Fitting conics of specific types to data

For each finite set of points in the Euclidean plane, and for each type of conic section—elliptic, hyperbolic, or parabolic––the algorithm presented here determines all the algebraically best fitting conics of the selected type: the best ellipse, the best hyperbola, or the best parabola. The supporting theory expands on Golub, Hoffman, and Stewart’s generalization of the Schmidt–Mirsky matrix approximation theorem, on Bookstein’s and Pratt’s methods to fit conics, and on Gander, Golub, and Strebel’s method to fit ellipses. Because neither the set of ellipses nor the set of hyperbolae is closed, the algorithm and its supporting theory must accommodate their boundary, which consists of the parabolic conics. The corresponding optimization problem consists in minimizing a quadratic form with two quadratic constraints, which an orthogonal change of variables transforms into a least-squares problem with one quadratic constraint. Hence analogies with geodetic coordinates identify geometric causes of numerical instability. For each type of conic, the resulting best fitting conic remains invariant under Euclidean transformations. Applications include the theory and use of sundials in archaeology and astronomy.