Biarc Approximation Research Articles

Precise Hausdorff distance computation for freeform surfaces based on computations with osculating toroidal patches

We present an efficient algorithm for computing the precise Hausdorff Distance (HD) between two freeform surfaces. The algorithm is based on a hybrid Bounding Volume Hierarchy (BVH), where osculating toroidal patches (stored in the leaf nodes) provide geometric properties essential for the HD computation in high precision. Intrinsic features from the osculating geometry resolve computational issues in handling the cross-boundary problem for composite surfaces, which leads to the acceleration of HD algorithm with a solution (within machine precision) to the exact HD. The HD computation for general freeform surfaces is discussed, where we focus on the computational issues in handling the local geometry across surface boundaries or around surface corners that appear as the result of gluing multiple patches together in the modeling of generic composite surfaces. We also discuss how to switch from an approximation stage to the final step of computing the precise HD using numerical improvements and confirming the correctness of the HD computation result. The main advantage of our algorithm is in the high precision of HD computation result. As the best cases of the proposed torus-based approach, we also consider the acceleration of HD computation for freeform surfaces of revolution and linear extrusion, where we can support real-time computation even for deformable surfaces. The acceleration is mainly due to a fast biarc approximation to the planar profile curves of the simple surfaces, each generated by rotating or translating a planar curve. We demonstrate the effectiveness of the proposed approach using experimental results.

Minkowski sum computation for planar freeform geometric models using $$G^1$$-biarc approximation and interior disk culling

We present an efficient algorithm for computing the Minkowski sum of two planar geometric models bounded by B-spline curves. The boundary curves are first approximated by \(G^1\)-biarc splines within a given error bound \(\epsilon > 0\). A superset of Minkowski sum boundary is then generated using the biarc approximations. For non-convex models, the superset contains redundant arcs. An efficient and robust elimination of the redundancies is the main challenge of Minkowski sum computation. For this purpose, we use the Minkowski sum of interior disks of the two input models, which are again disks in the Minkowski sum interior. The majority of redundant arcs are eliminated by testing each against a small number of interior disks selected for efficiency. From the planar arrangement of remaining arcs, we construct the Minkowski sum boundary in a correct topology. We demonstrate a real-time performance and the stability of circle-based Minkowski sum computation using a large set of test data.

A generative sampling system for profile designs with shape constraints and user evaluation

A sampling approach is proposed for deriving profiles of an existing product design using profile similarities and primitive shapes, such as circles, triangles, and ellipses, as constraints. A common approach for a designer is to generate or collect sketches to create a design space. The design space can be explored to retrieve the samples directly based on requirements or crossing over them to obtain new design variations. In this stage, the proposed method can be used to derive a large number of samples from a design image to work with. First, the user clicks on the image to define each design feature as cubic Bézier curve segment. The primitive shapes are then constructed for each segment and used as constraints, such that the control points can move only to where these shapes allow. Moreover, the design similarities are computed using the triangles based on their anisotropy ratios, which are measures of the deformation between corresponding triangles and which are used to ensure that they are highly related samples in the example design. Modified Hausdorff distances are also computed between the biarc approximations of the samples. There is diversity provided that these distances are large enough. A customized sampling algorithm that fulfills our constraints is executed several times synchronously via parallel programming to create a design space. Finally, the user-specified number of distinct samples is retrieved from the design space by minimizing the Audze–Eglais potential energy. We provide an additional tool with which the users can adjust the weights for each control point to guide the sampling process and record the chosen samples. The main goal of this study is to generate samples, which are diverse but still convey the key features of the supplied shape. Using the settings related to similarity, a designer can derive samples sticking to their initial idea to explore its better version or more creative results that they could not imagine on their own. The proposed system offers simplicity automating every step that must be done by the user. Thus even inexperienced users without needing any programming skill can vary shapes defined using a set of points in a matter of seconds, and experienced users can even focus on specific shape features weighting them without dealing any constraint or rule definition requirements. Besides, the proposed system allows partial modifications, in which users can select a specific region to modify while the rest of the shape maintains the original appearance and functionality of the exemplary design.

