Closed Loop Contour: Finding Threshold W Values

[JulieK]

I have a multivariable function, z = f(x, y, w), represented by a surface plot in 3D (z versus xy) for each value of w. As w varies, the function z varies (goes up and down and changes shape) over a given rectangular xy region. As z varies with w, contour lines with given constant values of z form and change shape. Some of these contour lines are open while others are closed. However, as w increases the open path contours usually become closed paths (closed loops).

I have two related problems:

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

(2) I want to find the threshold value of w at which a certain curve with the condition, z ≥ c where c is constant, turns from being open to closed.

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

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?