A Novel Approach to Orthogonal Distance Least Squares Fitting of General Conics

Fitting of conics to a set of points is a well researched area and is used in many fields of science and engineering. Least squares methods are one of the most popular techniques available for conic fitting and among these, orthogonal distance fitting has been acknowledged as the ’best’ least squares method. Although the accuracy of orthogonal distance fitting is unarguably superior, the problem so far has been in finding the orthogonal distance between a point and a general conic. This has lead to the development of conic specific algorithms which take the characteristics of the type of conic as additional constraints, or in the case of a general conic, the use of an unstable closed form solution or a non-linear iterative procedure. Using conic specific constraints produce inaccurate fits if the data does not correspond to the type of conic being fitted and in iterative solutions too, the accuracy is compromised. The method discussed in this paper aims at overcoming all these problems, in introducing a direct calculation of the orthogonal distance, thereby eliminating the need for conic specific information and iterative solutions. We use the orthogonal distances in a fitting algorithm that identifies which type of conic best fits the data. We then show that this algorithm requires less accurate initializations, uses simpler calculations and produces more accurate results.