Optimal biarc-curve fitting

doi:10.1016/0010-4485(91)90009-L

Computer-Aided Design

Volume 23, Issue 6, July–August 1991, Pages 411-419

https://doi.org/10.1016/0010-4485(91)90009-L Get rights and content

Abstract

The determination of the optimum biarc curve through a given set of points with given end gradients is considered. The method adopted is that of finding the optimum gradients at all the given points such that the integral of the square of the local curvature along the curve is a minimum. It is shown that this is equivalent to the simultaneous solution of the same problem for a set of 2-panel (3-point) curves, and such solutions are then investigated.

A linearized method of solution for the required gradients is proposed that results in a set of linear equations that can be inverted directly, and that should be capable of solution on a CNC machine control unit, following the direct input of the coordinates of the given points and relevant end slopes. Initial results are compared with those obtained from a rigorous numerical optimization procedure for the biarc curve and with a cubic-spline function. It is shown that instabilities may be avoided, and a smoother curve obtained, by the effective decoupling of segments of the whole curve by the specification of gradients at additional, intermediate, points.

References (7)

M.A. Malcolm
On the computation of non linear spline functions
SIAM J. Numer. Anal.
(1977)
G. Birkhoff
Non-linear interpolation by splines, pseudosplines and elastica
J.M. Glass
Smooth curve interpolation: a generalised spline-fit procedure
Bit
(1966)

