Elaine Cohen

Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics

Abstract The relevant theory of discrete B-splines with associated new algorithms is extended to provide a framework for understanding and implementing general subdivision schemes for nonuniform B-splines. The new derived polygon corresponding to an arbitrary refinement of the knot vector for an existing B-spline curve, including multiplicities, is shown to be formed by successive evaluations of the discrete B-spline defined by the original vertices, the original knot vector, and the new refined knot vector. Existing subdivision algorithms can be seen as proper special cases. General subdivision has widespread applications in computer-aided geometric design, computer graphics, and numerical analysis. The new algorithms resulting from the new theory lead to a unification of the display model, the analysis model, and other needed models into a single geometric model from which other necessary models are easily derived. New sample algorithms for interference calculation, contouring, surface rendering, and other important calculations are presented.

A non-photorealistic lighting model for automatic technical illustration

Phong-shaded 3D imagery does not provide geometric information of the same richness as human-drawn technical illustrations. A non-photorealistic lighting model is presented that attempts to narrow this gap. The model is based on practice in traditional technical illustration, where the lighting model uses both luminance and changes in hue to indicate surface orientation, reserving extreme lights and darks for edge lines and highlights. The lighting model allows shading to occur only in mid-tones so that edge lines and highlights remain visually prominent. In addition, we show how this lighting model is modified when portraying models of metal objects. These illustration methods give a clearer picture of shape, structure, and material composition than traditional computer graphics methods. CR Categories: I.3.0 [Computer Graphics]: General; I.3.6 [Computer Graphics]: Methodology and Techniques.

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.

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.

A framework for efficient minimum distance computations

We present a framework for minimum distance computations that allows efficient solution of minimum distance queries on a variety of surface representations, including sculptured surfaces. The framework depends on geometric reasoning rather than numerical methods and can be implemented straightforwardly. We demonstrate performance that compares favorably to other polygonal methods and is faster than reported results for other methods on sculptured surfaces.

Volumetric parameterization and trivariate B-spline fitting using harmonic functions

We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate B-spline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [Hughes, T.J., Cottrell, J.A., B., Y., 2005. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 4135-4195]. Input data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate B-spline geometric and attribute representations are generated from the resulting parameterization, creating trivariate B-spline material property representations over the same parameterization in a way that is related to [Martin, W., Cohen, E., 2001. Representation and extraction of volumetric attributes using trivariate splines. In: Symposium on Solid and Physical Modeling, pp. 234-240] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a B-spline representation with guaranteed quality of approximation to the original data. Then we focus attention on a model of simulation interest, a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate B-spline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis.

Direct haptic rendering of sculptured models

A new tracing algorithm is described that supports haptic rendering of NURBS surfaces without the use of any intermediate representation. By using this tracing algorithm in conjunction with algorithms for surface proximity testing and surface transitions, a complete haptic rendering system for sculptured models has been developed. The system links an advanced CAD modeling system with a Sarcos force-reflecting exo-skeleton arm. A method for measuring the quality of the tracking component of the haptic rendering separately from the haptic device and force computation is also described. CR Descriptors: H.1.2 [Models and Principles] User/Machine Systems; C.3 [Special-Purpose and Application-Based Systems] Real-Time Systems; I.3.7 [Computer Graphics] Three-Dimensional Graphics and Realism; I.6.4 [Simulation and Modeling] Types of Simulation - Distributed; F.2.2 [Analysis of Algorithms and Problem Complexity] Nonnumerical Algorithms and Problems; J.6 [Computer-Aided Engineering].

Practical ray tracing of trimmed NURBS surfaces

Abstract A system is presented for ray tracing trimmed NURBS surfaces. While approaches to components are drawn largely from existing literature, their combination within a single framework is novel. This paper also differs from prior work in that the details of an efficient implementation are fleshed out. Throughout, emphasis is placed on practical methods suitable to implementation in general ray-tracing programs.

ERROR BOUNDED VARIABLE DISTANCE OFFSET OPERATOR FOR FREE FORM CURVES AND SURFACES

Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset.

Algorithms for degree-raising of splines

Stable and efficient algorithms for degree-raising of curves (or surfaces) represented as arbitrary B-splines are presented as a application of the solution to the theoretical problem of rewriting a curve written as a linear combination of mth order B-splines as a linear combination of (m + 1)st order B-splines with a minimal number of knot insertions. This approach can be used to introduce additional degrees of freedom to a curve (or surface), a problem which naturally arises in certain circumstances in constructing mathematical models for computer-aided geometric design.