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We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals.By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is G 1 , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a G 2 construction and various possible extensions of the method.

We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region-growing-type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

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