Closed Loop Contour: Finding Threshold W Values

In summary, the conversation discusses a multivariable function represented by a 3D surface plot, with varying z values as w changes. Contour lines are formed at constant z values, some of which are open and others closed. The individual has two related problems: finding the minimum value of w at which a contour line becomes closed and finding the threshold value of w at which a curve with a given condition becomes closed. They are seeking an analytical method to determine these threshold values.
  • #1
JulieK
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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?
 
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  • #2
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?
 

FAQ: Closed Loop Contour: Finding Threshold W Values

What is a closed loop contour?

A closed loop contour is a path or line that connects points of equal value on a map or graph. It forms a closed shape, meaning that the starting and ending points are connected, and all points on the contour have the same value or characteristic.

Why is finding threshold W values important in closed loop contour analysis?

Finding threshold W values is important because it helps determine the boundary or extent of a certain characteristic or value on a map or graph. This is crucial in many scientific fields, such as environmental science, where identifying the threshold of certain pollutants can help in making informed decisions for resource management.

How is the threshold W value determined in closed loop contour analysis?

The threshold W value is determined by analyzing the data points and their values. It is usually set at a point where there is a significant change or break in the data, indicating a shift in the characteristic being studied. Different methods, such as statistical analysis or visual inspection, can be used to determine the threshold W value.

What factors can affect the accuracy of threshold W values in closed loop contour analysis?

Several factors can affect the accuracy of threshold W values, including the quality and quantity of data, the method used to determine the threshold, and the scale or resolution of the map or graph being analyzed. It is important to carefully consider these factors and use multiple techniques to ensure the accuracy of the threshold W value.

How can closed loop contour analysis be applied in practical situations?

Closed loop contour analysis can be applied in various practical situations, such as identifying pollution sources, predicting the spread of diseases, and determining the optimal location for natural resource extraction. It can also be used in urban planning to identify areas with high crime rates or to map the distribution of certain amenities in a city.

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