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I want to defined a function that solves some equation numerically:

h[r_, R_, l_, n_] := NSolve[l^2 == r^2 + R^2 - 2 h^2 - 2 Sqrt[(r^2 - h^2) (R^2 - h^2)] Cos[2 Pi/n] && h >= 0, h, Reals]

I then want to plot this function via

DensityPlot[h[r,1,l,6], {r,0,1}, {l,0,1}]

Of course, this does not work, as NSolve does not return a numeric value, but a list of lists of rules, which might even be empty.

My intention would be to define h in such a way, so that it extracts the desired numerical value (the unique positive solution, if there is one), and otherwise returns, ..., well, no idea actually. What should it return otherwise? What I want in the end, is that DensityPlot works well with that function, so that regions where h is not well-defined are just blank. The region where it is not well-defined are not explicitly known, though.

I thought about defining h so that it returns Indeterminate whenever the list is empty, but then DensityPlot seems to throw $Failure. Also

RegionPlot[h[r,1,l,6] =!= Indeterminate, {r,0,1}, {l,0,1}]

does not to work for showing the region in which h is well-defined.

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Try ?NumericQ and assign the NSolve result to h

h[r_?NumericQ, R_?NumericQ, l_?NumericQ, n_?NumericQ] :=h /. NSolve[
l^2 == r^2 + R^2 - 2 h^2 -2 Sqrt[(r^2 - h^2) (R^2 - h^2)] Cos[2 Pi/n] && h >= 0, h,Reals][[1]]
DensityPlot[h[r, 1, l, 4], {r, 0, 1}, {l, 0, 1}]

enter image description here

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  • $\begingroup$ Does this also work for h[r,1,l,6]? I will edit the question, as the case with 4 wasn't actually the problematic one. Sorry for this! But in general, how does ?NumericQ solve the issue? $\endgroup$ – M. Winter Sep 24 at 12:37
  • $\begingroup$ I tested it, and it cannot handle the case h[r,1,l,6], as then the NSolve might find no solutions, and [[1]] fails. $\endgroup$ – M. Winter Sep 24 at 12:44
  • $\begingroup$ Try ContourPlot3D[ l^2 == 0.0025 - 2 h^2 + r^2 - Sqrt[(0.0025 - h^2) (-h^2 + r^2)], {r, 0, 1}, {l, 0, 1}, {h, 0, 0.05}, AxesLabel -> Automatic] to check the possible solutions ! $\endgroup$ – Ulrich Neumann Sep 24 at 12:52

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