De Casteljau Research Articles

(q, γ)-Bernstein Basis Functions on the Interval [a ; b

In this paper, we present a novel set of (q, γ)-Bernstein basis functions that are parameterized by q and and defined over an interval. We give detailed proofs for several key properties of these functions, including the partition of unity, recurrence relations, degree elevation, and the Marsden identity. Additionally, we introduce and validate the (q, γ)-De Casteljau algorithm, providing comprehensive examples to illustrate its implementation. These results are analyzed to highlight the theoretical and practical implications of these (q, γ)-Bernstein basis functions in various fields, such as polynomial approximation, numerical methods, and Computer Aided Geometric Design (CAGD). Furthermore, we discuss potential extensions and applications of these functions, considering their impact on future research and developments in the domain. By exploring these aspects, we aim to offer a robust framework for understanding and utilizing (q, γ)-Bernstein basis functions.

A tour d'horizon of de Casteljau's work

Whilst Paul de Casteljau is now famous for his fundamental algorithm of curve and surface approximation, little is known about his other findings. This article offers an insight into his results in geometry, algebra and number theory.Related to geometry, his classical algorithm is reviewed as an index reduction of a polar form. This idea is used to show de Casteljau's algebraic way of smoothing, which long went unnoticed. We will also see an analytic polar form and its use in finding the intersection of two curves. The article summarises unpublished material on metric geometry. It includes theoretical advances, e.g., the 14-point strophoid or a way to link Apollonian circles with confocal conics, and also practical applications such as a recurrence for conjugate mirrors in geometric optics. A view on regular polygons leads to an approximation of their diagonals by golden matrices, a generalisation of the golden ratio.Relevant algebraic findings include matrix quaternions (and anti-quaternions) and their link with Lorentz' equations. De Casteljau generalised the Euclidean algorithm and developed an automated method for approximating the roots of a class of polynomial equations. His contributions to number theory not only include aspects on the sum of four squares as in quaternions, but also a view on a particular sum of three cubes. After a review of a complete quadrilateral in a heptagon and its angles, the paper concludes with a summary of de Casteljau's key achievements.The article contains a comprehensive bibliography of de Casteljau's works, including previously unpublished material.

Hermite subdivision schemes for manifold-valued Hermite data

This paper introduces a family of subdivision schemes that generate curves over manifolds from manifold-Hermite data. This data consists of points and tangent directions sampled from a curve over a manifold. Using a manifold-Hermite average based on the De Casteljau algorithm as our main building block, we show how to adapt a geometric approach for curve approximation over manifold-Hermite data. The paper presents the various definitions and provides several analysis methods for characterizing properties of both the average and the resulting subdivision schemes based on it. Demonstrative figures accompany the paper's presentation and analysis.

De Casteljau's algorithm in geometric data analysis: Theory and application

For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing and discrete geometry, to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology.

An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation

A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm.In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the Bézier basis functions on the manifold SO(3).The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.

A Survey of De Casteljau Algorithms and Regular Iterative Constructions of Bézier Curves with Control Mass Points

Drawing a curve on a computer actually involves approximating it by a set of segments. The De Casteljau algorithm allows to construct these piecewise linear curves which approximate polynomial Bézier curves using convex combinations. However, for rational Bézier curves, the construction no longer admits regular sampling. To solve this problem, we propose a generalization of the De Casteljau algorithm that addresses this issue and is applicable to Bézier curves with mass points (a weighted point or a vector) as control points and using a homographic parameter change dividing the interval [0, 1] into two equal-length intervals [0, 1/2] and [1/2 , 1] . If the initial Bézier curve is in standard form, we obtain two curves in standard form, unless the mass endpoint of the curve is a vector. This homographic parameter change also allows transforming curves defined over an interval [α, +∞], α ∈ R, into Bézier curves, which then enables the use of the De Casteljau algorithm. Some examples are given: three-quart of circle, semicircle and a branch of a hyperbola (degree 2), cubic curve on [0; +∞] and loop of a Descartes Folium (degree 3) and a loop of a Bernouilli Lemniscate (degree 4).

Paul de Casteljau: The story of my adventure: From an autobiographical letter

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.

Subdivision algorithms with modular arithmetic

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 ate2πki/m=cos(2kπ/m)+isin(2kπ/m)↔(cos(2kπ/m),sin(2kπ/m)).Given a sequence of integers (s0,…,sm)mod m, we connect consecutive values sjsj+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 (0,1,…,m)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 2k=±1modm. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2−1modm. 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.

