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.

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.

. 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...