Computer Aided Geometric Design最新文献

Paul de Faget de Casteljau (19.11.1930 - 24.3.2022) has left us an extensive autobiography, written in 1997. In 19 sections, he takes us through his eventful life which he describes with wit and humor. We read about his youth in occupied France and his education at the Ecole Normale Supérieure. He describes in detail various episodes from his time at Citroën, the situation during and after the discovery of his now famous algorithm, the takeover by Peugeot, his ban from working on CAD and his corporate rehabilitation thanks to his advances in polar forms and quaternions. His memoirs end with his departure from Citroën and his first invited talks at academic conferences.

The paper contains the transcribed French original, its English translation and numerous notes and annotations. The handwritten text is available as a digital supplement.

New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees.

This paper proposes two, geodesic-curvature based, criteria for shape-preserving interpolation on smooth surfaces, the first criterion being of non-local nature, while the second criterion is a local (weaker) version of the first one. These criteria are tested against a family of on-surface $C^2$ splines obtained by composing the parametric representation of the supporting surface with variable-degree (≥3) splines amended with the preimages of the shortest-path geodesic arcs connecting each pair of consecutive interpolation points. After securing that the interpolation problem is well posed, we proceed to investigate the asymptotic behaviour of the proposed on-surface splines as degrees increase. Firstly, it is shown that the local-convexity sub-criterion of the local criterion is satisfied. Second, moving to non-local asymptotics, we prove that, as degrees increase, the interpolant tends uniformly to the spline curve consisting of the shortest-path geodesic arcs. Then, focusing on isometrically parametrized developable surfaces, sufficient conditions are derived, which secure that all criteria of the first (strong) criterion for shape-preserving interpolation are met. Finally, it is proved that, for adequately large degrees, the aforementioned sufficient conditions are satisfied. This permits to build an algorithm that, after a finite number of iterations, provides a $C^2$ shape-preserving interpolant for a given data set on a developable surface.

In this paper, we present an algorithm to compute the intersection between a rational curve and a rational surface. Evaluating the parametric curve into the matrix representation of the parametric surface for implicitization, we get a matrix with one variable. We find the intersection from the matrix with the theory of real root isolation of univariate functions without computing its determinant as we have done in Jia et al. (2022).

We compare our method with the state-of-the-art methods in Gershon (2022); Luu Ba (2014). The given examples show that our algorithms are efficient.

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where $m>2$ is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle at$e^{2\pi \mathrm{ki}/m}=\cos (2k\pi /m)+i\sin (2k\pi /m)\leftrightarrow (\cos (2k\pi /m),\sin (2k\pi /m)).$Given a sequence of integers $\left(s_0,\dots ,s_m\right)$ mod m, we connect consecutive values $s_j{s}_{j+1}$ on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence $\left(0,1,\dots ,m\right)$ mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that $2^k=\pm 1modm$. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue $2^{-1}modm.$ We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.

We revisit the rational cubic Bézier representation of conics, simplifying and expanding previous works, elucidating their connection, and making them more accessible. The key ingredient is the concept of conic associated with a given (planar) cubic Bézier polygon, resulting from an intuitive geometric construction: Take a cubic semicircle, whose control polygon forms a square, and apply the perspective that maps this square to the given polygon. Since cubic conics come from a quadratic version by inserting a base point, this conic admitting the polygon turns out to be unique. Therefore, detecting whether a cubic is a conic boils down to checking out whether it coincides with the conic associated with its control polygon. These two curves coincide if they have the same shape factors (aka, shape invariants) or, equivalently, the same oriented curvatures at the endpoints. Our results hold for any cubic polygon (with no three points collinear), irrespective of its convexity. However, only polygons forming a strictly convex quadrilateral define conics whose cubic form admits positive weights. Also, we provide a geometric interpretation for the added expressive power (over quadratics) that such cubics with positive weights offer. In addition to semiellipses, they encompass elliptical segments with rho-values over the negative unit interval.

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

Given an algebraic surface implicitly defined by an irreducible polynomial, we present a method that decides whether or not this surface can be parametrized by a polynomial parametrization without base points and, in the affirmative case, we show how to compute this parametrization.

We describe a new de Casteljau–type algorithm for complex rational Bézier curves. After proving that these curves exhibit the maximal possible circularity, we construct their points via a de Casteljau–type algorithm over complex numbers. Consequently, the line segments that correspond to convex linear combinations in affine spaces are replaced by circular arcs. In difference to the algorithm of Sánchez-Reyes (2009), the construction of all the points is governed by (generically complex) roots of the denominator, using one of them for each level. Moreover, one of the bi-polar coordinates is fixed at each level, independently of the parameter value. A rational curve of the complex degree n admits generically n! distinct de Casteljau–type algorithms, corresponding to the different orderings of the denominator's roots.

Moving planes have been widely recognized as a potent algebraic tool in various fundamental problems of geometric modeling, including implicitization, intersection computation, singularity calculation, and point inversion of parametric surfaces. For instance, a matrix representation that inherits the key properties of a parametric surface is constructed from a set of moving planes. In this paper, we present an efficient approach to computing such a set of moving planes that follow the given rational parametric surface. Our method is based on the calculation of Dixon resultant matrices, which allows for the computation of moving planes with simpler coefficients, improved efficiency and superior numerical stability when compared to the direct way of solving a linear system of equations for the same purpose. We also demonstrate the performance of our algorithm through experimental examples when applied to implicitization, surface intersection, singularity computation as well as inversion formula computation.