Recent advance of Additive Manufacturing technologies allows us to manufacture various parts used in real-world products. Consequently, product tracking of such 3D printed parts is an important issue. Quick Response (QR) code which is a two-dimensional matrix barcode invented by Denso, a Japanese automotive industry, in 1994, can be used for this purpose. It can store more data than the 1D barcode in a smaller space, and using a smartphone as a scanner, one can directly visit a website where all the information of the parts is stored. However, QR codes require secondary procedures to add them to products and are also vulnerable to wear and tear. Moreover, QR codes cannot be added to freeform surfaces, but only to developable surfaces. In this paper we propose a novel technique to embed QR codes onto CAD models consisting of freeform surfaces represented by B-spline surfaces, which produces 3D QR codes. 3D QR codes work similar to 2D QR codes and can be read by existing QR scanners, but are designed by grooving the surface to obtain light and dark regions caused by ambient occlusion. Unlike conventional QR codes, 3D QR codes do not fall off from the part and can even be painted if necessary. Furthermore, we do not need to prepare dark-colored and light-colored materials for 3D printing as the dark color is provided by the grooving. We demonstrate the effectiveness of our technique with various examples.

Reverse engineering of 3D industrial objects such as automobiles and electric appliances is typically performed by fitting B-spline surfaces to scanned point cloud data with a fairing term to ensure smoothness, which often smooths out sharp features. This paper proposes a radically different approach to constructing fair B-spline surfaces, which consists of fitting a surface without a fairing term to capture sharp edges, smoothing the normal field of the constructed surface with feature preservation, and reconstructing the B-spline surface from the smoothed normal field. The core of our method is an image processing based feature-preserving normal field fairing technique. This is inspired by the success of many recent research works on the use of normal field for reconstructing mesh models, and makes use of the impressive simplicity and effectiveness of bilateral-like filtering for image denoising. In particular, our approach adaptively partitions the B-spline surface into a set of segments such that each segment has approximately uniform parameterization, generates an image from each segment in the parameter space whose pixel values are the normal vectors of the surface, and then applies a bilateral filter in the parameter domain to fair the normal field. As a result, our approach inherits the advantages of image bilateral filtering techniques and is able to effectively smooth B-spline surfaces with feature preservation as demonstrated by various examples.

Geometric iterative methods (GIM), including the progressive–iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, present the local properties and accelerating techniques, and show their convergence. Moreover, because it is easy to integrate geometric constraints in the iterative procedure, GIM has been widely applied in geometric design and related areas. We survey the successful applications of geometric iterative methods, including applications in geometric design, data fitting, reverse engineering, mesh and NURBS solid generation.

We introduce a framework for modeling of heterogeneous objects in terms of trivariate B-spline functions and a method for slicing them directly for additive manufacturing. We first fit volumetric attribute data associated with the geometry in terms of trivariate B-spline functions under the assumption that the geometric volume is already defined by the trivariate B-spline functions. Then, the B-spline volume and the associated attribute data are directly sliced without converting them to stereo-lithography format, resulting in a tool path with fewer errors. Furthermore, adaptive ray shooting is introduced in the slicing plane so that the zigzag tool path passes through all the tangential intersection points of the heterogeneous objects to represent all the feature points in the fabricated model. Complex examples illustrate the effectiveness of our method.

We study the effects of curvature on the energy absorption characteristics of cylindrical corrugated tubes under compression by isogeometric analysis and experiments. The corrugated volume is constructed by revolving a wavelike B-spline profile surface about the vertical axis. The curvature at peaks of the profile curve is gradually increased from a smaller value to a larger value while the wavelength and amplitude are kept unchanged. This leads to three curvature distribution patterns, which are found to significantly affect the energy absorption characteristics of the corrugated tube. Examples show that the tube with the largest curvature pattern absorbs around 21% more energy than that with the smallest curvature pattern. (C) 2016 Elsevier B.V. All rights reserved.

When drilling circular holes into metal circular pipes, burr is generated at the hole entrance as well as at the hole exit. The burrs generated at the edge curves associated with the outer and inner pipe surfaces must be removed by constant chamfering. Geometrically, the two edge curves can be defined as cylinder to-pipe intersection curves. In this paper, we employ differential geometric properties of the surface-to-surface intersection curves in order to generate an interference-free tool path with constant chamfering for a ball-end cutter. We demonstrate the effectiveness of our method by conducting experiments with physical pipe models. (C) 2016 Elsevier Ltd. All rights reserved.

