I have a multivariable function,(adsbygoogle = window.adsbygoogle || []).push({}); z = f(x, y, w),represented by a surface plot in 3D (zversusxy) for each value ofw. Aswvaries, the functionzvaries (goes up and down and changes shape) over a given rectangularxyregion. Aszvaries withw,contour lines with given constant values ofzform and change shape. Some of these contour lines are open while others are closed. However, aswincreases the open path contours usually become closed paths (closed loops).

I have two related problems:

(1) I want to find the threshold value ofwat which a certain contour,z =cwherecis a given constant, turns from being open to closed (i.e. what is the minimum value ofwat which the contour curve becomes closed loop).

(2) I want to find the threshold value ofwat which a certain curve with the condition,z ≥cwherecis constant, turns from being open to closed.

Is there an analytical way for finding the threshold minimum values ofwat which these two curves first become closed loops?

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# Closed loop contour

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