In this paper, we survey the state of the art in collision detection between general geometric models. The set of models include polygonal objects, spline or algebraic surfaces, CSG models, and deformable bodies. We present a number of techniques and systems available for contact determination. We also describe several N-body algorithms to reduce the number of pairwise intersection tests. 1 Introduction The goal of collision detection (also known as interference detection or contact determination) is to automatically report a geometric contact when it is about to occur or has actually occurred. The geometric models may be polygonal objects, splines, or algebraic surfaces. The problem is encountered in computer-aided design and machining (CAD/CAM), robotics and automation, manufacturing, computer graphics, animation and computer simulated environments. Collision detection enables simulationbased design, tolerance verification, engineering analysis, assembly and dis-assembly, motion pla...

In a geometric context, a collision or proximity query reports information about the relative configuration or placement of two objects. Some of the common examples of such queries include checking whether two objects overlap in space, or whether their boundaries intersect, or computing the minimum Euclidean separation distance between their boundaries. Hundreds of papers have been published on di#erent aspects of these queries in computational geometry and related areas such as robotics, computer graphics, virtual environments, and computer-aided design. These queries arise in di#erent applications including robot motion planning, dynamic simulation, haptic rendering, virtual prototyping, interactive walkthroughs, computer gaming, and molecular modeling. For example, a large-scale virtual environment, e.g., a walkthrough, creates a model of the environment with virtual objects. Such an environment is used to give the user a sense of presence in a synthetic world and it s

We present an efficient algorithm to compute the intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algebraic formulation. In particular, we propose a matrix representation for the intersection curve and compute it accurately using matrix computations. We present algorithms to compute a start point oneach component of the intersection curve (both open and closed components), detect the presence of singularities, and find all the curve branches near the singularity. We also suggest methods to compute the step size during tracing to prevent component jumping. The algorithm runs an order of magnitude faster than previously published robust algorithms. The complexity of the algorithm is output sensitive.

: The problem of computing the intersection of parametric and algebraic curves arises in many applications of computer graphics and geometric and solid modeling. Previous algorithms are based on techniques from elimination theory or subdivision and iteration. The former is however, restricted to low degree curves. This is mainly due to issues of efficiency and numerical stability. In this paper we use elimination theory and express the resultant of the equations of intersection as a matrix determinant. The matrix itself rather than its symbolic determinant, a polynomial, is used as the representation. The problem of intersection is reduced to computing the eigenvalues and eigenvectors of a numeric matrix. The main advantage of this approach lies in its efficiency and robustness. Moreover, the numerical accuracy of these operations is well understood. For almost all cases we are able to compute accurate answers in 64 bit IEEE floating point arithmetic. Keywords: Intersection, curves, a...

5.4.2 Non-uniform di erencing operator:::::::::::::::::::: 47 5.4.3 Derivative schemes:::::::::::::::::::::::::::: 48 5.5 Parametric analysis:::::::::::::::::::::::::::::::: 50 6 Multi-variate subdivision over regular grids 53

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In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class C4 to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement automatically generates conforming variable-resolution meshes in contrast to face and vertex split methods which require a postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4–8 scheme are C^4 continuous almost everywhere, except at extraordinary vertices where they are is C¹-continuous.

Detailed analysis of many mathematical properties of sculptured models has been hindered by the fact that the properties do not have the same representation as the surface. For example, unit tangents, surface normals, and principal curvatures are typically computed at predefined discrete sets of points on the surface. As such, aliasing can occur and features between samples can be missed. Synthesizing information about the shape of an object and operating on the model, whether by physical machining tools, graphics display programs, or mathematical analysis, has been treated as either a discrete or local problem in general. The researchbeing reported on here has focused on another approach, that of creating algorithms that construct the mathematical properties in closed form, or construct approximations to those mathematical properties through symbolic computation. Global analysis can then be applied while an accurate error bound is obtained.

Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a given set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces non-uniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consequently we show how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.

The representation of freeform surfaces by sufficiently refined polygonal meshes has become common in many geometric modeling applications where complicated objects have to be handled. While working with triangle meshes is flexible and efficient, there are difficulties arising prominently from the lack of infinitesimal smoothness and the prohibitive complexity of highly detailed 3Dmodels. In this paper we discuss the generation of fair triangle meshes which are optimal with respect to some discretized curvature energy functional. The key issues are the proper definition of discrete curvature, the smoothing of high resolution meshes by filter operators, and the efficient generation of optimal meshes by solving a sparse linear system that characterizes the global minimum of an energy functional. Results and techniques from differential geometry, variational surface design (fairing), and numerical analysis are combined to find efficient and robust algorithms that generate smooth meshes of...