Casteljau Algorithm Research Articles

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.

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.

H-Bernstein basis functions over a triangular domain

In this paper, we introduce and study new h-Bernstein basis functions over a triangular domain. In particular, after defining the h-Bernstein polynomial functions of degree n, we prove their algebraic and geometric properties, such as partition of unity and degree elevation and we show that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. Then, we propose the h-de Casteljau algorithm and we prove the Marsden identity.

Defining a curve as a Bezier curve

A Bezier curve is significant with its control points. When control points are given, the Bezier curve can be written using De Casteljau's algorithm. An important property of Bezier curve is that every coordinate function is a polynomial. Suppose that a curve is a curve which coordinate functions are polynomial. Can we find points that make the curve as Bezier curve? This article presents a method for finding points which present as a Bezier curve.

A Numerical Algorithm for $C^2$-Splines on Symmetric Spaces

Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, require the solution of a coupled set of non-linear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De~Casteljau's algorithm, which leads to generalized B\'ezier curves. To construct C2-splines from such curves is a complicated non-linear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel, thus suitable for multi-core implementation. We demonstrate the algorithm for three geometries of interest: the $n$-sphere, complex projective space, and the real Grassmannian.

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

AbstractWe introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ℝn to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.

Computing the minimum distance between a point and a curve on mesh

Computation of the distance between two objects is an important problem in computer aided geometric design. To make up the deficiencies in the existing algorithm, a calculation method of the closet distance between point and curve on mesh surfaces is proposed. The curves on surfaces are represented by geodesic B-spline, and the knot insertion algorithm in classic B-spline curve is expand to curved space, so that the geodesic B-spline curve can be decomposed into combination of sub-Bezier curves; using the intermediate results of the expand De Casteljau's algorithm, derivative of curve can be calculated. On this basis, the orthogonal projection point calculation algorithm, which is the projection point from point to curve, in Euclidean space is extended to curved space, and the point calculation method of the orthogonal projection on curved space is given, that is to say, the geodesic distance between point and its orthogonal projection point is the shortest distance from point to curve. Experimental results show that, the proposed method is robust, effective, and with a good adaptability to the shape variation of surface.

Theoretical p-y Curves for Laterally Loaded Single Piles in Undrained Clay Using Bezier Curves

An alternative method was introduced for predicting the nonlinear p-y curves for monotonic unidirectional laterally loaded single piles in uniform undrained clay. On the basis of numerical studies, closed-form solutions were developed for locating the start of yield (y e ); the ultimate yield point (y u ); and the initial stiffness, K i of the p-y curve. The nonlinear section of the curve between the start of the yield and the ultimate yield point was represented by Bezier polynomials (also known as de Casteljau's algorithm). Using these relationships, a direct method of constructing the p-y curves was presented considering either tension failure or no tension failure of soils. For a typical pile configuration, the resulting load-deflection response was observed to compare favorably with the predictions from FLAC analysis and Matlock.

Hammerstein model identification algorithm using Bezier–Bernstein approximation

A new identification algorithm is introduced for the Hammerstein model consisting of a nonlinear static function followed by a linear dynamical model. The nonlinear static function is characterised by using the Bezier-Bernstein approximation. The identification method is based on a hybrid scheme including the applications of the inverse of de Casteljau's algorithm, the least squares algorithm and the Gauss-Newton algorithm subject to constraints. The related work and the extension of the proposed algorithm to multi-input multi-output systems are discussed. Numerical examples including systems with some hard nonlinearities are used to illustrate the efficacy of the proposed approach through comparisons with other approaches.

Finding the point on Bezier curves with the normal vector passing an external point

Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau's algorithm for non-rational Bezier curve or Farin's recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov's method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Rational Wang?Ball curves

Although rational Bézier curves have been commonly used as a base model for curve modelling, rational Wang?Ball curves have been found to be more efficient. The present study introduces rational Wang?Ball curves which are derived from non-rational Wang?Ball curves. It is shown that a rational Wang?Ball curve is a rational Bézier curve of the same degree and vice versa. The relationships between two rational curves established in this paper, allow conversion from one curve to the other. A recursive algorithm for plotting rational Bézier and Wang?Ball curves is obtained from the de Casteljau and Wang algorithm using homogeneous coordinates. An efficient algorithm for evaluating a rational Wang?Ball curve is attained and has a linear time complexity. Finally, another algorithm for rational Bézier curves which has linear time computation is also devised.

De Casteljau's algorithm is an extrapolation method

One of the most important recursive schemes in CAGD is De Casteljau's algorithm for the evaluation of Bézier curves and surfaces. Within the theory of triangular recursive schemes we discuss the De Casteljau's algorithm as a particular case, i.e. we prove that it is identical to the E-algorithm (or GNA-algorithm) in a particular frame. This result is of theoretical interest since it leads to some classification of recurrence relations in CAGD. Furthermore, it may be regarded as a model example how to obtain known and possibly new recursive schemes in CAGD as examples of the theory of general extrapolation algorithms.

An affine representation of de Casteljau's and de Boor's rational algorithms

The well-known algorithm of de Casteljau was already known in 1870 by I.C.F. Haase. Haase's idea is strongly related to an affine representation of de Casteljau's algorithm for rational curves and surfaces, and even to de Boor's algorithm for rational splines.

A multisided generalization of Bézier surfaces

In this paper we introduce a class of surface patch representations, called S-patches, that unify and generalize triangular and tensor product Bézier surfaces by allowing patches to be defined over any convex polygonal domain; hence, S-patches may have any number of boundary curves. Other properties of S-patches are geometrically meaningful control points, separate control over positions and derivatives along boundary curves, and a geometric construction algorithm based on de Casteljau's algorithm. Of special interest are the regular S-patches, that is, S-patches defined on regular domain polygons. Also presented is an algorithm for smoothly joining together these surfaces with C k continuity.

Uniform refinement of curves

We propose and analyze a class of algorithms for the generation of curves and surfaces. These algorithms encompass some well-known methods of subdivision for Bernstein-Bézier curves (de Casteljau's algorithm) and B-spline curves (Lane and Riesenfeld's algorithm). Several results concerning properties of the limiting curves as well as related questions are discussed.

Computing curves invariant under halving

We consider functions that are produced by subdivision algorithms which have the same structure as de Casteljau's algorithm, Chaikin's algorithm and Lane/Riesenfeld's algorithm. Results from a forthcoming paper concerning the differentiability and polynomial structure of the functions produced by these algorithms will be described here.