In the manufacturing process of machine products, it is important to automate the manual visual inspection in detecting microscopic surface defects in order to improve the efficiency and eliminate human errors as well. The basic hardware system consists of a high-resolution camera equipped with a telecentric lens, and a device to change light directions using 60 LED light bulbs. We introduce an accurate automatic system to detect such surface defects based on the novel hardware system and the iterative photometric stereo techniques, which iteratively improve the quality of the estimation of the surface shape. Complex examples are provided to demonstrate the effectiveness of the proposed system.

We introduce a system to reconstruct large scale LEGO models from multiple two dimensional images of objects taken from different views. We employ a unit voxel with an edge length ratio of 5:5:6 for the shape from silhouette method that reconstructs an octree voxel-based three dimensional model with color information from images. We then convert the resulting voxel model with color information into a LEGO sculpture. In order to minimize the number of LEGO bricks, we use a stochastic global optimization method, simulated annealing, to hollow the model as much as possible but keep its strength for portability. Several real complex LEGO models are provided to demonstrate the effectiveness of the proposed method. (C) 2015 Elsevier Ltd. All rights reserved.

This paper studies the recovery of the height field function of an object from its surface normal vectors in terms of uniform bi-quadratic B-spline surfaces. The study is motivated by two modern technological needs namely, the recovery of the height field function of an object from its surface normal vectors obtained by photometric stereo techniques, and the introduction of new design methodology of streamlined aesthetic surfaces through the direct editing of normal vectors. Our mathematical formulation for explicit surface reconstruction is based on differential geometry and geometric modeling techniques. Complex examples are provided to demonstrate the effectiveness of the proposed method. (c) 2015 Elsevier Ltd. All rights reserved.

This paper presents a novel method for modifying the shapes of existing uniform bi-cubic B-spline surfaces by interactively editing the curvatures along isoparametric curves. The method allows us to edit the curvatures of the two intersecting isoparametric curves at each knot with specified positions, unit tangents, and unit normals. The user adjusts the radii of circles, representing the radii of curvature in the u and v isoparametric directions directly via a GUI without having to work with control points and knots. Such shape specifications are converted into iterative repositionings of the control points on the basis of geometrical rules. Using these point-based curvature-editing techniques, we successfully embedded log-aesthetic curves into existing surfaces along their isoparametric curves. Moreover, we were able to distribute the cross curvature with log-aesthetic variation along the isoparametric curves. We applied our technique to the design of automobile hood surfaces to demonstrate the effectiveness of our algorithms. (C) 2014 Elsevier Ltd. All rights reserved.

We present algorithms for computing the differential geometry properties of lines of curvature of parametric surfaces. We derive a unit tangent vector, curvature vector, binormal vector, torsion, and algorithms to evaluate their higher-order derivatives of lines of curvature of parametric surfaces. Among these quantities, it is shown that the curvature and its first derivative of the lines of curvature lend a hand for the formation of curved plates in shipbuilding. We also visualize the twist of lines of curvature, which enables us to observe how much the osculating plane of the line of curvature turns about the tangent vector. (C) 2014 Elsevier Inc. All rights reserved.

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline's control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms. (C) 2012 Elsevier Ltd. All rights reserved.

We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms. The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors or normal vectors, and it has C-2 continuity. We compare our method with the conventional curve interpolation methods, namely, the standard point interpolation method, the method introduced by Piegl and Tiller, which interpolates points as well as the first derivatives at every point, and the piecewise cubic Hermite interpolation method. Examples are provided to demonstrate the effectiveness of the proposed algorithms. (C) 2009 Elsevier Ltd. All rights reserved.

We present a method of reconstructing a 3D model from several 2D images of an object taken from different views. The data acquisition system consists of a camera on a tripod, and a computer-controlled turntable. We employ the Shape-From-Silhouette (SFS) method, which constructs a voxel-based 3D model from silhouette images. The concave shapes are further carved by using the space carving technique. In order to make the resulting model compatible with the CAD/CAM system, the voxel model is converted into a triangular mesh using the Marching Cubes algorithm. Since the mesh generated from the voxel model by using the Marching Cubes algorithm inherits the staircase effect, the mesh is adjusted to recover the object precisely by using silhouette images. Finally, we evaluate the accuracy of the proposed method. The reconstructed models of complex objects show the effectiveness of the proposed inexpensive 3D shape reconstruction system.© 2009 CAD Solutions, LLC.

