Gershon Elber

Geometric modeling with splines : an introduction

Written by researchers who have helped found and shape the field, this book is a definitive introduction to geometric modeling. The authors present all of the necessary techniques for curve and surface representations in computer-aided modeling with a focus on how the techniques are used in design. They achieve a balance between mathematical rigor and broad applicability. Appropriate for readers with a moderate degree of mathematical maturity, this book is suitable as an undergraduate or graduate text, or particularly as a resource for self-study.

Geometric constraint solver using multivariate rational spline functions

We present a new approach to building a solver for a set of geometric constraints represented by multivariate rational functions. The constraints are formulated using inequalities as well as equalities. When the solution set has dimension larger than zero, we approximate it by fitting a hypersurface to discrete solution points. We also consider a variety of constraint solving problems common in geometric modeling. These include computing ray-traps, bisectors, sweep envelopes, and regions accessible during 5-axis machining.

Inferring 3D models from freehand sketches and constraints

Abstract This paper describes ‘Quick-sketch’, a 2D and 3D modelling tool for pen-based computers. Users of this system define a model by simple pen strokes, drawn directly on the screen of a pen-based PC. Exact shapes and geometric relationships are interpreted from the sketch. The system can also be used to sketch 3D solid objects and B-spline surfaces. These objects may be refined by defining 2D and 3D geometric constraints. A novel graph-based constraint solver is used to establish the geometric relationships, or to maintain them when manipulating the objects interactively. The approach presented here is a first step towards a conceptual design system. Quick-sketch can be used as a hand sketching front-end to more sophisticated modelling, rendering or animation systems.

Toolpath generation for freeform surface models

Abstract The generation of optimal NC code to drive milling machines for models defined by freeform trimmed surfaces is a difficult problem. In practice, two main approaches are used to generate toolpaths for surfaces, neither of which is optimal, in general. The first exploits the parametric representation, and generates isocurves that are uniformly distributed across the parametric domain. This approach is not optimal if the surface mapping into Euclidean space is not isometric. The second approach contours the models by intersecting the surfaces with planes equally spaced in Euclidean space, resulting in a piecewise-linear toolpath approximation which is nonadaptive to the local surface geometry. Further, the toolpath generated by contouring is suitable for 3-axis milling, but is inappropriate for 5-axis milling. In the paper, an algorithm developed to extract isocurves for rendering adaptively is modified and enhanced to generate milling toolpaths for models consisting of trimmed surfaces, and it can be used in both -3 and 5-axis milling. The resulting toolpaths do not gouge locally, and they combine the advantages of both prior approaches. The output toolpath is appealing, since it is composed of isoparametric curves, and is therefore compact, exact, and easy to process. Further, it is more optimal than the previous methods in that the resulting toolpath is shorter, and it provides a direct quantitative bound on the resulting scallop height. This algorithm has been used to compute gouge-avoiding toolpaths for the automatic milling of freeform surfaces, without the introduction of auxiliary check and drive surfaces being required.

Comparing offset curve approximation methods

Offset curves have diverse engineering applications, spurring extensive research on various offset techniques. In a paper on offset curve approximation (Lee et al., 1996), we suggested a new approach based on approximating the offset circle instead of the offset curve itself. To demonstrate the effectiveness of this approach, we compared it extensively with previous methods. To our surprise, Tiller and Hansons (1984) simple method outperforms other methods for offsetting (piecewise) quadratic curves, even though its performance is not as good for high degree curves. The experimental results revealed other interesting facts, too. Had these details been reported several years ago, we believe offset approximation research might have developed somewhat differently. This article is intended to fill an important gap in the literature. We conducted qualitative as well as quantitative comparisons employing various contemporary offset approximation methods for freeform curves in the plane. We measured the efficiency of the offset approximation in terms of the number of control points generated while making the approximations within a prescribed tolerance.

