Geometric least-squares fitting of spheres, cylinders, cones and tori G.Luk´acs * , A. D. Marshall, R. R. Martin Dept of Computer Science University of Wales, Cardiff PO Box 916, Cardiff, UK, CF2 3XF May 23, 1997 Abstract This paper considers a problem arising in the reverse engineering of boundary represen- tation solid models from three-dimensional depth maps of scanned objects. In particular, we wish to identify and fit surfaces of known type wherever these are a good fit, and we briefly outline a segmentation strategy for deciding to which surface type the depth points should be assigned. The particular contributions of this paper are methods for the least-squares fitting of spheres, cylinders, cones and tori to three-dimensional data. While plane fitting is well understood, least-squares fitting of other surfaces, even of such simple geometric type, has been much less studied; we review previous approaches to the fitting of such surfaces. Our method has the particular advantage of being robust in the sense that as the principal curvatures of the surfaces being fitted decrease (or become more equal), the results which are returned naturally become closer and closer to the surfaces of “simpler type”, i.e. planes, cylinders, or cones (or spheres) which best describe the data, unlike other methods which may diverge as various parameters or their combination become infinite. Keywords. Non-linear least squares. Geometric distance. Cylinder, cone, sphere, torus, surface fitting. 1 Introduction This paper considers the problem of least-squares fitting of spheres, cylinders, cones and tori to three-dimensional point data. The motivation for this problem lies in reverse engineering of geometric shape. A laser scanner or similar device is used to capture three-dimensional point data sampled from the surface of an object. From this we wish to construct a boundary representation solid model of the object’s shape. In particular, we wish to identify and fit simple surfaces of known type to portions of the boundary wherever these are in good agreement with the point data. The problem can be decomposed into two logical steps: segmentation, where the data points are grouped into sets each belonging to a different surface, and fitting, where the best surface of an appropriate type is fitted to each set of points. The new results in this paper mainly concern the latter problem, but we first outline the method we use for solving the former, as it has a significant effect on the final model created. While plane fitting is well understood, least-squares fitting of other surfaces, even of simple geometric type, has been relatively much less studied. We review previous approaches to the fitting of spheres, cylinders, cones and tori, and then present some new results on fitting these surfaces. Our method has the particular advantage of being robust in the sense that as the * Visiting from the Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1518 Budapest, POB 63 1