We present techniques for creating an approximate implicit representation of space curves and of surfaces of revolution. In both cases, the proposed techniques reduce the problem to that of implicitization of planar curves. For space curves, which are described as the intersection of two implicitly defined surfaces, we show how to generate an approximately orthogonalized implicit representation. In the case of surfaces of revolution, we address the problem of avoiding unwanted branches and singular points in the region of interest.

We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization. It is shown that these techniques are able to handle general free-form surfaces, and they can therefore be used to exploit the duality of implicit and parametric representations. In addition, we discuss several applications of these techniques, such as detection of self-intersections, raytracing, footpoint computation and parameterization of scattered data for parametric curve or surface fitting.

We compare several methods for approximate implicitization by piecewise polynomials which have been developed by the authors, and a linear-algebrabased numerical method for implicitization which is provided as a part of MAPLE. We investigate both quantitative criteria (such as computing time, memory use, and the error of the approximation) and qualitative criteria. As demonstrated by the results, piecewise approximate implicitization is able to handle surfaces arising in industrial applications. However, special care has to be taken to avoid additional branches and unwanted singularities.

We present a new method for constructing a low degree C 1 implicit spline representation of a given parametric planar curve. To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces. Adaptive knot removal, which is based on spline wavelets, is used to reduce the number of segments. The B-spline segments are implicitized. After multiplying the implicit B-spline segments by suitable polynomial factors the resulting bivariate functions are joined along suitable transversal lines. This yields to a globally C 1 bivariate function.

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