Apollonian de Casteljau–type algorithms for complex rational Bézier curves

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.

Endpoint Geodesic Formulas on Graßmannians Applied to Interpolation Problems

Simple closed formulas for endpoint geodesics on Graßmann manifolds are presented. In addition to realizing the shortest distance between two points, geodesics are also essential tools to generate more sophisticated curves that solve higher order interpolation problems on manifolds. This will be illustrated with the geometric de Casteljau construction offering an excellent alternative to the variational approach which gives rise to Riemannian polynomials and splines.

B/Surf: Interactive Bézier Splines on Surface Meshes.

We present a practical framework to port Bzier curves to surfaces. We support the interactive drawing and editing of Bzier splines on manifold meshes with millions of triangles, by relying on just repeated manifold averages. We show that direct extensions of the de Casteljau and Bernstein evaluation algorithms to the manifold setting are fragile, and prone to discontinuities when control polygons become large. Conversely, approaches based on subdivision are robust and can be implemented efficiently. We implement manifold extensions of the recursive de Casteljau bisection, and an open-uniform Lane-Riesenfeld subdivision scheme. For both schemes, we present algorithms for curve tracing, point evaluation, and approximated point insertion. We run bulk experiments to test our algorithms for robustness and performance, and we compare them with other methods at the state of the art, always achieving correct results and superior performance. For interactive editing, we port all the basic user interface interactions found in 2D tools directly to the mesh. We also support mapping complex SVG drawings to the mesh and their interactive editing.

An evaluation algorithm for [formula omitted]-Bézier triangular patches formed by convex combinations

An extension to triangular domains of the univariate q-Bernstein basis functions is introduced and analysed. Some recurrence relations and properties such as partition of unity and degree elevation are proved for them. It is also proved that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. In addition, it is presented a de Casteljau type evaluation algorithm whose steps are all linear convex combinations.

Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces

In the past few decades polynomial curves with Pythagorean hodograph (PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining, and robotics. This work deals with classes of PH curves built upon exponential-polynomial spaces (EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the Bézier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau--like algorithm and two recently proposed methods: Woźny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong.

Approximation of conic sections by weighted Lupaş post-quantum Bézier curves

Abstract This paper deals with weighted Lupaş post-quantum Bernstein blending functions and Bézier curves constructed with the help of bases via ( p , q ) \left(p,q) -integers. These blending functions form normalized totally positive bases. Due to the rational nature of weighted Lupaş post-quantum Bézier curves and positive weights, they help in investigating from geometric point of view. Their degree elevation properties and de Casteljau algorithm have been studied. It has been shown that quadratic weighted Lupaş post-quantum Bézier curves can represent conic sections in two-dimensional plane. Graphical analysis has been presented to discuss geometric interpretation of weight and conic section representation by weighted Lupaş post-quantum Bézier curves. This new generalized weighted Lupaş post-quantum Bézier curve provides better approximation and flexibility to a particular control point as well as control polygon due to extra parameter p p and q q in comparison to classical rational Bézier curves, Lupaş q q -Bézier curves and weighted Lupaş q q -Bézier curves.

Intrinsic spherical smoothing method based on generalized Bézier curves and sparsity inducing penalization

This study examines an intrinsic penalized smoothing method on the 2-sphere. We propose a method based on the spherical Bézier curves obtained using a generalized de Casteljau algorithm to provide a degree-based regularity constraint to the spherical smoothing problem. A smooth Bézier curve is found by minimizing the least squares criterion under the regularization constraint. The de Casteljau algorithm constructs higher-order Bézier curves in a recursive manner using linear Bézier curves. We introduce a local penalization scheme based on a penalty function that regularizes the velocity differences in consecutive linear Bézier curves. The imposed penalty induces sparsity on the control points so that the proposed method determines the number of control points, or equivalently the order of the Bézier curve, in a data-adaptive way. An efficient Riemannian block coordinate descent algorithm is devised to implement the proposed method. Numerical studies based on real and simulated data are provided to illustrate the performance and properties of the proposed method. The results show that the penalized Bézier curve adapts well to local data trends without compromising overall smoothness.

