. We discuss adaptive enumeration and rendering methods for implicit surfaces, using octrees computed with affine arithmetic, a new tool for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and sub-formulas, generally providing much tighter bounds for the computed quantities. The resulting octrees are accordingly much smaller, and the rendering faster. We also describe applications of affine arithmetic to intersection and ray tracing of implicit surfaces. keywords: cellular models, interval analysis, rendering, implicit surfaces. 1 Introduction Implicit surfaces have recently become popular in computer graphics and solid modeling. In order to exploit existing hardware and algorithms, it is often necessary to approximate such surfaces by models with simpler geometry, such as polygonal meshes or voxel arrays. Let S be a surface defined implicitly by the equation h(x; y; z) = 0. A simple and general techn...

This paper describes a simple and efficient polygonization algorithm that gives a practical way to construct adapted piecewise linear representations of implicit surfaces. The method starts with a coarse uniform polygonal approximation of the surface and subdivides each polygon recursively according to local curvature. In that way, the inherent complexity of the problem is tamed by separating structuring from sampling and reducing part of the full three dimensional search to two dimensions.

In this paper we present a framework for the approximate conversion of parametric to implicit surfaces. It takes as input a parametric description and generates a piecewise analytic implicit representation. The conversion process consists of three steps: 1) the parametric surface is rasterized into a volumetric characteristic function. 2) this binary function is converted into a volume array corresponding to samples of a smooth implicit function. 3) this volumetric representation is converted into a multiscale B-spline model. This method is based on wavelet analysis and synthesis techniques and is very general. It can convert to implicit form any geometric object that is suitable for discretization into a characteristic function. 1. Introduction The two main forms of describing the geometry a solid object are the parametric and implicit descriptions. In the parametric description, the solid is specified through its bounding surface, usually in a piecewise manner. In the implic...

. After reviewing three classical sampling methods for implicit objects, we describe a new sampling method that is not based on scanning the ambient space. In this method, samples are "randomly" generated using physically-based particle systems. Introduction In computer graphics, an object is described either by a set of sample points or by an analytic scheme that uses mathematical equations to define its geometry and topology. Descriptions based on samples occur in areas such as medical images and terrain models. Analytical descriptions are usually found in applications of geometric modeling, such computeraided design and manufacture. When an object is described by samples, a reconstruction scheme is needed to recover its geometry and topology from the samples. This problem, called structuring, consists of providing a combinatorial structure to the samples in order to (ideally) recover the exact topology of the object and an approximation of its geometry. When the object is describe...

. This paper describes a simple algorithm for the adaptive polygonization of implicit surfaces. It gives a practical way to construct optimal piecewise linear representations. The method starts with a coarse uniform polygonal approximation of the surface and subdivides each polygon recursively according to local curvature. In that way, the inherent complexity of the problem is tamed by reducing part of the full three dimensional search to two dimensions. 1 Introduction Implicit models constitute a powerful mathematical description of the geometry of three dimensional objects [10]. Under this framework, a surface is defined as the set of points which satisfy the equation f(x; y; z) = 0. Simple primitive implicit shapes can be specified by algebraic functions, such as quadrics, [5]. More complex implicit shapes can be specified by combining primitives using point set or blend operations that are the basis of, respectively, CSG, [13], and Blobby models, [4]. The implicit descriptio...

We discuss the problem of adaptive polygonization of regular surfaces of the euclidean 3D space, and present effective algorithms for computing optimal polygonizations of surfaces described in parametric or implicit form. Keywords: Surface approximation, polygonization, parametric surfaces, implicit surfaces, geometric modeling. 1 Introduction The polygonization of surfaces is a classical problem in computer graphics and geometric modeling that has many practical applications. The problem is computing a piecewise linear approximation for a smooth surface described either in parametric or implicit form. In this paper, we present a conceptual framework for the piecewise linear approximation of surfaces and also a methodology for computing good polygonal approximations while keeping the number of polygons low. Based on the general principles in this methodology, we describe two specific new algorithms for the adaptive polygonization of parametric and implicit surfaces. 1.1 Importance...

We divide the computation of polygonal approximations of implicit objects into two phases: sampling and structuring. Unlike classical treatments, we study each phase separately. Classical sampling methods are reviewed, and a new sampling method is proposed; this method uses physically-based particle systems, and is more robust than classical enumeration and continuation methods. We describe a broad taxonomy of structuring problems. According to this taxonomy, the presence of an independent structuring phase casts the modeling problem as a problem in non-parametric scattered data interpolation, which we propose to solve using computational morphology. The use of global structuring is described in detail for curves: we prove that minimal spanning trees are good morphological tools for arcs. A couple of methods are also suggested for surfaces.

This tutorial paper gives an overview of the area of implicit surfaces. The history of the technique is presented and various approaches categorised and compared. Existing problems with modelling, rendering and animation of implicit surfaces are discussed and available solutions evaluated. The paper provides a comprehensive introduction for novice and experienced users of implicit surfaces. Keywords: Implicit Surfaces, Deformable Objects, Modelling Techniques, Animation Control, Review 1. Introduction The most common methods of modelling the surface of an object are to explicitly represent it as a set of polygons or parametric patches. Many Computer-Aided Design (CAD) and computer graphics systems utilise these modelling techniques. However, they have their limitations, especially when smooth, deformable objects need to be represented and animated. Then, it is time for implicit surfaces to step into the limelight. As their name suggests the surface of an object is not modelled explic...

Objects can be represented at different levels of detail within a required precision. A given level of detail can be defined based on the concept of scale. Not rarely applications need to deal with a single model represented at different scales. Two main approaches are used in Computer Graphics to generate different versions of an object: intrinsic and extrinsic. Both techniques are based on removing details from an object. The main difference between them is that intrinsic methods work directly on a surface whereas extrinsic methods use ambient space. This paper proposes a new way to represent objects in different levels of detail by combining intrinsic and extrinsic methods.

This paper introduces a unified and general tesselation algorithm for parametric and implicit surfaces. The algorithm produces a hierarchical mesh that is adapted to the surface geometry and has a multiresolution and progressive structure. This representation can be exploited with advantages in several applications.