In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a Bezier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.

We herein propose a novel method for removing irregularities of B-spline surfaces via smoothing circular highlight lines. A circular highlight line is defined as a set of points on a surface such that the distance between a circular light source and an extended surface normal to be zero. Circular highlight lines allow us to capture the surface fairness in all directions, whereas conventional method, which uses a family of parallel straight lines for light sources, can capture the surface irregularity only in one direction. This method of correcting surface irregularities through circular highlight lines is intuitive and allows non-skilled persons to generate surfaces that can satisfy requirements imposed by downstream applications. Nonlinear equations that relate the difference between the circular highlight lines of the current surface and the target curves in the parameter space are formulated in terms of control points of the surface to be modified. The nonlinear governing equations are solved by Newton’s method. The effectiveness of these algorithms is demonstrated through examples. © 2007 by CAD Solutions.

A free-form object matching problem is addressed in this paper. Two methods are proposed to solve a partial matching problem with scaling effects and no prior information on correspondence or the rigid body transformation involved. The first method uses umbilical points, which behave as fingerprints of a surface and their qualitative properties can be used for matching purposes. The second method uses an optimization scheme based on the extension of the KH curvature matching method [Comput. Aided Design 35 (2003) 913], first introduced in the context of a matching problem without scaling effects. Two types of curvatures, the Gaussian and the mean curvatures, are used to establish correspondences between two objects. The curvature matching method is formulated in terms of minimization of an objective function depending on the unknown scaling factor, and the rigid body transformation parameters. The accuracy and complexity of the proposed methods as well as the convergence for the optimization approach are analyzed. Examples illustrate the two methods. (C) 2004 Elsevier Inc. All rights reserved.

This paper presents a parametric and feature-based methodology for the design of solids with local composition control (LCC). A suite of composition design features are conceptualized and implemented. The designer can use them singly or in combination, to specify the composition of complex components. Each material composition design feature relates directly to the geometry of the design, often relying on user interaction to specify critical aspects of the geometry. This approach allows the designer to simultaneously edit geometry and composition by varying parameters until a satisfactory result is attained. The identified LCC features are those based on volume, transition, pattern, and (user-defined) surface features. The material composition functions include functions parametrized with respect to distance or distances to user-defined geometric features; and functions that use Laplace's equation to blend smoothly various boundary conditions including values and gradients of the material composition on the boundaries. The Euclidean digital distance transform and the Boundary Element Method are adapted to the efficient computation of composition functions. Theoretical and experimental complexity, accuracy and convergence analyses are presented. The representations underlying the composition design features are analytic in nature and therefore concise. Evaluation for visualization and fabrication is performed only at the resolutions required for these purposes, thereby reducing the computational burden. (C) 2003 Elsevier Ltd. All rights reserved.

This paper presents an overview of surface intersection problems and focuses on the rational polynomial parametric/rational polynomial parametric surface intersection case including transversal and tangential intersections. Emphasis is placed on marching methods with a discussion of the problems with conventional tracing algorithms. An approach using a validated interval ordinary differential equation system solver is outlined and illustrated with examples, which offers significant advantages in robustness over conventional marching schemes. © 2004 by CAD Solutions Company Limited.

Despite extensive research and rapid increase of computing power, free-form object matching still remains a challenging problem in CAD/CAM areas. In this paper, various object features are discussed, and matching methods which use these features are introduced along with robust computational algorithms for umbilical points and intrinsic wireframes. The similarity of matched objects is assessed with three proposed tests. Each algorithm is demonstrated with examples.

A novel method of matching for 3-D-free-form objects (points vs. surface and surface vs. surface) is proposed. The method formulates the problem in terms of solution of a non-linear polynomial equation system, which can be solved robustly by the Interval Projected Polyhedron (IPP) algorithm. Two intrinsic surface properties, the Gaussian and the mean curvatures, are used as object features for matching. The related iso-curvature lines are used to establish the correspondence between two objects. The intersection points of these iso-curvature lines are calculated and sorted out to satisfy the Euclidean constraints from which the translation and rotation transformations are estimated. The performance of the proposed algorithm is also analyzed. This approach can cover global and partial matching, and be applied to automated inspection, copyright protection of NURBS models, and object recognition. Examples illustrate our technique. (C) 2003 Elsevier Science Ltd. All rights reserved.

