We present a shape representation, the multi-level partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.

We introduce a new method of creating smooth implicit surfaces of arbitrary manifold topology. These surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is created from these constraints using a variational scattered data interpolation approach. We call the iso-surface of this function a variational implicit surface. Like other implicit surface descriptions, these surfaces can be used for CSG and interference detection, may be interactively manipulated, are readily approximated by polygonal tilings, and are easy to ray trace. A key strength is that variational implicit surfaces allow the direct specification of both the location of points on the surface and surface normals. These are two important manipulation techniques that are difficult to achieve using other implicit surface representations such as sums of spherical or ellipsoidal Gaussian functions ("blobbies"). We show that these properties make variational implicit surfaces particularly attractive for interactive sculpting using the particle sampling technique introduced by Witkin and Heckbert in [30]. Our formulation also yields a simple method for converting a polygonal model to a smooth implicit model.

. Morse theory describes the relationship between a function's critical points and the homotopy type of the function's domain. The theorems of Morse theory were developed specifically for functions on a manifold. This work adapts these theorems for use with parameterized families of implicit surfaces in computer graphics. The result is a theoretical basis for the determination of the global topology of an implicit surface, and supports the interactive modeling of implicit surfaces by direct manipulation of a topologically-correct triangulated representation. 1 Introduction Implicit surfaces provide a powerful and versatile shape model in computer graphics by representing geometry as the zero-set of a function over threespace, although displaying such surfaces requires a search through space. The display of an implicit surface is hastened by maintaining a triangulation that can be quickly rendered on modern graphics workstations. However, when the implicit surface changes topological t...

In this chapter, we present distance mapping, a technique for adding small-scale displacement mapping to objects in a pixel shader. We treat displacement mapping as a ray-tracing problem, beginning with texture coordinates on the base surface and calculating texture coordinates where the viewing ray intersects the displaced surface. For this purpose, we precompute a three-dimensional distance map, which gives a measure of the distance between points in space and the displaced surface. This distance map gives us all the information necessary to quickly intersect a ray with the surface. Our algorithm significantly increases the perceived geometric complexity of a scene while maintaining real-time performance. Cook (1984) introduced displacement mapping as a method for adding small-scale detail to surfaces. Unlike bump mapping, which affects only the shading of surfaces, displacement mapping adjusts the positions of surface elements. This leads to effects not possible with bump mapping, such as surface features that occlude each other and

We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions.

In this paper a general method is given for combining CSG modeling with soft blending using implicit surfaces. A class of various blending functions sharing some desirable properties like differentiability and intuitive blend control are given. The functions defining the CSG objects satisfy the Lipschitz condition which gives the possibility of fast root-finding, but can also prove useful in the field of collision detection and adaptive triangulation. Introduction Methods for defining smooth surfaces can be divided into two categories: ffl Parametric functions Parametric functions are functions of the form f(u; v) = (f x (u; v); f y (u; v); f z (u; v)). Typical examples are Bezier or B-spline patches (See [Bohm84] for an overview). The surface is defined by control points and the surface can be adjusted by moving the control points. The surface can be rendered by evaluating the function for different well-chosen values of u and v. ffl Implicit functions Implicit functions have th...

Abstract. The ray-casting of implicit surfaces on GPU has been explored in the last few years. However, until recently, they were restricted to second degree (quadrics). We present an iterative solution to ray cast cubics and quartics on GPU. Our solution targets efficient implementation, obtaining interactive rendering for thousands of surfaces per frame. We have given special attention to torus rendering since it is a useful shape for multiple CAD models. We have tested four different iterative methods, including a novel one, comparing them with classical tessellation solution. Fig. 1. The faces of two bounding boxes are used to trigger the fragment shader responsible for rendering the tori. 1

by Barton Talbot Stander, Ph.D. Washington State University May 1997 Chair: John C. Hart An interactive modeling system for implicit surfaces is presented. The display consists of a polygonal approximation which is guaranteed to have the same topology as the implicit surface. The current work focuses on blended ellipsoids, but could be extended to include any smooth, bounded implicit surface. A polygonization algorithm and an incremental repolygonization algorithm are provided. Treating an implicit surface as a gradient system allows theorems from Morse theory to describe implicit surface topology. An implicit surface changes topology only when a critical value of its defining function changes sign. These critical points may be found using interval analysis. Techniques for modifying the polygonization to accommodate such changes in topology are given. iv Contents 1 Introduction 1 1.1 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Implicit Surfaces...

We present a new definition of an implicit surface over a noisy point cloud, based on the weighted least squares approach. It can be evaluated very fast, but artifacts are significantly reduced. We propose to use a different...

A piecewise smooth surface, possibly with boundaries, sharp edges, corners, or other features is defined by a set of samples. The basic idea is to model surface patches, curve segments and points explicitly, and then to glue them together based on explicit connectivity information. The geometry is defined as the set of stationary points of a projection operator, which is generalized to allow modeling curves with samples, and extended to account for the connectivity information. Additional tangent constraints can be used to model shapes with continuous tangents across edges and corners.