fitting an ellipse to a data

Question

I want to fit an ellipse model to a data. This sample data I extracted from a simple parametric plot of 2 Sin[t], Cos[t](Let's say, I don't know that) and I want to fit my model of an ellipse to this data. This is what I have tried:

Sample data:

data = Flatten[ Cases[ParametricPlot[{Cos[t], 2 Sin[t]}, {t, 0, 2 Pi}], Line[data_] :> data, Infinity], 1];

Model:

x^2/a^2 + y^2/b^2 == 1
f[x_]:=Sqrt[(1 - x^2/a^2) b^2];

and,

fit = NonlinearModelFit[data,f[x], {a,b}, x]

It throws me complex infinity error and I am not able to solve it. Any help is appreciated.

Answer 1

Since the model is of the form f(x,y) = 1, arrange the data as {{x,y,1} . . } for the fit:

(* make some data *)
(* the data is of the form { {x,y,1}, . . . } *)

eq = x^2/4 + y^2/9 == 1;

y[xx_] := y /. Solve[eq /. x -> xx, y]

points = Union[
   Flatten[Table[{x, y[x]}, {x, 
       Range[-2, 2, .1]}] /. {x_, {y1_, y2_}} -> {{x, y1, 1}, {x, y2, 
        1}}, 1]];

(* plot the data *)
ListPlot[points[[All, {1, 2}]]]

(* the general model *)

model = x^2/a^2 + y^2/b^2;

(* fit the data *)

fit = NonlinearModelFit[points, model, {a, b}, {x, y}];

fit["BestFitParameters"]

(* {a\[Rule]2.`,b\[Rule]2.999999999953799`} *)

Answer 2

Here is a SingularValueDecomposition approach following @Danial Lichtblau's answer to 51549.

mean = Mean[data];
newpts = Map[# - mean &, data];
{uu, ww, vv} = SingularValueDecomposition[newpts, 2];
ListPlot[uu, AspectRatio -> Automatic]
rsqr = Mean[Map[#.# &, uu]];
{nx, ny} = Inverse[vv.ww].({x, y} - mean);
expr = Expand[nx^2 + ny^2] == rsqr;
expr = MultiplySides[expr, 1/expr[[2]]] // Expand
reg = ImplicitRegion[expr[[1]] <= expr[[2]] 1.1, {x, y}];
ContourPlot[Evaluate@expr, {x, y} \[Element] reg, 
 Epilog -> {Red, PointSize[Medium], Point[data]}, 
 AspectRatio -> Automatic]