Bezier Segments Research Articles

Конструктивные алгоритмы формирования составных кубических кривых Безье в пространстве и на плоскости

The article considers two graph-analytical algorithms. The first algorithm makes it possible to form a spatial curve passing through given nodal points with direction vectors specified at these points. The second algorithm allows you to generate a flat compound curve passing through the given nodal points with direction vectors and curvature radii specified at these points. The constructed curves are formed by cubic Bezier segments. Direction vectors are treated as a control for the shape of the curve being constructed. At the nodal points, second-order geometric smoothness is ensured, due to the continuity of the slope and curvature. A distinctive feature of the proposed algorithms is the significant use of constructive means of computer graphics.

Конструирование G2-гладкой составной кривой на основе кубических сегментов безье

The theory and practice of forming composite G2-smooth (two-continuously differentiable) curves, used in technical design since the mid-60s of the 20th century, is still not reflected in any way in the curriculum of technical universities or in Russian textbooks in engineering and computer graphics. Meanwhile, such curves are used in modeling a wide variety of geometric objects and physical processes.
 The article deals with the problem of constructing a composite G2-smooth curve passing through given points and touching at these points pre-specified straight lines. To solve the problem, cubic Bezier segments are used. The main problem in constructing a smooth compound curve is to ensure the continuity of curvature at the joints of the segments. The article shows that for parametrized cubic curves, this problem is reduced to solving a quadratic equation. A software module has been compiled that allows one to construct a plane G2-smooth curve passing through predetermined points and tangent at these points with predetermined straight lines. The shape of the curve (“completeness” of its segments) is adjusted by the user in the dialog mode of the program module.
 Solved the problem of constructing a cubic curve smoothly connecting unconnected Bezier segments. An algorithm for constructing a Bezier segment with given tangents and given curvature at its boundary points is proposed.
 Some properties of the cubic Bezier segment are considered. In particular, it was shown that for the case of parallel tangents, the curvature at the end of a segment is determined by the position of only one control point (Theorem 1). Cases are considered when the curvature at the ends of the Bezier segment is equal to zero (Theorem 2).
 An approximation of a three-point physical spline is performed using Bezier segments. The approximation error was less than 2%, which is comparable to the error in processing the experimental data.
 A method is proposed for modeling a spatial G2-smooth curve passing through points set in advance in space and touching at these points arbitrarily oriented lines in space.
 The article is of an educational nature and is intended for an in-depth study of the basics of computational geometry and computer graphics.

Cyclographic images of Bezier Splines

In the present paper the analysis of impact of smoothness of connection of spatial curve segments on smoothness of connection of its cyclographic images constituting a cyclographic projection of the curve is presented. The spatial curve is represented by a Bezier spline consisting of Bezier segments of constant order connected with a certain order of smoothness. Formation of Bezier splines consisting of segments of equal order (from two to five) and formation of respective cyclographic projections is considered. Reduction of order of smoothness of connection of segments of cyclographic projection with respect to the order of smoothness of connection of corresponding segments of the Bezier spline is discovered. Numerical experiments were conducted, their results confirm the theoretical conclusion. Recommendations on Bezier spline selection in practical tasks based on connection of segments of cyclographic projection of spline with certain order of smoothness are given.

Aspheric surface lens for LED collimating illumination with low Fresnel loss

An aspheric surface lens is presented to realize collimating illumination with low Fresnel loss based on Fresnel equations and Snell’s law. The smooth 2D contour of refractive optical surface is constructed from a set of cubic Bezier segments, whose control points are computed by deCasteljau algorithm. Simulation results show that the optical efficiency of 90.82% is achieved under a divergence angle of ±2.87° for an extended light-emitting diode (LED) source with chip size of 1 mm × 1 mm and the Fresnel loss is only 8.76%, whose optical efficiency has improved 14.3% than traditional collimating lens. By employing this proposed surface construction method, the largest divergence angle of collimating lens for point source is only 0.26° with 15 feature points on each refractive surface, while more than 2° for the traditional method. Therefore, the beams are well controlled with fewer feature data points. Tolerance analyses are also conducted in detail.

