Shape-fitting Using Pivotal Models

A general method is given for fitting a shape to points measured with statistical error on a surface of known “blueprint” shape. The results have application in standards for the quality of engineered surfaces. Pivotal models are an extension of the structured models devised by Fraser (1968). The method is developed here only for the fitting of circles and ellipses. The shapes of the megalithic stone rings at Brogar and Avebury are used to illustrate the procedure. An introduction is given briefly to the extension to multi-pivotal models, which are a step-by-step combination of pivotal models. This paper revisits work carried out by Paul James Scott (Scott, 1983). MSC2000: 60B15, 62F15. 1 Estimating engineered surfaces This study developed out of an engineering problem in optics. Engineered surfaces can be measured to nanometric precision, a much greater precision than engineering can achieve, an exception being the mirrors and lenses required for telescopes such as the Hubble telescope. The problem is that of estimating the model surface (the “blueprint shape”) from a series of points on its engineered surface. Sometimes, as with artificial joints, the surfaces are required to be rough for mechanical reasons, but the fit of ball to socket must still be precise. A particular problem is when the surface to be measured can only with difficulty be taken to a laboratory for measurement. An example of this is a ship’s propellor shaft that is suspected of being out of true. The method shown here allows the circularity to be fitted and tested from data collected in situ from just an arc of the circle. The aim of this paper is to bring to a wider audience in a brief and elementary way the methodology used by Paul J. Scott in his unpublished PhD thesis. The passage of nearly twenty years permits the further exploitation of his treatment taking advantage of the more sophisticated computing now available. In section 2 we consider roundness and the structural probability circle. In section 3 pivotal models are described. In section 4 these are generalised to include the composition of transformations, giving the multi-pivotal model. We conclude with examples of circle and ellipse fitting. SHAPE-FITTING USING PIVOTAL MODELS 206