Efficient Voronoi diagram construction for planar freeform spiral curves

We present a real-time algorithm for computing the Voronoi diagram of planar freeform piecewise-spiral curves. The efficiency and robustness of our algorithm is based on a simple topological structure of Voronoi cells for spirals, which also enables us a direct construction of Voronoi structure without relying on intermediate polygonal or biarc approximations to the given planar curves. Using a Möbius transformation, we provide an efficient search for maximal disks. The correct topology of Voronoi diagram is computed by sampling maximal disks systematically, which entails subdividing spirals until each belongs to a pair/triple of spirals under a certain matching condition. The matching pairs and triples serve as the basic building blocks for bisectors and bifurcations, and their connectivity implies the Voronoi structure. We demonstrate a real-time performance of our algorithm using experimental results including the medial axis computation for planar regions under deformation with non-trivial self-intersections and the Voronoi diagram construction for disconnected planar freeform curves.

Approximation of parabola and quadratic Bézier curve, by fewest circular-arcs within a tolerance-band

This paper presents a new approximation method for parabola and quadratic Bezier curves by circular-arc within a tolerance-band, and it fully proves that the proposed method supports the fewest number of tangent circular-arcs. Thus, it proposed an approximation method on a specified type of general curve as parabola and QB-curve, to concentrate more on optimization techniques which is an interesting property in CAD/CAM for more efficiency. The approximation method defines tolerance-band by the exterior- and interior-offsets, and it defines some special circular-arcs within the tolerance-band which are used in the approximation by fewest circular-arcs. This article analyzes the proposed method with the possible arc approximations to prove and find out the conditions and features of a sequence of tangent arcs which consists of the minimized number of arcs. Finally, based on the proven minimized conditions and features, two algorithms represent the approximation methods for parabola and QB-curve which guarantee fewest number of arcs and they are not based on Bisection structure and biarc approximation, unlike most existing methods.

Biarc Approximations Tool-Path Generation for Polyhedral Models

An effective biarc approximate tool path generation algorithm is presented for polyhedral models. A series of sections intersect the offset surface of polyhedral models to obtain the initial tool paths. The path interval is calculated by the curvature radiuses and scallop height for offset surface models. Within the range of fitting precision, based on the biarc fitting constraint condition and adjustment parameter of curvature, iteration method is used to determine adjustment parameter of curvature radius. So the last radius for biarc is obtained for fitting the initial tool paths. Example shows the method can keep G1 biarc approximations tool path and fewer biarc segments, so the machining efficiency is improved.

Projecting points onto planar parametric curves by local biarc approximation

This paper proposes a geometric iteration algorithm for computing point projection and inversion on planar parametric curves based on local biarc approximation. The iteration begins with initial estimation of the projection of the prescribed test point. For each iteration, we construct a biarc that locally approximates a segment on the original curve starting from the current projective point. Then we compute the projective point for the next iteration, as well as the parameter corresponding to it, by projecting the test point onto this biarc. The iterative process terminates when the projective point satisfies the required precision. Examples demonstrate that our algorithm converges faster and is less dependent on the choice of the initial value compared to the traditional geometric iteration algorithms based on single-point approximation.

Efficient offset trimming for planar rational curves using biarc trees

We present an efficient algorithm for trimming both local and global self-intersections in planar offset curves. The algorithm is based on a G1-continuous biarc approximation of the given planar curves. We first consider an implementation that employs a distance map which can be stored in the graphics hardware depth buffer. The depth-buffer approach is easier to implement than a different approach that is based on a biarc-tree, a hierarchical data structure for the biarc approximation of the given planar curves. Though more involved technically, the biarc-tree algorithm is more efficient both in computing time and in memory space needed for storing the data structure. We demonstrate the real-time performance of our algorithm using several experimental results.

