Cococubed.com Fitting to Conic Sections
Home Astronomy Research    2024 Radiative Opacity    2024 Neutrino Emission from Stars    2023 White Dwarfs & 12C(α,γ)16O    2023 MESA VI    2022 Earendel, A Highly Magnified Star    2022 Black Hole Mass Spectrum    2021 Skye Equation of State    2021 White Dwarf Pulsations & 22Ne    Software Instruments      Stellar equation of states      EOS with ionization      EOS for supernovae      Chemical potentials      Stellar atmospheres      Voigt Function      Jeans escape      Polytropic stars      Cold white dwarfs      Adiabatic white dwarfs      Cold neutron stars      Stellar opacities      Neutrino energy loss rates      Ephemeris routines      Fermi-Dirac functions      Polyhedra volume      Plane - cube intersection      Coating an ellipsoid      Nuclear reaction networks      Nuclear statistical equilibrium      Laminar deflagrations      CJ detonations      ZND detonations      Fitting to conic sections      Unusual linear algebra      Derivatives on uneven grids      Pentadiagonal solver      Quadratics, Cubics, Quartics      Supernova light curves      Exact Riemann solutions      1D PPM hydrodynamics      Hydrodynamic test cases      Galactic chemical evolution      Universal two-body problem      Circular and elliptical 3 body      The pendulum      Phyllotaxis      MESA      MESA-Web      FLASH      Zingale's software      Brown's dStar      GR1D code      Iliadis' STARLIB database      Herwig's NuGRID      Meyer's NetNuc AAS Journals    2024 AAS YouTube    2024 AAS Peer Review Workshops 2024 ASU Energy in Everyday Life 2024 MESA Classroom Outreach and Education Materials Other Stuff:    Bicycle Adventures    Illustrations    Presentations Contact: F.X.Timmes my one page vitae, full vitae, research statement, and teaching statement.	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. 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.