A simplified thermo-mechanical model for the prediction of angular deformations of metal plates due to line heating is presented. The model utilizes a semi-analytically determined temperature distribution, which incorporates the effects of heat loss and a distributed moving heat source, to find the dimensions of a critical heat-affected region. The dimensions of this region are used to find the angular deformation by an analytic solution method. Predicted values of angular deformation compare well with corresponding values from experimental results. This model is also compared with thermo-mechanical analysis based on the three-dimensional finite element method. The comparisons indicate that the simplified model is fairly accurate and much more efficient in comparison to the finite element solution. (C) 2001 Elsevier Science Ltd. All rights reserved.

A finite element model is developed for thermo-mechanical analysis of the process of metal plate forming by laser line heating. A rezoning technique is adopted to greatly reduce the simulation time. The effects of the refinement of mesh size on temperature distribution and final distortion are studied. Comparison between numerical and experimental results shows a good agreement in final distortion of the formed plate. Finally, edge effects in the laser forming process are studied.

We present an accurate and efficient method to generate a CNC toot path for a smooth free-form surface in terms of planar cubic B-spline curves which will be fed into a free-form curve interpolator. We assume the use of a three-axis CNC machine tool with a ball end-mill cutter. We first interpolate break points, which art: generated by computing the offset surface-driving plane intersection curve reflecting the curvature, by a planar cubic B-spline curve. We then evaluate the maximum scallop height along a scallop curve by computing the stationary points of the distance function between the scallop curve and the design surface. Furthermore, we compute the maximum pick feed such that the maximum scallop height along a scallop curve coincides with the prescribed tolerance. illustrative examples show the substantial improvements this method achieves over conventional methods where the tool path consists of linear or circular paths. (C) 2001 Elsevier Science Ltd. All rights reserved.

This paper presents algorithms for optimal development (flattening) of a smooth continuous curved surface embedded in three-dimensional space into a planar shape. The development process is modeled by in-plane strain (stretching) from the curved surface to its planar development. The distribution of the appropriate minimum strain field is obtained by solving a constrained nonlinear programming problem. Based on the strain distribution and the coefficients of the first fundamental form of the curved surface, another unconstrained nonlinear programming problem is solved to obtain the optimal developed planar shape. The convergence and complexity properties of our algorithms are analyzed theoretically and numerically. Examples show the effectiveness of the algorithms. (C) 2000 Elsevier Science B.V. All rights reserved.

We present algorithms for computing the differential geometry properties of intersection curves of two surfaces where the combination of two surfaces can be parametric-parametric, implicit-implicit and parametric-implicit. We derive unit tangent vector, curvature vector, binormal vector, curvature, torsion, and algorithms to evaluate the higher-order derivatives for transversal as well as tangential intersections for all three types of intersection problems. (C) 1999 Elsevier Science B.V. All rights reserved.

A literature survey on offset curves and surfaces up to 1992 was carried out by Pham (Pham B, Offset curves and surfaces: a brief survey. Computer Aided Design 1992; 24(4): 223-229). The objective of this article is to overview the literature after 1992 and those which were not cited in aforementioned paper. The article focuses on five active areas of research on offsets: (1) representing exact offsets in Bezier/B-spline format, (2) approximations, (3) self-intersections, (4) geodesic offsets and (5) general offsets. (C) 1999 Elsevier Science Ltd. All rights reserved.

A developable surface can be formed by bending or rolling a planar surface without stretching or tearing; in other words, it can be developed or unrolled isometrically onto a plane. This paper describes two main differential geometry properties of developable surfaces that have industrial applications. First, a method for identifying the inflection line of a developable surface is described. Second, geodesic lines between two points on a developable surface are found as an initial value problem, which is easier to solve than the boundary value problem that is normally required.

A developable surface can be formed by bending or rolling a planar surface without stretching or tearing; in other words. it can be developed or unrolled isometrically onto a plane. Developable surfaces are widely used in the manufacture of items that use materials that are not amenable to stretching such as the formation of ducts, shoes, clothing and automobile parts including upholstery and body panels (Frey & Bindschadler 1993). Designing a ship hull entirely of developable surfaces would allow production of the hull using only rolling or bending. Heat treatment would only be required for removal of distortion, thus greatly reducing the labor required to form the hull. Although developable surfaces play an important role in various manufacturing applications, little attention has been paid to implementing developable surfaces from the onset of a design. This paper investigates novel. user friendly methods to design complex objects using B-spline developable surfaces based on optimization techniques. Illustrative examples show the substantial improvements this method achieves over previously developed methods.