Divide-and-conquer for Voronoi diagrams revisited

We show how to divide the edge graph of a Voronoi diagram into a tree that corresponds to the medial axis of an (augmented) planar domain. Division into base cases is then possible, which, in the bottom-up phase, can be merged by trivial concatenation. The resulting construction algorithm—similar to Delaunay triangulation methods—is not bisector-based and merely computes dual links between the sites, its atomic steps being inclusion tests for sites in circles. This guarantees computational simplicity and numerical stability. Moreover, no part of the Voronoi diagram, once constructed, has to be discarded again. The algorithm works for polygonal and curved objects as sites and, in particular, for circular arcs, which allows its extension to general free-form objects by Voronoi diagram preserving and data saving biarc approximations. The algorithm is randomized, with expected runtime O ( n log n ) under certain assumptions on the input data. Experiments substantiate an efficient behavior even when these assumptions are not met. Applications to offset computations and motion planning for general objects are described.

Using a biarc filter to compute curvature extremes of NURBS curves

A method to compute curvature minima and maxima of parametric curves (represented in NURBS format) is presented in this paper. Since the curvature changes vary rapidly along the path of (even smooth) curves, a biarc filter is employed to approximate the curvature function with a piecewise constant function. This allows the isolation of curvature extreme values that are found within-engineering tolerances via repeated biarc approximation followed by golden section search. Because the derivative of the curvature is numerically very unstable, only optimization without derivatives is feasible. However, given the excellent isolation property of biarc filters, curvature extremes are found within 10–20 steps even for high accuracy requirements ranging from 10−4 to 10−6.

A smooth spiral tool path for high speed machining of 2D pockets

We introduce a new algorithm for generating a spiral tool path for high-speed machining of pockets without islands. The pocket may be an arbitrary simply-connected 2D shape bounded by straight-line segments and circular arcs. The tool path is generated by interpolating growing disks placed on the medial axis of the pocket. It starts inside the pocket and spirals out to the pocket boundary. The tool path is guaranteed to be free of self-intersections and allows machining of the pocket without tool retractions. The start point of the spiral may be chosen freely by the user anywhere within the pocket. Most importantly, the spiral tool path complies with a user-specified maximum cutting width. The output of our algorithm is a G 1 -continuous spiral path. However, in a post-processing step, a properly adapted variant of the recent “ PowerApx” package [Heimlich M, Held M. Biarc approximation, simplification and smoothing of polygonal curves by means of Voronoi-based tolerance bands. International Journal of Computational Geometry & Applications 2008;18(3):221–50] can be used to boost the tool path to C 2 -continuity. Our new algorithm was implemented and tested successfully on real-world data. We conclude our paper by an analysis of sample tool paths produced by our implementation.

Medial axis computation for planar free–form shapes

We present a simple, efficient, and stable method for computing—with any desired precision—the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.

3D Reconstruction from Orthographic Image and Optimization Processing of NC Machining for Reconstructed Part

For inverse constructing object from its orthographic image, a new method of 3D reconstruction based on image vectorization and bi-arc approximation of cubic B-spline is investigated. A 3D CAD model is automatically acquired from 2D views by means of 2D primitive recognition, 3D reconstruction of primitive and combination of 3D primitive bodies. With the help of CimatronE8.0 software, reasonable procedure design and optimization toolpath planning are applied to increase machining accuracy and productivity during automatic programming for reconstructed 3D solid. Reconstructed parts are machined on the VMC-600 type machine center with FANUC system. The results of application indicate that this method has many advantages, such as quickly obtaining the CAD model of part, getting smooth machined surface, efficient machining without invalid or reduplicate feed and so on. As a result, it can be applied widely in NC machining field.

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

Extensive research on G 1 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 G 1 biarc curves for CNC machining.

Biarc approximation of polygons within asymmetric tolerance bands

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.

