Finding the arc length of a region defined implicitly with an interpolated function

Question

I begin by defining an interpolation of some data I have stored at this link for brevity. The interpolating function is a function of two variables, x and y

S = (*data*)
f = Interpolation[S]
Plot3D[f[x, y], {x, -5, 4}, {y, -5, 4}]

I then want to define a one-dimensional surface by when this interpolated function intersects zero

R = ImplicitRegion[f[x, y] == 0.0]
RegionPlot[R, PlotRange -> {{-5, 4}, {-5, 4}}

This looks like it has worked quite well, but now when I try to calculate ArcLength[R], I don't get an answer, even when I try restricting the domain of the implicit region

R = ImplicitRegion[f[x, y] == 0.0, {{x, -5, 4}, {y, -5, 4}}]

The ellipse looks pretty well-defined, is there a way of getting its arc Length?

Answer 1

I don't know how accurately you want the arc length, but going along the lines you started, one can do this:

R = ImplicitRegion[f[x, y] == 0.0, {{x, -5, 4}, {y, -5, 4}}];
DiscretizeRegion[R, MaxCellMeasure -> 0.01] // ArcLength

(*  19.7927  *)

A more accurate determination:

Module[{tf},
 NDSolveValue[{{x'[t], y'[t]} == 
    Cross@D[f[x[t], y[t]], {{x[t], y[t]}}],
   x[0] == (x /. FindRoot[f[x, 0], {x, 2}]), y[0] == 0,
   s'[t] == Sqrt[x'[t]^2 + y'[t]^2], s[0] == 0,
   WhenEvent[y[t] == 0 && x[t] > 0, {tf = t; "StopIntegration"}]},
  s[tf], {t, 0, 10000}]
 ]

(*  19.7955  *)

It takes a bit of work to get there with DiscretizeRegion:

DiscretizeRegion[R, MaxCellMeasure -> 0.0001] // ArcLength

(*  19.7954  *)

Answer 2

This is a bit of a hack. Assume we have your plot: pl from the data:

R = ImplicitRegion[f[x, y] == 0.0, {x, y}]
pl = RegionPlot[R, PlotRange -> {{-5, 4}, {-5, 4}}]

We may no dissect pl to get the relevant information. Unfortunately, the line is not stored directly, but first the coordinates of the points are stored and then the line is specified by the indices of the coordinates. To get the coordinates of the line and to check if it is o.k. we write:

coor = g[[1, 1, 1]];
ind = g[[1, 1, 2, 1, 1, 3, 1]];
line = coor[[ind]];
ListLinePlot[line]

Now the length of the line is easily calculated by taking the successive differences and the norm therefrom and finally adding up:

Total[Norm /@ Differences[line]]
(* 19.7268 *)