A realtime curvature-smooth interpolation scheme and motion planning for CNC machining of short line segments

G01 codes generated by CAM (Computer Aided Manufacturing) system are the most common form of tool path in CNC (Computer Numerical Control) machining. For the piecewise linear path, tangential and curvature discontinuities bring about large fluctuation of feedrate and acceleration, which produces vibration of machine tool. In recent studies, the methods for G2 (curvature-continuous) tool-path smoothing and jerk-limiting feedrate scheduling were developed. However there still exist the deficiencies when these methods are employed in CNC machining. It is difficult to simultaneously ensure that the tool path is chord-error-constrained and G01-point-interpolated in real time. In addition, heavy computational load hinders realtime processing in CNC system. Recently the scholars experimentally found the potential of G3 (curvature-smooth) trajectory and jerk-continuous motion in reducing the vibration of machinery. This work proposes a realtime tool-path smoothing algorithm, generating G3 interpolative tool path composed by mixed linear and quartic Bezier segments. The purpose of the smoothing scheme is the simultaneous considerations of G3 continuity, confined chord error, G01 points interpolated, and realtime performance. And the tool path generated is optimized in curvature variation energy (CVE) and analytical curvature extrema is available. To reduce the vibration, a high-efficient algorithm of jerk-continuous (JC) feedrate scheduling for G3 tool path is provided. Finally, a realtime tool-path processing scheme is developed, including G3 interpolation and motion planning functions. As shown in the simulation, the contour error, curvature of tool path, feedrate fluctuation and machining time are reduced compared with G2 transition scheme. The experiment on a machine tool is conducted to demonstrate the advantages of the proposed algorithm in vibration reduction and surface quality, compared with G2 transition scheme.

An Algorithm for Representing Planar Curves in B-Splines

An algorithm for representing planar curves in B-splines is presented in this paper. The representing problem is different from the approximation to data points; planar curve provided more information than data points. To make full use of the information, we propose a three-step representing approach: 1.Sample data points along with their tangent vectors from the planar curve according to the given accuracy. 2. Fit the sampled points by Bezier segments using local interpolation; compose these segments to an interpolation curve. 3. Approximate the interpolation curve using the best least approximation to get the final B-spline curve. Tangent information is used in the second step to construct the interpolation curve. In the third step, the system is always positive because of using the best least square approximation, so we can get more freedoms to approximate the interpolation curve. Finally, some examples of this algorithm demonstrate its usefulness and quality.

B-Spline Curve Approximation with Nearly Arc-Length Parameterization

An algorithm of B-spline curve approximation with the three-dimensional data is presented in this paper. In this algorithm, we will get a smooth curve which is nearly arc-length parameterization. The smoothness and uniform parameterization are key factors of the approximating curve, specifically in skinning surface and surface approximation. Firstly, the data points are fitted using local interpolation, this local fitting algorithm yields n Bezier segments, each segment having speed equal to 1 at their end and midpoints. Then segments are composed of a C1 continuous cubic B-spline curve which named controlling curve. But the controlling curves control points are redundancy, so we find another curve to approximate the controlling curve using least square approximation with smoothness

Curve watermarking technique for protecting copyright of digital maps

In order to protect the copyright of digital maps, a curve-based watermarking technique for fingerprinting digital maps was proposed. First the embeddable sample points were selected according to their curvatures, and then the spread spectrum based watermark sequence (also called fingerprint sequence) was embedded into the coordinates of the embeddable sample points. To achieve high quality watermarked curve and robust watermark, the Bezier segments were used to reconstruct the watermarked curve. Our watermarked curve suffers smaller embedding distortions and is robust against common geometrical distortions (e.g., translation, rotation, and scaling), collusions, and printing-scaling attacks.

Hermite interpolation of space curves using the symmetric algebra

Conditions are presented which describe the construction of a degree 5 Bezier curve which interpolates given positional, tangent, curvature, and torsion data at end points. The result is extended to provide technique for interpolation of a given curve by a sequence of Bezier segments that interpolate positional, tangent, curvature, and torsion data associated with intermediate points on the curve. It is shown that the order of approximation is O(h8), 0 < h < 1. This corroborates a conjecture of Hollig and Koch. Techniques which simplify calculation are used. The techniques involve the consideration of a class of fundamental Bezier curves which lie in the graded symmetric algebra associated with R2.

Application of BEM and sensitivity analysis to the solution of the governing diffusion–convection equation for a continuous casting process

This paper presents a two-dimensional diffusion–convection boundary problem which models a continuous casting process of a pure substance (e.g. copper). The boundary problem is formulated for both liquid and solid subdomains separated by a phase change front. Its location is unknown and has to be updated during the iteration process utilising the so-called front tracking algorithm. The solution procedure is based on the Boundary Element Method and involves the total derivative of the temperature and relevant sensitivity coefficients to find a correct position of the phase change front. In order to considerably reduce the number of degrees of freedom in the front tracking algorithm, the position and shape of the phase change front is interpolated by a small number of Bezier segments. Numerical examples are included to demonstrate the main features of the developed algorithms.

Darstellung von Bézier-Kurven als B-Spline-Kurven

In this paper we determine the coefficients and knots of a B-spline curve, which has the same shape as a given segmented Bezier curve. The order of continuity in different joins of Bezier segments may be different.