Error-bounded biarc approximation of planar curves

Presented in this paper is an error-bounded method for approximating a planar parametric curve with a G 1 arc spline made of biarcs. The approximated curve is not restricted in specially bounded shapes of confined degrees, and it does not have to be compatible with non-uniform rational B-splines (NURBS). The main idea of the method is to divide the curve of interest into smaller segments so that each segment can be approximated with a biarc within a specified tolerance. The biarc is obtained by polygonal approximation to the curve segment and single biarc fitting to the polygon. In this process, the Hausdorff distance is used as a criterion for approximation quality. An iterative approach is proposed for fitting an optimized biarc to a given polygon and its two end tangents. The approach is robust and acceptable in computation since the Hausdorff distance between a polygon and its fitted biarc can be computed directly and precisely. The method is simple in concept, provides reasonable accuracy control, and produces the smaller number of biarcs in the resulting arc spline. Some experimental results demonstrate its usefulness and quality.

Biarc approximation of NURBS curves

An algorithm for approximating arbitrary NURBS curves with biarcs is presented. The main idea is to approximate the NURBS curve with a polygon, and then to approximate the polygon with biarcs to within the required tolerance. The method uses a parametric formulation of biarcs appropriate in geometric design using parametric curves. The method is most useful in numerical control to drive the cutter along straight line or circular paths.

Numerically Controlled Machining of Freeform Curves Using Biarc Approximation

To machine a shape incorporating freeform curves, the freeform curves must be approximated and represented in a form suitable for numerically controlled (NC) tool-path generation. Instead of the commonly used line segment approximation, a new method using biarc approximation is presented here. The purpose is to reduce sudden changes in tool movement and increase continuity and smoothness. A biarc can be described as two consecutive circular arcs with an identical tangent at the connecting point. Therefore, tangent continuity can be maintained. Furthermore, curvature continuity can be improved if the difference between the two curvatures is minimised. In this research, biarc segments are used to approximate the freeform curves by controlling the largest deviation distance between the curves and the biarcs. The test results show that the biarc approximation method generates fewer tool-path segments than the traditional line approximation. The definitions of the biarcs are used to generate NC tool paths. A new tool-path generation method is developed for processing rough cutting and finish cutting paths. Example parts modelled with B-spline curves are tested and machined.

Three dimensional biarc approximation of freeform surfaces for machining tool path generation

In typical methods for machining freeform surfaces, the machining tool paths are generated by approximating the surface curves using line segments. While the approximated shape of the surface can be produced using cutters traversing along the line segments, the final surface produced may lack smoothness and continuity due to the zigzag patterns of the line segments. To reduce the difference and increase smoothness and continuity, the arc splines can be used to approximate the freeform curves and surfaces. A biarc is composed of two consecutive circular arcs with an identical tangent at the connecting point. Since the tangents at the connecting node are the same, the C1 continuity property can be preserved. In addition, if the difference between the curvatures at the connecting node is minimized, then the C2 smoothness property can be enhanced. In this research, the biarc segments are used to approximate the 2D and 3D freeform curves and surfaces using the largest deviation distance between the curves and the biarcs as the approximation criterion. The tool contact points generated and biarcs can be used to generate NC tool paths for machining 2D curves and 3D curves and surfaces. The methodology presented is implemented on a personal computer. Example parts modelled with B-spline curves and surfaces are tested and discussed. The test results show that the number of tool contact points and the number of segments are fewer than the typical linear approximation methods.

Numerical methods for approximating digitized curves by piecewise circular arcs

Two-dimensional digitized curves are often approximated by some piecewise linear or high-order curves. The approximation usually results in a compact and effective representation for further applications. In this paper, we propose new algorithms to approximate digitized curves by piecewise circular arcs with geometric continuity G 0 or G 1. First, iterative methods are proposed to solve the best single arc and biarc approximation problems of a digitized curve with respect to the maximum norm. The basic notion of our methods is to show that the error functions of two approximations are unimodal functions, so techniques developed in optimization theory can be used to search for the best approximation. Then the piecewise arcs approximation problems can be solved by applying these approximation algorithms to construct an arcwise curve such that each join point between two arcs is G 0 or G 1 continuity and its maximum approximation error is no more than a given tolerance ε > 0. Some experimental results are given to demonstrate the quality and the performance of the proposed algorithms.