A comparison of Gaussian and mean curvatures estimation methods on triangular meshes

Estimating intrinsic geometric properties of a surface from a polygonal mesh obtained from range data is an important stage of numerous algorithms in computer and robot vision, computer graphics, geometric modeling, industrial and biomedical engineering. This work considers different computational schemes for local estimation of intrinsic curvature geometric properties. Five different algorithms and their modifications were tested on triangular meshes that represent tessellations of synthetic geometric models. The results were compared with the analytically computed values of the Gaussian and mean curvatures of the non-uniform rational B-spline (NURBs) surfaces, these meshes originated from. This work manifests the best algorithms suited for that indeed different algorithms should be employed to compute the Gaussian and mean curvatures.

Planar curve offset based on circle approximation

An algorithm is presented to approximate planar offset curves within an arbitrary tolerance �> 0. Given a planar parametric curve C(t) and an offset radius r, the circle of radius r is first approximated by piecewise quadratic Bezier curve segments within the tolerance � . The exact offset curve Cr(t) is then approximated by the convolution of C(t) with the quadratic Bezier curve segments. For a polynomial curve C(t )o f degree d, the offset curve Cr(t) is approximated by planar rational curves, C a r (t)s, of degree 3d − 2. For a rational curve C(t) of degree d, the offset curve is approximated by rational curves of degree 5d − 4. When they have no self-intersections, the approx- imated offset curves, C a (t)s, are guaranteed to be within � -distance from the exact offset curve Cr(t). The effectiveness of this approximation technique is demonstrated in the offset computation of planar curved objects bounded by polynomial/rational parametric curves.

Polynomial/rational approximation of Minkowski sum boundary curves

Given two planar curves, their convolution curve is defined as the set of all vector sums generated by all pairs of curve points which have the same curve normal direction. The Minkowski sum of two planar objects is closely related to the convolution curve of the two object boundary curves. That is, the convolution curve is a superset of the Minkowski sum boundary. By eliminating all redundant parts in the convolution curve, one can generate the Minkowski sum boundary. The Minkowski sum can be used in various important geometric computations, especially for collision detection among planar curved objects. Unfortunately, the convolution curve of two rational curves is not rational, in general. Therefore, in practice, one needs to approximate the convolution curves with polynomial/rational curves. Conventional approximation methods of convolution curves typically use piecewise linear approximations, which is not acceptable in many CAD systems due to data proliferation. In this paper, we generalize conventional approximation techniques of offset curves and develop several new methods for approximating convolution curves. Moreover, we introduce efficient methods to estimate the error in convolution curve approximation. This paper also discusses various other important issues in the boundary construction of the Minkowski sum.

Line art illustrations of parametric and implicit forms

A technique is presented for line art rendering of scenes composed of freeform surfaces. The line art that is created for parametric surfaces is practically intrinsic and is globally invariant to changes in the surface parameterization. This method is equally applicable for line art rendering of implicit forms, creating a unified line art rendering method for both parametric and implicit forms. This added flexibility exposes a new horizon of special, parameterization independent, line art effects. Moreover, the production of the line art illustrations can be combined with traditional rendering techniques such as transparency and texture mapping. Examples that demonstrate the capabilities of the proposed approach are presented for both the parametric and implicit forms.

Three dimensional freeform sculpting via zero sets of scalar trivariate functions

This paper presents a three dimensional interactive sculpting paradigm that employs a collection of scalar uniform trivariate Bspline functions. The sculpted object is evaluated as the zero set of the sum of the collection of the trivariate functions defined over a three dimensional working space, resulting in multi-resolution control capabilities. The continuity of the sculpted object is governed by the continuity of the trivariates. The manipulation of the objects is conducted by modifying the scalar control coefficients of the meshes of the participating trivariates. Real time visualization is achieved by incrementally computing a polygonal appra’ximation via the Marching Cubes algorithm. The exploitation of trivariates in this context benefits from the different properties of the Bspline’s representation such as subdivision, refinement and convex hull containment. A system developed using the presented approach has b’een used in various modeling applications including reverse engineering.