Computer Aided Geometric Design 19 (2002) 19–42 www.elsevier.com/locate/comaid Advanced surface fitting techniques V. Weiss a,∗ , L. Andor b , G. Renner a , T. Várady a a Computer and Automation Research Institute, Budapest, Hungary b CADMUS Consulting and Development Ltd., Budapest, Hungary Received July 2001; revised August 2001 Abstract In spite of extensive research on fitting parametric surfaces, the published ‘standard’ solutions often fail, when data points are irregularly distributed over topologically irregular domains, high accuracy is required and the free quantities of least squares fitting—such as the number and placement of knots, the weights of the smoothing functionals and the best parametrisation of the data points—must be set without user assistance. Further difficulties arise when the fitted surface needs to be extended in a natural way and hole loops without underlying point data need to be covered smoothly. This paper attempts to analyse the above difficulties and provide practical solutions to overcome these. Main results include algorithms to compute a good initial parametrisation, a fitting strategy to maintain tight tolerances and smoothness simultaneously, to handle weakly defined control points and a shape dependent knot refinement procedure. A few examples and suggestions for future work conclude the paper.  2001 Elsevier Science B.V. All rights reserved. Keywords: Parametrization; Surface fitting 1. Introduction The problem of surface fitting can be stated in the following way. Given a finite set of data points p r ∈ R n and associated tolerances ε r ∈ R + . We search for a ‘pleasing’ surface S, which approximates each point p r within tolerance ε r ‖S p r - p r ‖  ε r , r ∈{1,...,m}. (1) S p r denotes a point on the surface associated with the data point p r . This abstract formu- lation does not lead to any direct algorithm for the computation of S, and there are various * Corresponding author. E-mail address: weiss@sztaki.hu (V. Weiss). 0167-8396/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0167-8396(01)00086-3