Editing Curve fitting (section)

==Geometric fitting of plane curves to data points{{anchor|Plane curves|Geometric}}==
If a function of the form <math>y=f(x)</math> cannot be postulated, one can still try to fit a [[plane curve]].

Other types of curves, such as [[conic sections]] (circular, elliptical, parabolic, and hyperbolic arcs) or [[trigonometric functions]] (such as sine and cosine), may also be used, in certain cases.  For example, trajectories of objects under the influence of gravity follow a parabolic path, when air resistance is ignored.  Hence, matching trajectory data points to a parabolic curve would make sense.  Tides follow sinusoidal patterns, hence tidal data points should be matched to a sine wave, or the sum of two sine waves of different periods, if the effects of the Moon and Sun are both considered.

For a [[parametric curve]], it is effective to fit each of its coordinates as a separate function of [[arc length]]; assuming that data points can be ordered, the [[chord distance]] may be used.<ref>p.51 in Ahlberg & Nilson (1967) ''The theory of splines and their applications'', Academic Press, 1967  [https://books.google.com/books?id=S7d1pjJHsRgC&pg=PA51]</ref>

===Fitting a circle by geometric fit{{anchor|Circles}}===
[[File:Regression circulaire coope arc de cercle.svg|thumb|Circle fitting with the Coope method, the points describing a circle arc, centre (1 ; 1), radius 4.]]
[[File:Wp ellfitting.png|thumb|different models of ellipse fitting]]
[[File:Regression elliptique distance algebrique donnees gander.svg|thumb|Ellipse fitting minimising the algebraic distance (Fitzgibbon method).]]

Coope<ref>{{cite journal|author=Coope, I.D.|title=Circle fitting by linear and nonlinear least squares|journal=Journal of Optimization Theory and Applications |volume =76|issue =2|year=1993|doi=10.1007/BF00939613|pages=381–388|hdl=10092/11104|s2cid=59583785 |hdl-access=free}}</ref> approaches the problem of trying to find the best visual fit of circle to a set of 2D data points. The method elegantly transforms the ordinarily non-linear problem into a linear problem that can be solved without using iterative numerical methods, and is hence much faster than previous techniques.

===Fitting an ellipse by geometric fit{{anchor|Ellipses}}===
The above technique is extended to general ellipses<ref>Paul Sheer, [http://wiredspace.wits.ac.za/bitstream/handle/10539/22434/Sheer%20Paul%201997.pdf?sequence=1&isAllowed=y A software assistant for manual stereo photometrology], M.Sc. thesis, 1997</ref> by adding a non-linear step, resulting in a method that is fast, yet finds visually pleasing ellipses of arbitrary orientation and displacement.