Fitting elliptical arcs to points or other curves

Question

I'm struggling to fit an elliptical arc to some points. The points are either from line segments or all part of the same polyline or they are generated from a bezier curve. I basically have two issues:

1. I would like to fit an elliptical arc to the points if it is reasonable and i straight line if that is a better fit. I'm thinking about starting at the first point and then constructing a line and a ellip_arc to the next points. The one with the lowest error wins. Or something like that at least. The problem is that an ellipse has many free parameters as opposed to a straight line. So how could I fit an ellipse arc to the points? It also has to start and stop at the points.

2. Fitting seems to be easiest when the ellipse arc is parameterized. Even though I find it hard to define where to start and stop the arc. But, as output, I need to have the ellipse described like in SVG format (Center Point, Minor and Major Axis, rotation to x-axis etc. https://www.w3.org/TR/SVG2/paths.html#PathDataEllipticalArcCommands). I'm not sure how to convert to this representation, or maybe I can fit it like that?

Answer 1

see:

• Circular approximation of polygon (or its part)

If you compute this for your curve samples from local change of radius and center you could group and estimate parts of curve that belongs to the same ellipse and also its eccentricity and or a,b semiaxises sizes , center and orientation to ease up your fitting... Even if not get the precise value it will be a start point and range for ellipse parameter fitting hugely improving speed and stability of fitting

Some of the ellipse parameters might be obtained directly if your data has enough large chunks of ellipses see:

• Algorithms: Ellipse matching

Fitting SVG like parameters fully will be very slow easier would be to fit just the ellipses first (center,a,b,rotation) and then convert/fit to SVG form elliptic arc. See:

• Converting an svg arc to lines
• Express SVG arc as series of curves

And finally you can use any fitting algorithm my favorite is this one:

• approximation search

I do not code in C# so I have no idea about any existing packages for tasks like this.