Fitting to Conic Sections


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Contact: F.X.Timmes
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Lots of tools exist for fitting a set a data to a straight line. How about fits to a general conic $$ a x^2 + b x y + c y^2 + d x + e y + f = 0 \ , \label{eq1} \tag{1} $$ perhaps with a constraint that enforces a specific conic section? References I found useful include Thomas & Chan 1989, Halir & Flusser 1998, Fitzgibbon et al 1999, and Harker et al 2008.

For direct (i.e., not iterative) least-squares fitting of data to an ellipse, fit_ellipse.f90.tbz will generate noisy data for an ellipse (an interesting problem itself!), fit the noisy data to equation (1) with the constraint $b^2 - 4 a c \lt 0$ that assures an ellipse, and report key geometric attributes such as the center coordinates, foci coordinates, lengths of the semi-major and semi-minor axes, and rotatation angle of the conic. This tool currently depends on the lapack routines dgetrf and dgetri to invert a 3x3 matrix and the lapack routine dgeev to obtain the eigensystem of a 3x3 matrix.


For direct, least-squares fitting of data to a hyperbola, fit_hyperbola.tbz will do the same as above, except with the constraint $b^2 - 4 a c \gt 0$ that assures a hyperbola.

Parabolas, defined by the constraint $b^2 - 4 a c = 0$ with zero to near machine precision, lie on the boundary surface between ellipses and hyperbolas. Two of the figures below show that $b^2 - 4 a c = 0$ is an elliptical cone with its vertex at the origin. All ellipses lie within the cone, hyperbolas lie outside the cone. Direct, least-squares fitting of data to a parabola is thus more involved than for ellipses or hyperbola. The tool fit_parabola.tbz will do the same exercises as its ellipse and hyperbola cousins above.

image image

For direct, least-squares fitting of data to a circle where the measure of fit goodness is an area and not a length, the tool fit_circle.tbz will do the same as its cousins above.


For iterative least-squares fitting of data to a non-specific conic, the tool fit_nonspecific_conic.tbz will do a traditional singular value decomposition on equation (1) ala e.g., Gander, Golub & Strebel 1994. Here is an example of noisy parabolic data with the returned best fit coefficients suggesting the geometery is an ellipse.

Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.