Arc Spline Research Articles

Reliability-based G1 continuous arc spline approximation

This paper introduces an algorithm for approximating a set of data points with G1 continuous arcs, leveraging covariance data associated with the points. Prior approaches to arc spline approximation typically assumed equal contribution from all data points, resulting in potential algorithmic instability when outliers are present. To address this challenge, we propose a robust method for arc spline approximation, taking into account the 2D covariance of each data point. Beginning with the definition of models and parameters for single-arc approximation, we extend the framework to support multiple-arc approximation for broader applicability. Finally, we validate the proposed algorithm using both synthetic noisy data and real-world data collected through vehicle experiments conducted in Sejong City, South Korea.

Interpolation by polygon rolling motions for approximate sweep computation

We interpolate planar rigid body motions by piecewise rotational and translational motions. The trajectories of all points are then arc splines, i.e., curves composed of circular arcs or line segments. For objects with arc spline boundaries, the boundary of the volume swept by the object and the motion is then again an arc spline. We prove that the distance between the original motion and its approximation as between curves in a suitable kinematic image space converges quadratically to zero. This implies the same rate of convergence between point trajectories under the two motions.

Geometry reconstruction based on arc splines with application to wheel-rail contact simulation

PurposeWhen establishing a mathematical model to simulate solid mechanics, considering realistic geometries, special tools are needed to translate measured data, possibly with noise, into idealized geometrical entities. As an engineering application, wheel-rail contact interactions are fundamental in the dynamic modeling of railway vehicles. Many approaches used to solve the contact problem require a continuous parametric description of the geometries involved. However, measured wheel and rail profiles are often available as sets of discrete points. A reconstruction method is needed to transform discrete data into a continuous geometry.Design/methodology/approachThe authors present an approximation method based on optimization to solve the problem of fitting a set of points with an arc spline. It consists of an initial guess based on a curvature function estimated from the data, followed by a least-squares optimization to improve the approximation. The authors also present a segmentation scheme that allows the method to increment the number of segments of the spline, trying to keep it at a minimal value, to satisfy a given error tolerance.FindingsThe paper provides a better understanding of arc splines and how they can be deformed. Examples with parametric curves and slightly noisy data from realistic wheel and rail profiles show that the approach is successful.Originality/valueThe developed methods have theoretical value. Furthermore, they have practical value since the approximation approach is better suited to deal with the reconstruction of wheel/rail profiles than interpolation, which most methods use to some degree.

Curve approximation by G1 arc splines with a limited number of types of curvature and length

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.

Automatic $\rm~G_{~}^{1}$ arc spline approximation via sparse representation

Arc spline is popularized in describing the tool path for a certain CNC machine,and it is desirable to approximate curves with as few arcs as possible.Most of existing methods are heuristic; researchers have seldom paid attention to the piecewise constant property of the curvature of arc spline.Based on the intrinsic sparse property,this paper proposes an automatic $\\rm~G_{~}^{1}$ arc spline approximation method via sparse representation. There are two main steps: global initialization using a sparse optimization problem for automatically detecting the joint points between adjacent arcs,and local modification by readjusting the positions of joint points to ensure the $\\rm~G_{~}^{1}$ continuity.The experimental results show that the new method is sensitive to symmetry. Some sharp features and self-intersecting data can be approximated well, which fully shows the advantages of the new method.

Tool-path control for singularity avoidance of 5-axis machine tool by combining circular arc and cubic spline

Tool-path control for singularity avoidance of 5-axis machine tool by combining circular arc and cubic spline

Cutting coal path planning of shearer up-drum based on double arc spline curve

Cutting coal path planning of shearer up-drum based on double arc spline curve

Mitered Offsets and Skeletons for Circular Arc Polygons

The offsetting process that defines straight skeletons of polygons is generalized to arc polygons, i.e., to planar shapes with piecewise circular boundaries. The offsets are obtained by shrinking or expanding the circular arcs on the boundary in a co-circular manner, and tracing the paths of their endpoints. These paths define the associated shape-preserving skeleton, which decomposes the input object into patches. While the skeleton forms a forest of trees, the patches of the decomposition have a radial monotonicity property. Analyzing the events that occur during the offsetting process is non-trivial; the boundary of the offsetting object may get into self-contact and may even splice. This leads us to an event-driven algorithm for offset and skeleton computation. Several examples (both manually created ones and approximations of planar free-form shapes by arc spline curves) are analyzed to study the practical performance of our algorithm.

