Converting 2D density (circle) to 3D density (sphere)

[Jimbo2]

Hi All,
I'm looking for help in converting 2D density (objects/area) in a circle to 3D density (objects/volume) in a sphere, the circle and sphere having the same radius and distribution of objects being uniform.

To make this problem more intuitive, here's a sample application: both crabs and fish forage on the same plankton in the water column. The crab can only access plankton as he walks on the ground (i.e., he forages in 2D space), but the fish can access the plankton in the water column (i.e., he forages in 3D space). The crab has a detection area of ∏*acuity^2 while the fish has a detection volume of (4/3)*∏*acuity^3. Given a prey density, can the number of prey within the 2D detection zone be converted to the number of prey within the 3D detection zone (or vice versa)? I may be overthinking this, but help is much appreciated! Thanks!

 

[haruspex]

In your concrete example, the crab's feeding zone isn't really 2D - it's a thin sheet. Since the crab's detection range in the vertical direction can taken to extend right through th sheet you can treat it as 2D. But maybe this isn't really the scenario that you want to apply it to.

 

[chiro]

In addition to what haruspex mentioned, one thing you should consider is the differences in the situations of depth (like water) and area (like land) with regards to the densities.

For example, the water based situation will have densities that will most likely be a lot sparser and weirdly distributed in land.

It's called the "Curse Of Dimensionality":

http://en.wikipedia.org/wiki/Curse_of_dimensionality