURBS solid generation.
•Several geometric iterative methods are introduced.•Convergence analysis of the geometric iterative methods is presented.•Practical applications of the geometric iterative methods are overviewed.
•Construction of G1 continuous PH cubic biarcs is done.•Thorough analysis of system of nonlinear equations is given.•An algorithm enabling an easy implementation is provided.
The interpolation of two ...points and two tangent directions by planar parametric cubic curves with prescribed arc lengths is considered. It is well known that this problem is highly nonlinear if standard cubic curves are used. However, if Pythagorean-hodograph (PH) curves are considered, the problem simplifies due to their distinguished property that the arc length is a polynomial function of its coefficients. Since a single segment of a PH cubic curve does not provide enough free parameters, the so called PH cubic biarcs are used. A detailed and thorough analysis of the resulting system of nonlinear equations is provided and closed form solutions are given for any possible configuration of given data. The lookup table of the solutions is constructed enabling an easy implementation of the described method. Some quantities arising from geometric properties of the resulting curves are suggested in order to select the most appropriate one and the bending energy is confirmed as the most promising selection criterion. Several numerical examples are presented which confirm theoretical results. Finally, an example of approximation of an analytic curve by G1 PH cubic biarc spline curve is presented and the approximation order is numerically established.
•The optimal geometric approximation of an equilateral spherical triangle.•The optimal geometric approximation of the sphere tiled by equilateral spherical triangles.•A detailed analysis of ...particular cases arising from a tetrahedron, an octahedron and an icosahedron.
A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal approximation of equilateral spherical triangles by parametric polynomial patches if the measure of quality is the (simplified) radial error. As a consequence, optimal approximations of the unit sphere by parametric polynomial spline patches underlying on particular regular spherical triangulations arising from a tetrahedron, an octahedron and an icosahedron inscribed in the unit sphere are provided. Some low total degree spline patches with corresponding geometric smoothness are analyzed in detail and several numerical examples are shown confirming the quality of approximants.
In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. ...They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig–Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.
We propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves. The approach is based on a constrained uniform approximation of an error function by ...scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the Bézier form is derived and a detailed analysis provided for some low degree cases which were not studied yet. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the simplified radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.
In this paper, we introduce latent embedded graphs, a simple approach for shape and image interpolation using generative neural network models. A latent embedded graph is defined as a topological ...structure constructed over a set of lower-dimensional embedding (latent space) of points in a high-dimensional dataset learnt by a generative model. Given two samples in the original dataset, the problem of interpolation can simply be re-formulated as traversing through this embedded graph along the minimal path. This deceptively simple method is based on the fundamental observation that a low-dimensional space induced by a given sample is typically non-Euclidean and in some cases may even represent a multi-manifold. Therefore, simply performing linear interpolation of the encoded data may not necessarily lead to plausible interpolation in the original space. Latent embedded graphs address this issue by capturing the topological structure within the spatial distribution of the data in the latent space, thereby allowing for approximate geodesic computations in a robust and effective manner. We demonstrate our approach through variational autoencoder (VAE) as the method for learning the latent space and generating the topological structure using k-nearest-neighbor graph. We then present a systematic study of our approach by applying it to 2D curves (superformulae), image (Fashion-MNIST), and voxel (ShapeNet) datasets. We further demonstrate that our approach performs better than the linear case in preserving geometric and topological variations during interpolation.
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•Latent Embedded Graph (LEG) is proposed to interpolate images and shapes.•Adaptive k-NN graphs are created in latent space learnt by Variational Autoencoders.•LEG is compared with linear interpolation in latent spacce for images and shapes.•Gradient Projection Metric (GPM) is proposed to evaluate LEG interpolation.
•We extend known results on cubic interpolation for a circular arc of an inner angle ≤π, to an arbitrary circular arc.•We constructed the best quartic parametric polynomial interpolant of a circular ...arc of an arbitrary inner angle.•This is the first optimization of an circular arc with a three-parametric family of interpolants.
The aim of this paper is a construction of quartic parametric polynomial interpolants of a circular arc, where two boundary points of a circular arc are interpolated. For every unit circular arc of an inner angle not greater than 2π we find the best interpolant, where the optimality is measured by the simplified radial error.
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