Blended Elliptic Arc Splines

Arcs of conics are natural looking, smooth and gradual in curvature transition. The method in this paper proposes piecewise interpolating splines, which fit elliptic arcs between consecutive points in a given set of planar data points. Tangents of the curve at the data points are determined by a local procedure inspired by Akima’s formulation. Elliptic arcs that suitably fit the orientation of these tangents are fit at all segments between control points, giving a G1-continuous curve fit. In another option, the curve segments on both sides of the control points are made curvature continuous by blending with circular arcs; resulting in a G2-continuous curve. These curves avoid unwanted inflections and undulations; and are capable of recognizing and generating straight-line paths and sharp corners; and also have an option to specify tension of curve at individual segments. Moreover, this is computationally less expensive. Having these desirable properties, this method can be used as an effective curve fitting tool for multiple-valued functional approximation, motion path tracing, 2D geometric modeling (including typography), etc.

Numerically robust computation of circular visibility

We address the question of whether a point inside a domain bounded by a simple closed arc spline is circularly visible from a specified arc from the boundary. We provide a simple and numerically stable linear time algorithm that solves this problem. In particular, we present an easy-to-check criterion that implies that a point is not visible from a specified boundary arc.

Sharp corner transitional trajectory planning based on arc splines in glass edge grinding

The sharp corners on final contours in flat glass grinding can cause broken wheel center trajectories. A corner transitional trajectory planning algorithm based on arc splines is proposed in this paper to control the wheel center trajectory G1 continuity and the grinding depth at a corner. The radii of arc splines are derived first to meet design requirements based on the geometrical features of a sharp corner. A three-phase feedrate planning scheme is introduced to generate the arc splines. Corner classification and treatment are presented to make the proposed algorithm flexible, and steps for algorithm application are summarized. Experimental results demonstrate that the proposed algorithm can achieve satisfactory shape accuracy at corners without any breakage, rounding, or flattening.

Storage-efficient reconstruction framework for planar contours

A storage-efficient reconstruction framework for cartographic planar contours is developed. With a smaller number of control points, we aim to calculate the area and perimeter as well as to reconstruct a smooth curve. The input data forms an oriented contour, each control point of which consists of three values: the Cartesian coordinates (x, y) and tangent angle θ. Two types of interpolation methods are developed, one of which is based on an arc spline while the other one is on a cubic Hermite spline. The arc spline-based method reconstructs a G1 continuous curve, with which the exact area and perimeter can be calculated. The benefit of using the Hermite spline-based method is that it can achieve G2 continuity on most control points and can obtain the exact area, whereas the resulting perimeter is approximate. In a numerical experiment for analytically defined curves, more accurate computation of the area and perimeter was achieved with a smaller number of control points. In another experiment using a digital elevation model data, the reconstructed contours were smoother than those by a conventional method.

Numerical study of the flow in the Yellow River with non-monotonous banks

A 2-D depth averaged RNG k – ɛ model is developed to simulate the flow in a typical reach of the Upper Yellow River with non-monotonic banks. In order to take account of the effect of the secondary flow in a bend, the momentum equations are modified by adding an additional source term. A comparison between the numerical simulation and the field measurements indicates that the improved 2-D depth averaged RNG k – ɛ model can improve the accuracy of the numerical simulation. An arc spline interpolation method is developed to interpolate the non-monotonic river banks. The method can also be reasonably applied for the 2-D interpolation of the river bed level. Through a comparison of the water surface gradients simulated in the seven bends of the studied reach, some analytical formulae are improved to reasonably calculate the longitudinal and transverse gradients in meandering river reaches. Furthermore, the positions of the maximum water depth and the maximum velocity in a typical bend are discussed.

