Approximation Of Circular Arcs Research Articles

Circular arc approximation by hexic polynomial curves

Circular arc approximation by hexic polynomial curves

Road-Map Aided Gaussian Mixture Labeled Multi-Bernoulli Filter for Ground Multi- Target Tracking

Ground multi-target tracking (MTT) is one of the core tasks of airborne ground moving target indicator (GMTI) radar and automotive radar. However, ground MTT still remains a challenging issue especially in complex traffic scenarios, since it often suffers from high clutter, dense targets, low visibility, etc. To enhance the tracking performance for ground multiple targets, in this paper, we present a comprehensive solution named road-map aided Gaussian mixture labeled multi-Bernoulli filter (RA-GMLMB) filter, which incorporates road-map information into the labeled multi-Bernoulli (LMB) filter. Specifically, we first propose a hybrid circular arc and line segments approximation approach to extract road-map information, which alleviates approximation errors in the procedure of road-map approximation. Then, we deduce the RA-GMLMB filter by integrating the extracted road-map information into the LMB filter with Gaussian mixture implementation. Simulation experiments are conducted and experimental results show that the proposed RA-GMLMB filter outperforms the state-of-the-art methods.

On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G0, cubic G1, quartic G2 and quintic G3 interpolation. Several numerical examples are presented which confirm theoretical results.

A circular arc approximation algorithm for cucumber classification with image analysis

Abstract The cucumber is one of the most important consumer vegetables, and due to high freshness and appearance standards, it must be graded according to quality. One key metric for grading cucumbers is curvature (arch height) relative to length. To date, this classification standard has not been implemented in a commercial automatic system because the curvature of the cucumber is related to its length. This paper presents and tests a circular arc approximation algorithm for measuring the curvature of a cucumber using image processing. By fitting the central axis of each cucumber to establish a curve equation, the approximate eigenvalue radius, R, and the radius of the section perpendicular to the curvature of cucumber, r, could be obtained quickly and efficiently. The curve of the cucumber was then transformed into a new characteristic, R' (R' = R – r), which was used as a threshold to classify each cucumber. The method was verified by theory and experiment, and the average error of 0.71 % corresponding to the coefficient R2 = 0.9958 of the similarity index between the curve and central axis of the cucumber was verified. It was found that the fitted curve and the central axis of the cucumber were coincident. Moreover, we designed an algorithm that can detect S-shaped cucumbers based on the analysis of the contour, which can avoid the influence of S-shaped cucumbers on the results. In our tests, 148 cucumbers were classified using the circular arc approximation method, the elliptic approximation method, and a manual method. The results of the classifications were compared. According to China's standard evaluation, the classification error rate of the circular arc approximation method was 10.1 % lower than that of the elliptic approximation method. The incidence of cucumbers being classified into an adjacent grade was 0.7 % using the circular approximation method, and there were no cases of cucumbers being classified into non-adjacent grades. The classification error rate of the elliptic approximation was 10.8 %, of which 2.7 % was classified into non-adjacent grades. Under the European standard, the cucumber classification accuracy using the circular arc approximation algorithm reached 100 %. Therefore, the proposed method offers a more accurate classifier than the elliptic approximation method, and the circular arc approximation algorithm can be fully applied in the commercial cucumber classification process.

The best quintic Chebyshev approximation of circular arcs of order ten

<p>Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is of degree $10$ rather than $6$; the Chebyshev error function equioscillates $11$ times rather than $7$; the approximation order is $10$ rather than $6$. The method approximates more than the full circle with Chebyshev uniform error of $1/2^{9}$. The examples show the competence and simplicity of the proposed approximation, and that it can not be improved.</p>

Trajectory generation in 5-axis milling of freeform surfaces using circular arc approximation and its influence in surface roughness

Current methods used to generate toolpath in 5-axis machining, multiply the program lines when the precision is sought, producing a deficient approach to the points and a highly variable feed rates. These factors cause geometric imprecision and poor surface finishes. In this work, a new method for approximating trajectories through circular arcs between selected points has been developed. It has been checked through the machining of an elliptical toroid surface and its results have been compared with those produced by other CAM strategy. The part produced by the new method shows a considerable improvement in the surface roughness.

A general framework for the optimal approximation of circular arcs by parametric polynomial curves

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.

Circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials

The quality of the approximation of circular arcs by parametric polynomials is usually measured by the Hausdorff distance. It is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property. In this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs. A general approach to this problem is considered and corresponding (nonlinear) equations are derived. For the approximants possessing the maximal order of geometric smoothness, a scalar nonlinear equation is analyzed in detail for the parabolic, the cubic and the quartic case. The existence of the admissible solution is confirmed. Moreover, the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved. Theoretical results are confirmed by numerical examples.

Circular arc approximation by quartic H-Bézier curve

The quartic H-Bezier curve is used for the approximation of circular arcs. It has five control points and one positive real free parameter. The four control points are carried out by G^1-approximation constraints and the remaining control point is dividing the line segment joining the second and fourth control points in the ratio 1:2. Optimized value of free parameter α is obtained by minimizing the maximum value of absolute radius error of the recommended approximation scheme. The developed approximation scheme is found considerably better than the existing approximation schemes for these computed values of control points and optimized value of the free parameter.

Approximation of conic section by quartic Bézier curve with endpoints continuity condition

A new method for approximation of conic section by quartic Bezier curve is presented, based on the quartic Bezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G 2 continuous spline approximation of conic section when using the subdivision scheme, and the effectiveness of this method is demonstrated by some numerical examples.

