Volume Bound By Multiple Solids

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SUMMARY

The discussion focuses on methods for calculating volumes bound by multiple geometric solids, specifically using cylindrical coordinates. The user seeks efficient techniques to determine the overlapping volume of a cylinder, three spheres, and three cones, with the goal of finding the overlapping density (OD). The user currently employs MATLAB for calculations but finds the process laborious. The problem involves complex geometric relationships and requires advanced mathematical approaches.

PREREQUISITES
  • Understanding of geometric solids, specifically cylinders, spheres, and cones.
  • Proficiency in cylindrical coordinates and their applications in volume calculations.
  • Familiarity with MATLAB for mathematical modeling and computational geometry.
  • Knowledge of overlapping volume concepts and density calculations in geometry.
NEXT STEPS
  • Research algorithms for calculating volumes of intersection among geometric solids.
  • Explore MATLAB functions for geometric computations, such as 'polyshape' and 'intersect'.
  • Learn about computational geometry techniques, including the use of Venn diagrams for volume analysis.
  • Investigate software tools like Wolfram Alpha or GeoGebra for visualizing and calculating overlapping volumes.
USEFUL FOR

This discussion is beneficial for undergraduate students in mathematics or engineering, researchers in computational geometry, and anyone involved in designing detectors or analyzing geometric relationships in three-dimensional space.

NullSpaceMan
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Hi,

I have a question regarding appropriate methods of finding volumes bound by geometric solids. I can work through the math in MatLab by finding points in common within each solid volume...but it is very laborious and I thought that I'd ask you math people how you would tackle this problem. I am just a poor helpless fizzer.

I am working on designing a detector for my undergrad thesis, and the cones represent a sensor's aperture and the spheres an event.


A basic example of the type of problem I am looking at in cylindrical co-ords:

A cylinder's axis is parallel to \hat{z}-direction and is placed at the origin where its radius = 'R'. The centres of three sphere are placed in the cylinder all with different radii (a1, a2, a3), all of which can exceed 'R'. At the surface of the cylinder are the apex of three cones and normal vectors of each cone base point in the -\hat{r}-direction.

I am interested in finding the volume of the overlapping shapes and the "overlapping density" (OD)..i.e how many solids are binding a bound volume...

..e.g
see: http://www.daviddarling.info/images/Venn_diagram.gif
See the centre area of three overlapping circles, the OD = 3
 
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