De Casteljau Algorithm 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.

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.

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).

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.

A RATIONAL PARAMETRIZATION OF BÉZIER LIKE CURVES

In this paper, we construct a family of Bernstein functions using a class of rational parametrization. The new family of rational Bernstein basis on an index $\alpha \in {\left(-\infty \, , \, 0 \right)}\cup {\left(1 \, , \, +\infty\right)}$, and for a given degree $k\in \mathbb{N}^*$, these basis functions are rational with a numerator and a denominator are polynomials of degree k. All of the classical properties as positivity, partition of unity are hold for these rational Bernstein basis and they constitute approximation basis functions for continuous functions spaces. The B\'ezier curves obtained verify the classical properties and we have the classical computational algorithms like the deCasteljau Algorithm and the algorithm of subdivision with the similar accuracy. Given a degree k and a control polygon points all of these algorithms converge to the same B\'ezier curve as the classical case. That means the B\'ezier curve is independent of the index $\alpha$. The classical polynomial Bernstein basis seems a asymptotic case of our new class of rational Bernstein basis.

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.

Design of Novel Mobility and Obstacle-Aware Algorithm for Optimal MANET Routing

Existence obstacles in terrain area and the mobility of the nodes in the network are the vital constrains that degrade the performance in Mobile Adhoc Network (MANET). To diminish the effect of such obstacles and mobility on the performance of MANET, the protocol used for routing must be efficient enough to handle both the factors. In the present work, we have considered the issues of Mobility and obstacle aware multipath communication in MANET. For avoiding obstacles, DeCasteljau Algorithm with Bezier curve concept has been applied. To the minimize impact of mobility, a speed based mobility prediction concept is used. Here optimal routing path across the obstacle is selected depending on its mobility prediction indicator, link duration, network connectivity and path availability. The performance of the suggested protocol is evaluated with Network Simulator-2. Simulation results confirm that the suggested protocol minimizes average energy consumption, network overhead, average delay and improves data deliverance in MANET.

Compensated Evaluation of Tensor Product Surfaces in CAGD

In computer-aided geometric design, a polynomial surface is usually represented in Bézier form. The usual form of evaluating such a surface is by using an extension of the de Casteljau algorithm. Using error-free transformations, a compensated version of this algorithm is presented, which improves the usual algorithm in terms of accuracy. A forward error analysis illustrating this fact is developed.

De Casteljau Algorithm and Degree Elevation of Toric Surface Patches

De Casteljau algorithm and degree elevation of Bezier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation of toric surface patches, which include tensor product and triangular rational Bezier surfaces as special cases. Some representative examples of toric surface patches with common shapes are illustrated to verify these two algorithms. Moreover, the authors also apply the degree elevation of toric surface patches to isogeometric analysis. And two more examples show the effectiveness of proposed method.

Lupa\u015f blending functions with shifted knots and q-B\xe9zier curves

In this paper, we introduce blending functions of Lupaş q-Bernstein operators with shifted knots for constructing q-Bézier curves and surfaces. We study the nature of degree elevation and degree reduction for Lupaş q-Bézier Bernstein functions with shifted knots for t in [frac{a}{[mu ]_{q}+b} , frac{[mu ]_{q}+a}{[mu ]_{q}+b} ]. For the parameters a=b=0, we get Lupaş q-Bézier curves defined on [0,1]. We show that Lupaş q-Bernstein functions with shifted knots are tangent to fore-and-aft of its polygon at end points. We present a de Casteljau algorithm to compute Bernstein Bézier curves and surfaces with shifted knots. The new curves have some properties similar to q-Bézier curves. Similarly, we discuss the properties of the tensor product for Lupaş q-Bézier surfaces with shifted knots over the rectangular domain.

Nonlinear Weighted Average and Blossoming

In this paper, we introduce a new averaging rule, the nonlinear weighted averaging rule. As an application, this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with this scheme we can also generate transcendental functions which cannot be generated by the classical de Casteljau algorithm. We also investigate the properties of the curves of the functions generated by blossoming, where the results show that these curves and the classical Bezier curves have some similar properties, including variation diminishing property and endpoint interpolation. However, the curves obtained by blossoming using nonlinear weighted averaging rules induced by certain functions violate some properties like convex hull property.

The cubic de Casteljau construction and Riemannian cubics

The classical de Casteljau construction of cubic polynomials in Euclidean space Em has been generalised to Riemannian settings since the 1980's. The generalisations replace line segments by geodesic arcs, yielding elegant methods for Hermite interpolation in Riemannian manifolds M, such as spheres and rotation groups. Unlike the classical algorithm however, it is not so easy to analyse the resulting curves. Even when M is the unit m-sphere Sm in Em+1, the mathematical properties of generalised cubic de Casteljau curves are not well understood.There is another class of curves called Riemannian cubics which can also be used for Hermite interpolation in Riemannian manifolds. Unlike generalised cubic de Casteljau curves, Riemannian cubics are defined as critical curves for a variational problem, and their mathematical properties are much better understood. Riemannian cubics are also more difficult to construct, whereas generalised cubic de Casteljau curves have a simple geometrical construction.It is well-known that, when M is curved, generalised cubic de Casteljau curves and Riemannian cubics are different. On the other hand their general appearance is somewhat similar. In the classical situation where M=Em, both cubic de Casteljau curves and Riemannian cubics reduce to curves that are cubic polynomial in each coordinate.The present paper analyses the differences between generalised cubic de Casteljau curves and Riemannian cubics. We also modify the generalised cubic de Casteljau algorithm to yield curves that are much closer to Riemannian cubics. In this way the elegant geometry of the de Casteljau algorithm is adjusted to better approximate a class of curves that are better understood from a mathematical point of view. Examples are given of the modified cubic de Casteljau algorithm for curves in the 2-dimensional unit sphere S2⊂E3, and in the special orthogonal group SO(3) with bi-invariant Riemannian metric.

Obstacle and mobility aware optimal routing for manet

In Mobile Adhoc Network (MANET), obstacles in terrain and nodes mobility are main constrains due which performance degrades. To mitigate these effects, the routing protocol should be mobility and obstacles aware. In this work, we proposed a novel Obstacle and Mobility Aware Routing (OMAR) protocol . In OMAR, for obstacles avoidance, DeCasteljau Algorithm based on Bezier curve is been used. And to reduce effects of mobility and energy shortage of a node, an Energy based Mobility Index (EMI) routing scheme is been developed. The path possesses high EMI is selected as route. The performance of proposed Mobility and Obstacle Aware Routing protocol is evaluated using NS2 simulator. Simulation results show that the proposed algorithm reduces energy consumption, overhead, delay and increases data delivery in the network.

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.