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Curve approximation by G1 arc splines with a limited number of types of curvature and length
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They have been studied mainly for the application to NC toolpaths of CNC machines because they have many advantages such as stability of CNC machines during machining processes and improvement of processing speed compared to toolpaths consisting of only straight-line segments (Meek and Walton, 1993; Yeung and Walton, 1994). Approximation methods for discrete points (Chen et al., 2004; Hoschek, 1992; Meek and Walton, 1992; Parkinson and Moreton, 1991; Schonherr, 1993; Yeung and Walton, 1994; Yong et al., 1999) and planar curves (Ahn et al., 1998; Maier, 2014; Meek and Walton, 1993, 1995; Ong et al., 1996; Piegl and Tiller, 2002; Walton and Meek, 1994; Yeung and Walton, 1994) have been investigated. Most of them use the biarc curve introduced in Bolton, 1975, which is a pair of two arcs connected in G1 manner, and it can interpolate arbitrary two points and tangent vectors at those two points, whereas one arc curve can only interpolate arbitrary two points and a tangent vector at one of them.
A method for approximating planar curves by G¹ continuous arc splines with a limited number of types of curvatures and lengths, named k-arc splines, is proposed. A k-arc spline is a particular arc spline, and it has only k types of arcs less than the total number of segments. Due to this property, k-arc splines require discrete variables that relate segments and k types of arcs for representation. Therefore k-arc spline approximation problems must deal with discrete variables in contrast to general arc spline approximation problems. The authors present a method to appropriately estimate these variables from curvature values of a target curve. The proposed method enables a formulation of the approximation problem as a continuous optimization problem that conventional optimization algorithms can solve efficiently. The estimation method also provides a suitable initial solution for the algorithms. Some examples to show the broad applicability of the proposed algorithm are shown.
A method for investigating the springback behavior of 3D tubes
2017, International Journal of Mechanical Sciences
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When we get a spatial product, we firstly should define the representation and connection method of the tube elements so that the forming parameters (ρ, ϕ) can be obtained (ρ refers to the bending radius and ϕ refers to the twisting angle that causes the torsion of the spatial curve). Generally, if the tube elements are small enough, they can be considered as a series of bi-arcs [1,30,31] or spirals. When using bi-arcs as tube element, two adjacent elements share the same tangent vector at the joint.
The objective of this paper is to investigate the springback of spatial tubes. The model for analyzing the springback of spatial tubes in [1] is extended to different loading modes and hardening materials. Comparisons among theoretical model, FEM model and experiments are performed to validate the effectiveness of the predictions. Results show that the new model is more effective to describe the springback behavior of spatial tubes, and the principles obtained by the new model are coincident with previous researches. Based on the theoretical model, the influence of loading modes on spatial tube springback is newly investigated. It is found that radii after springback and angles after springback will both increase accordingly with the increasing of loading index k. And evaluating the loading index k reasonably is beneficial for improving the precision of prediction results.
A semi-analytical method for the springback prediction of thick-walled 3D tubes
2016, Materials and Design
The aim of this paper is to investigate the deformation and springback behavior of 3D tubes under complex stress conditions. We proposed a new semi-analytical method to predict the springback behavior of 3D tubes. In the new method, discretization and approximation are implemented on the tube axis so that the deformation parameters (bending radiuses and twisting angles) can be derived. Based on the discretization and approximation, a theoretical model is proposed to calculate the stress distributions and residual deformations. The new method is applied on a typical 3D tube for springback prediction, and comparison between theoretical and experimental results shows a remarkable agreement. Additionally, it is also found that the radii after springback will decrease inversely with the increase of twisting angle, and this tendency will become clearer when the bending radius increases.
The family of biarcs that matches planar, two-point G1 Hermite data
2008, Journal of Computational and Applied Mathematics
The biarc is a curve made by joining two circular arcs in a $G^{1}$ fashion. There is a one-parameter family of biarcs that can match given planar, two-point $G^{1}$ Hermite data. This note considers the range of $G^{1}$ Hermite data that can be matched, identifies the region in which the members of the family of biarcs lie, and shows that there is exactly one member biarc passing through each point in that region.
A practicable approach to G1 biarc approximations for making accurate, smooth and non-gouged profile features in CNC contouring
2006, CAD Computer Aided Design
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Specifically, to use the circular interpolation in CNC machining, a popular strategy is to (1) fit the profile with continuous circular arcs in such a way that the fitting errors (the distances from the arcs and perpendicular to the profile) are within the specified tolerance, and (2) apply the tool radius compensation to the arcs in their CNC machining. In the past decades, many approaches [1–12] have been proposed to fit G1 arc splines (or biarc curves) to free-form curves for various purposes, including CNC contouring for smooth, accurate profile features. Meek and Walton [4] interpolated a G1 arc spline (a number of biarcs with continuous tangents at the joints) into a set of data points within a specified tolerance.
Extensive research on G¹ biarc approximations to free-form curves has been conducted for the production of accurate, smooth and non-gouged profile features in CNC contouring. However, all the published work has only focused on improving the fitting accuracy between the biarc curve and the nominal free-form curve of a profile and minimizing the biarc number; as a result, the radii of the concave arcs of some biarcs could be less than the pre-determined tool radius, and the tool would overcut these arcs in machining, eventually gouging the profile. In this work, a new, practicable approach is proposed to completely solve this problem. The main feature of this approach is to find the gouging-free parameter interval of a biarc family, among which the radii of all the concave arcs are larger than the tool radius, and then to search in this interval for a best fitting biarc so that its approximation accuracy is within the tolerance. This approach is robust and easy to implement and can substantially promote the use of G¹ biarc curves for CNC machining.
Biarc approximation of polygons within asymmetric tolerance bands
2005, CAD Computer Aided Design
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Several algorithms assume some freedom about the position of the points or about the tangents to the original curve. See, for instance, the work by Piegl [13], Rossignac and Requicha [14], Su and Liu [7], Parkinson and Moreton [15], Hoschek [16], Meek and Walton [17], Schönherr [18], Yeung and Walton [5], Ong et al. [1], and Poliakoff et al. [19] for more details. Reference is given to Marciniak and Putz [20] for a notable algorithm for a single-arc approximation.
We present an algorithm for approximating a simple planar polygon by a tangent-continuous approximation curve that consists of biarcs. Our algorithm guarantees that the approximation curve lies within a user-specified tolerance from the original polygon. If requested, the algorithm can also guarantee that the original polygon lies within a user-specified distance from the approximation curve. Both symmetric and asymmetric tolerances can be handled. In either case, the approximation curve is guaranteed to be simple. Simplicity of the approximation curve is achieved by restricting it to a ‘tolerance band’ which represents the user-specified tolerance and which takes into account bottlenecks of the input polygon. The tolerance band itself is computed by means of a regular grid and so-called k-dops. The basic algorithm is readily extended to compute biarc approximations of collections of polygonal curves simultaneously. Experimental results demonstrate that this algorithm computes biarc approximations of an n-vertex polygon with a close-to-minimum number of biarcs in roughly $O (n log n)$ time.

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Computer-Aided Design

Abstract

References (7)

On the computation of non linear spline functions

SIAM J. Numer. Anal.

Non-linear interpolation by splines, pseudosplines and elastica

Smooth curve interpolation: a generalised spline-fit procedure

Bit

Cited by (54)

Curve approximation by G<sup>1</sup> arc splines with a limited number of types of curvature and length

A method for investigating the springback behavior of 3D tubes

A semi-analytical method for the springback prediction of thick-walled 3D tubes

The family of biarcs that matches planar, two-point G<sup>1</sup> Hermite data

A practicable approach to G<sup>1</sup> biarc approximations for making accurate, smooth and non-gouged profile features in CNC contouring

Biarc approximation of polygons within asymmetric tolerance bands