Geometric Hermite interpolation by logarithmic arc splines

This paper considers the problem of G1 curve interpolation using a special type of discrete logarithmic spirals. A “logarithmic arc spline” is defined as a set of smoothly connected circular arcs. The arcs of a logarithmic arc spline have equal angles and the curvatures of the arcs form a geometric sequence. Given two points together with two unit tangents at the points, interpolation of logarithmic arc splines with a user specified winding angle is formulated into finding the positive solutions to a vector equation. A practical algorithm is developed for computing the solutions and construction of interpolating logarithmic arc splines. Compared to known methods for logarithmic spiral interpolation, the proposed method has the advantages of unbounded winding angles, simple offsets and NURBS representation.

Total curvature variation fairing for medial axis regularization

We present a new fairing method for planar curves, which is particularly well suited for the regularization of the medial axis of a planar domain. It is based on the concept of total variation regularization. The original boundary (given as a closed B-spline curve or several such curves for multiply connected domains) is approximated by another curve that possesses a smaller number of curvature extrema. Consequently, the modified curve leads to a smaller number of branches of the medial axis. In order to compute the medial axis, we use the state-of-the-art algorithm from [1] which is based on arc spline approximation and a domain decomposition approach. We improve this algorithm by using a different decomposition strategy that allows to reduce the number of base cases from 13 to only 5. Moreover, the algorithm reduces the number of conic arcs in the output by approx. 50%.

Optimal arc spline approximation

We present a method for approximating a point sequence of input points by a G1-continuous (smooth) arc spline with the minimum number of segments while not exceeding a user-specified tolerance. Arc splines are curves composed of circular arcs and line segments (shortly: segments). For controlling the tolerance we follow a geometric approach: We consider a simple closed polygon P and two disjoint edges designated as the start s and the destination d. Then we compute a SMAP (smooth minimum arc path), i.e. a smooth arc spline running from s to d in P with the minimally possible number of segments. In this paper we focus on the mathematical characterization of possible solutions that enables a constructive approach leading to an efficient algorithm.In contrast to the existing approaches, we do not restrict the breakpoints of the arc spline to a predefined set of points but choose them automatically. This has a considerably positive effect on the resulting number of segments.

Approximating Cornu spirals by arc splines

While modeling planar parametric curves, it is desirable to approximate them efficiently by arc splines with fewest segments within the prescribed tolerance. Planar arc splines are ideal for descri...

Approximating Cornu spirals by arc splines

While modeling planar parametric curves, it is desirable to approximate them efficiently by arc splines with fewest segments within the prescribed tolerance. Planar arc splines are ideal for describing the path of a numerically controlled cutting machine because they are easy to offset. In this paper, we derive a vector solution to interpolate a G1-continuous biarc curve between a given pair of points with specified start, joint and end tangents. Using this solution as the underlying idea, we present an efficient algorithm for approximating a Cornu spiral with a sequence of arcs within the prescribed maximum deviation. The method is simple and generic and can be applied to other types of planar parametric curves as well.

Fairing an arc spline and designing with G 2 PH quintic spiral transitions

This paper describes a method to smooth an arc spline. Arc splines are G 1 continuous segments made of circular arcs and straight lines. We have proposed a smooth version of the arc spline by replacing its parts with C-, S-, and J-shaped spiral transitions, stitched with G 2 continuity, by using a single segment of Pythagorean hodograph quintic function. Use of a single polynomial function rather than two has the benefit that designers have fewer entities to deal with. Spiral transitions are important in manufacturing industries because of their use in the cutting paths for numerically controlled cutting machinery, highway or railway designing, non-holonomic robot path planning and spur gear designing.

Approximation of a closed polygon with a minimum number of circular arcs and line segments

Optimal approximation of closed polygons differs from the case of open polygons in the sense that the location of a starting point must be determined in a suitable way. We present an algorithm which first computes a proper extremal arc as starting point and then approximates the input polygon with a minimum number of circular arcs and line segments. The resulting curve is called Cyclic Minimum Arc Path (CMAP). Our algorithm guarantees the CMAP staying inside a user-specified tolerance. In contrast to the existing approaches, we do not restrict the breakpoints of the arc spline to a predefined set of points but choose them automatically. This has considerable effects on the resulting number of segments.We can handle every type of tolerance zone representing the user-specified tolerance as long it is formed by piecewise restricted analytic curves. In case of polygonal tolerance zones the proposed algorithm takes O(n3) time for an original polygonal chain with n vertices. For generating a solution which has at most one additional segment we present an O(n2) algorithm.