Area between two closed curves

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SUMMARY

The discussion focuses on calculating the area between two closed curves, specifically through the use of line integrals. The user seeks a general solution for determining the area of intersection, denoted as Aij, between two parameterized closed loops, Ai and Aj. The conversation highlights the need to find intersections between the two boundaries to define the overlap area accurately. The complexity increases when there are infinite intersections, necessitating a more advanced approach to the problem.

PREREQUISITES
  • Understanding of line integrals in calculus
  • Familiarity with parameterization of curves
  • Knowledge of intersection points in geometry
  • Basic concepts of closed curves and their properties
NEXT STEPS
  • Research methods for calculating areas using line integrals
  • Explore techniques for finding intersection points of curves
  • Study advanced calculus topics related to multiple integrals
  • Investigate computational geometry algorithms for curve intersections
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Mathematicians, engineers, and students studying calculus or computational geometry who are interested in solving problems related to area calculations between curves.

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I have been trying to find an answer to this problem for some time. So I was hoping the community might be able to point me in the right direct.

https://drive.google.com/open?id=1Y2hYkRG94whLroK20Zmce07jslyRL3zZ

I couldn't get the image to load. So above is a link to an image of the problem. I am trying to find a general solution to the intersection of the two general closed loops.

In the image Ai and Aj are just the area of each closed loops and Aij is the area of the intersection of the two closed loops. ri and rj are just parameterizations of the positions around the closed loop.

Their is a general equation for the area of a closed loop defined by a line integral and I was wondering if there might be a line integral for the area of intersection between two closed loops.
 
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You can find the boundary of the overlap area by finding the intersections between the two boundaries and combining the sections between them to form a new line.
If there is an infinite set of intersections or similar things get more complicated.
 

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