Volume Bound By Multiple Solids

[NullSpaceMan]

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

 

[NullSpaceMan]

No ideas??

 

[tacman]

.

Hi there,

Thank you for reaching out with your question. Finding volumes bound by multiple solids can definitely be a challenging task, but there are a few approaches you can take to make the process more efficient.

One method is to use integration to find the volume of the overlapping shapes. This involves breaking down the overlapping region into smaller, simpler shapes (such as rectangles or triangles) and using the appropriate formula to find their volumes. Then, you can add up all the individual volumes to get the total volume of the overlapping region.

Another approach is to use the concept of "cavities". A cavity is a space within a solid that is not filled by another solid. By finding the volumes of these cavities and subtracting them from the total volume of the larger solid, you can find the volume of the overlapping region.

In terms of finding the "overlapping density", you can use a similar approach by dividing the total volume of the overlapping region by the volume of a single solid. This will give you the number of solids that are binding the bound volume.

I hope this helps and good luck with your undergrad thesis!