Developable surfaces are widely used in various engineering applications. However, little attention has been paid to implementing developable surfaces from the onset of a design. The first half of the paper describes a user friendly method of designing developable surfaces in terms of a B-Spline representation whose two directrices lie on parallel planes. The second half of the paper investigates a new method for development and tessellation of such B-Spline developable surfaces which is necessary for plate cutting and finite element analysis.

We present an efficient and reliable method for computing the unit-in-the-last-place (ulp) of a double-precision floating-point number, taking advantage of the standard binary representation for floating-point numbers defined by IEEE Std 754-1985. The ulp is necessary to perform software rounding for robust rounded-interval arithmetic (RIA) operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid modeling. (C) 1998 Elsevier Science Ltd. All rights reserved.

A pipe (or tubular) surface is the envelope of a one-parameter family of spheres with constant radii r and centers C(t). In this paper we investigate necessary and sufficient conditions for the nonsingularity of pipe surfaces. In addition, when C(t) is a rational function, we develop an algorithmic method for the rational parametrization of such a surface. The latter is based on finding two rational functions alpha(t) and beta(t) such that /C'(t)/(2) = alpha(2)(t) + beta(2)(t) (Lu and Portmann, 1996). (C) 1998 Elsevier Science B.V.

The paper investigates self-intersections of offsets of implicit quadratic surfaces, The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can de locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e, the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.

Although offset surfaces ape widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally mole complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature.
It is well known that any regular surface can be locally approximated in the neighborhood of a point p by the explicit quadratic surface of the form r(x, y) = [x, y, 1/2 (alpha x(2) + beta y(2))](T) to the second order where -alpha and -beta are the principal curvatures at point p. Therefore investigations of the self-intersecting mechanisms of the offsets of explicit quadratic surfaces die to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. in this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-intersection curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.

In this paper, we develop robust algorithms for computing interval polynomial curve-to-surface and surface-to-surface intersections. These include well-conditioned transversal intersections as well as ill-conditioned non-transversal intersections. Key components of our methods are the reduction of the intersection problems into solving balanced or unbalanced systems of non-linear interval polynomial equations. These systems are solved using an interval non-linear polynomial solver based on Bernstein subdivision coupled with rounded interval arithmetic, documented in a series of earlier papers. The solver provides results with numerical certainty and verifiability. Examples illustrate our techniques. We also provide a theoretical analysis of degenerate interval polynomial curve-to-surface and surface-to-surface ill-conditioned non-transversal intersections. (C) 1997 Elsevier Science Ltd.

Self-intersection of offsets of regular Bezier surface patches due to local differential geometry and global distance function properties is investigated. The problem of computing starting points for tracing self-intersection curves of offsets is formulated in terms of a system of nonlinear polynomial equations and solved robustly by the interval projected polyhedron algorithm. Trivial solutions are excluded by evaluating the normal bounding pyramids of the surface subpatches mapped from the parameter boxes computed by the polynomial solver with a coarse tolerance. A technique to detect and trace self-intersection curve loops in the parameter domain is also discussed. The method has been successfully tested in tracing complex self-intersection curves of offsets of Bezier surface patches. Examples illustrate the principal features and robustness characteristics of the method.

In this paper, we develop and study a robust algorithm for computing intersections of two planar interval polynomial curves. The intersection problems include well-conditioned transveral intersections as well as ill-conditioned cases such as tangential and overlapping intersections. Key components of our methods are the reduction of the intersection problems into solving systems of nonlinear interval polynomial equations which consist of m equations with n unknowns. An earlier interval nonlinear polynomial solver for balanced system based on Bernstein subdivision method coupled with rounded interval arithmetic is extended to solve unbalanced systems. The solver provides results with numerical certainty and verifiability. Examples illustrate our techniques. Copyright (C) 1996 Elsevier Science Ltd

The objective of this paper is to compute the singularities of a normal offset of a planar integral polynomial curve and the intersections of two specific normal offsets of two planar integral polynomial curves. Singularities include irregular points (such as isolated points and cusps) and self-intersections. The key element in the above techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The curves that we are investigating are described by polynomial functions, but their offset curve representations involve polynomials and square roots of polynomials. A methodology based on adaptive subdivision techniques to solve the resulting systems of nonlinear equations is investigated. Key components of our methods are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating point implementation. Examples illustrate our techniques.