Generalized B\'{e}zier curves based on Bernstein-Stancu-Chlodowsky type operators

In this paper, we use the blending functions of Bernstein-Stancu-Chlodowsky type operators with shifted knots for construction of modified Chlodowsky B\'{e}zier curves. We study the nature of degree elevation and degree reduction for B\'{e}zier Bernstein-Stancu-Chlodowsky functions with shifted knots for $t \in [\frac{\gamma}{n+\delta},\frac{n+\gamma}{n+\delta}]$. We also present a de Casteljau algorithm to compute Bernstein B\'{e}zier curves with shifted knots. The new curves have some properties similar to B\'{e}zier curves. Furthermore, some fundamental properties for Bernstein B\'{e}zier curves are discussed. Our generalizations show more flexibility in taking the value of $\gamma$ and $\delta$ and advantage in shape control of curves. The shape parameters give more convenience for the curve modelling.

Lupaş post quantum blending functions and Bézier curves over arbitrary intervals

In this paper, we extend the properties of rational Lupa?-Bernstein blending functions, Lupa?-B?zier curves and surfaces over arbitrary compact intervals [?,?] in the frame of post quantum-calculus and derive the de-Casteljau?s algorithm based on post quantum-integers. We construct a two parameter family as Lupa? post quantum Bernstein functions over arbitrary compact intervals and establish their degree elevation and reduction properties. We also discuss some fundamental properties over arbitrary intervals for these curves such as de Casteljau algorithm and degree evaluation properties. Further we construct post quantum Lupa? Bernstein operators over arbitrary compact intervals with the help of rational Lupa?- Bernstein functions. At the end some graphical representations are added to demonstrate consistency of theoretical findings.

Asymptotic Frame Fields of Rational Bézier Curves

Bézier curves are a type of curves used in computer aided design and related fields. The curves can be defined with the help of De Casteljau algorithm, which is one of the most basic elements of curve and surface design, and Bernstein polynomials, which facilitate theoretical developments. A rational Bézier curve may be evaluated by applying the de Casteljau algorithm to both numerator and denominator and finally dividing through. The curves are defined by suitable control points and corresponding scalar weights. In this work, we constitute the asymptotic orthonormal frame field of a spacelike quadratic rational Bézier curve at all points on 2 and 3-dimensional lightlike cones which are degenerate surfaces in Minkowski 3 and 4-spaces. We get the formulas of curvatures for a spacelike quadratic rational Bézier curve 2 and 3-dimensional lightlike cones.

Flexibility-Enhanced Continuous-Time Scheduling of Power System Under Wind Uncertainties

Flexibility helps accommodate uncertainties from the increasing renewables, such as wind power generation (WPG), but limited to scheduling methods, flexibility potentials of the power system have not been fully realized. This paper proposes the mathematical formulation and novel transformation for flexibility-enhanced continuous-time stochastic scheduling (FE-CTSS) of the power system under wind uncertainties. The Bernstein polynomial (BP) spline-based solution space transformation (SST) for solving continuous-time (CT) scheduling is revisited and the potential capacity and ramping flexibility, which are lost during the SST from function space to algebraic space, are then exposed for the first time. Developed from De Casteljau's algorithm (DCA), the enhancement matrix is derived to tighten the convex hull property and the enhanced SST is proposed to alleviate the feasible region narrowing to activate the potential flexibility in FE-CTSS. Case studies in the modified 6-bus system and the IEEE 118-bus system demonstrate the effectiveness and the promising role of FE-CTSS in the future high-renewable-penetrated and high-flexibility-required power system.

Based on the single-layer contour pattern, the comparison of the zigzag parallel hatch and the contour parallel hatch

Laser marking refers to marking on the surface of a product with a laser mark, text, symbol, image, etc. The hatch tool of laser marking machine can be used to hatch specified 2D-compound curve graph. The direction parallel hatch and the contour parallel hatch are two basic ways of hatching. Under the single-layer contour pattern, for the zigzag parallel hatch, we established an improved area Parity Check Filling Model. For the contour parallel hatch, we established an equidistant line generation model based on the Regular Bezier curve, and used the De Casteljau algorithm to generate equidistant lines with the given hatching line spacing. By running the relevant programs, we get the total length of hatch curves and the average elapsed time of the two hatch algorithms. According to the analysis of the result, the performance of the zigzag parallel hatch algorithm is higher than that of the contour parallel hatch algorithm.