# Obtain multiple coordinates from a function at a particular value

I have a function of the type:

F[{x_,y_}] := x^2 + y^2


One can plot this function using Plot3D:

Plot3D[F[{x,y}], {x, -2, 2}, {y, -2, 2}]


If one plots the function, it is clear that F[{x,y}] = 6 at more than one set of coordinates.

How can I obtain all sets of coordinates for which F[{x,y}] = 6?

Thanks, Stuart.

Well there is a nice way to extract points:

mYcp = ContourPlot[x^2 + y^2 == 6, {x, -3, 3}, {y, -3, 3}]


Short[Cases[Normal@mYcp, Line[x_] :> x, Infinity], 20]

{{{-1.07143,-2.20209},{-1.1106,-2.18203},{-1.17857,-2.14721},{-1.18634,-2.14286},{-1.24885,-2.10599},{-1.28571,-2.08271},{-1.38018,-2.02304},{-1.47143,-1.95714},{-1.5,-1.93609},{-1.50433,-1.9329},{-1.50952,-1.92857},{-1.62121,-1.8355},{-1.71429,-1.7479},{-1.7316,-1.7316},{-1.7479,-1.71429},<<246>>,{-0.289216,-2.43207},{-0.321429,-2.42778},{-0.363553,-2.42216},{-0.383423,-2.41914},{-0.428571,-2.41111},{-0.473545,-2.40212},{-0.634454,-2.36555},{-0.642857,-2.36335},{-0.647783,-2.36207},{-0.663265,-2.35714},{-0.857143,-2.29252},{-0.961538,-2.25275},{-0.965438,-2.25115},{-0.968045,-2.25},{-1.07143,-2.20209}}}

• wow great ! thanks for sharing it, +1 Mar 25, 2016 at 10:52
• Nice, thanks for the solution. Mar 25, 2016 at 11:01

Perhaps not a good job but the closest i can get to:

plot = Reap[ContourPlot[F[{x, y}], {x, -3, 3}, {y, -3, 3},
EvaluationMonitor :> If[5.95 < F[{x, y}] < 6, Sow[{x, y, F[{x, y}]}],]]][[2,1]] //
#[[All, {1, 2}]] & //ListPlot[#, PlotStyle -> Red] &


Show[ContourPlot[F[{x, y}] == 6, {x, -3, 3}, {y, -3, 3}], plot]


• There are certainly numerical errors associated with the way i extracted points from the plot. however, if accuracy is not a big issue, the answers will be quite reasonable Mar 25, 2016 at 8:55
• Hi Ali, thanks for your answer. The example I posted was a much simpler function that the one I am actually dealing with. I have now tried your approach on my function and the results don't look too bad. Do you have any more ideas on how to improve the accuracy? Thanks, Stuart. Mar 25, 2016 at 9:26
• Or alternatively, incorporate more points? Mar 25, 2016 at 9:27
• search for the question titled, "getting a list of accurate coordinates from a plot". I think someone used a mouse event to get all the coordinates by hovering the mouse over the plot. It may be more reasonable Mar 25, 2016 at 9:37
• Hi Ali, thanks for the help again. Looking back at your previous answer the contour plot is completely sufficient. Thanks for your help. Cheers, Stuart. Mar 25, 2016 at 10:31