Geometric least-squares fitting of spheres, cylinders, cones and tori

This paper considers a problem arising in the reverse engineering of boundary representation 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.