The Best Uniform Cubic Approximation of Circular Arcs with High Accuracy

In this article, the issue of the best uniform cubic approximation of circular arcs with parametrically defined polynomial curves is considered. By a proper choice of the Bezier points, the best uniform approximation of degree 3 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 6; the error function equioscillates 7 times; the approximation order is 6. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfying the properties of the best uniform approximation and yielding the best approximation of least deviation and the highest possible accuracy.

APPROXIMATION ORDER OF C3QUARTIC B-SPLINE APPROXIMATION OF CIRCULAR ARC

In this paper, we present a <TEX>$C^3$</TEX> quartic B-spline approximation of circular arcs. The Hausdorff distance between the <TEX>$C^3$</TEX> quartic B-spline curve and the circular arc is obtained in closed form. Using this error analysis, we show that the approximation order of our approximation method is six. For a given circular arc and error tolerance we find the <TEX>$C^3$</TEX> quartic B-spline curve having the minimum number of control points within the tolerance. The algorithm yielding the <TEX>$C^3$</TEX> quartic B-spline approximation of a circular arc is also presented.

The best uniform quadratic approximation of circular arcs with high accuracy

Abstract In this article, the issue of the best uniform approximation of circular arcs with parametrically defined polynomial curves is considered. The best uniform approximation of degree 2 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 4; the error function equioscillates five times; the approximation order is four. For θ = π/4 arcs (quarter of a circle), the uniform error is 5.5 × 10−3. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfy the properties of the best uniform approximation and yield the highest possible accuracy.

A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

A new approximation method for conic section by quartic Bezier curves is proposed. This method is based on the quartic Bezier approximation of circular arcs. We give the upper bound of Hausdorff distance between the conic section and the quartic Bezier curve, and also show that the approximation order is eight. And we prove that our approximation method has a smaller upper bound than previous quartic Bezier approximation methods. A quartic G2-continuous spline approximation of conic sections is obtained by using the subdivision scheme at the shoulder point of the conic section.

Curvature approximation of circular arcs by low-degree parametric polynomials

AbstractIn this paper some new methods for curvature approximation of circular arcs by low-degree Bézier curves are presented. Interpolation by geometrically continuous (

CIRCLE APPROXIMATION BY QUARTIC G2SPLINE USING ALTERNATION OF ERROR FUNCTION

In this paper we present a method of circular arc approximation by quartic Bezier curve. Our quartic approximation method has a smaller error than previous quartic approximation methods due to the alternation of the error function of our quartic approximation. Our method yields a closed form of error so that subdivision algorithm is available, and curvaturecontinuous quartic spline under the subdivision of circular arc with equal-length until error is less than tolerance. We illustrate our method by some numerical examples.

On polynomial approximation of circular arcs and helices

We present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two-point Taylor expansion. We obtain the degree-(2n+1) polynomial for the approximation problem in an efficient way, which is very convenient to increase the degree of polynomial by adding new terms. An upper bound on the approximation error is available, so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user-prescribed error tolerance.

CV circular-arc ordinary and (multi-) wedge elements for harmonic problems

The computational advantages of the complex variable (CV) are employed to derive recurrence quadrature rules and to develop robust subroutines for circular-arc boundary elements, both ordinary and singular. The elements provide (i) accurate approximation of the geometry of boundaries and contacts and (ii) accurate approximation of physical quantities along an element. The first, geometrical, gain is due to the fact that circular-arc approximation allows one to have a continuous tangent at the common points of neighbouring elements at smooth parts of a curve. The second, density approximation, gain is due to using trigonometric polynomials multiplied when appropriated by functions accounting for the asymptotic behaviour of a physical quantity. We focus on integrals containing the log-type kernels and real density. These integrals are of special need when influxes of heat, fluid, electricity or particle concentration to highly conductive contacts are considered. Other integrals, although involving complications when the density is real, are easily evaluated by properly adjusting the already-known technique. Numerical results illustrate the accuracy, specific features and potential of the developed CV boundary elements.

On Programming for NC Grinding of Grooved Cam with Discretional Plane Curve Profile

In grinding of grooved cam, the programming for NC machining can’t directly be accomplished according to track curve of roller center in working drawing because the diameter of wheel becomes smaller than diameter of cam’s follower roller after wheel dressing. Track curves of wheel center are equidistant curves of the track curve of roller center and they can be gained by equidistant offset. But track curves of wheel center can’t be gained by equidistant offset for grooved cam with discretional plane curve profile, due to the profile including aspheric curves whose equations are usually unknown. To solve above problem, circular arc approximation for the aspheric curve part was realized by using Approximate Double Circular Arc Interpolated Method (ADCAIM), then equidistant offset of circular arcs was carried out by “offset” way, and consequently track curves designed for wheel center of any diameter can expediently be gained on AutoCAD platform. Validation experiment of simulation grinding was fulfilled on a NC milling machine. The experiment results indicate: it is feasible to NC grind grooved cam with discretional plane curve profile using the method introduced in this paper for the example.

Approximation of circular arcs by parametric polynomial curves

In this paper the approximation of circular arcs by parametric polynomial curves is studied. If the angular length of the circular arc is h, a parametric polynomial curve of arbitrary degree \(n \in {\mathbb{N}}\) , which interpolates given arc at a particular point, can be constructed with radial distance bounded by h 2n . This is a generalization of the result obtained by Lyche and Morken for odd n.