Intersecting spheres of equal energy density and unequal volume:
I’m an accountant and good at math when it comes to a checkbook or a balance sheet. But can someone help me put some thoughts together mathematically to deal with the intersection of two spheres?
Refer to these diagrams: Spherical Cap, Sphere-sphere intersection, and circle-circle intersection.
The volume V of a Spherical cap = Vcap = 1/3 pi h^2 (3 R – h) and where two spheres intersect they form an overlap in the shape of a 3-D lens. The math for the volume of the lens uses the above formula for each cap and adds the volumes together.
My question adds an assumption that the spheres are spheres of energy density and that the overlap is a combination of the energy density of the two spheres. Using that premise, if the two spheres were equal in size and energy density, then the 3-D lens would have twice the energy density as the individual spheres.
But if the spheres were different sizes and have equal total energy but different energy density I wanted a formula that would work.
Since there are two spherical caps, one from each sphere, and each cap contains overlapping or combined energy from each sphere I thought that by using the volume of the cap as a percentage of the sphere and adding the percentages together would work. Since I am assuming different size spheres and different energy density but equal total energy in each sphere and since the lens is made up of energy from each sphere, I came up with this for the calculation of the combined percentage of energy in the 3-D lens shaped intersection:
$$\frac{V_{cap1}}{V_1} + \frac{V_{cap2}}{V_2} + \frac{V_{cap1}}{V_2} + \frac{V_{cap2}}{V_1} = \frac{1/3 \pi h^2 (3 R – h)}{4/3 \pi R^3} + \frac{1/3 \pi h^2 (3 r – h)}{4/3 \pi r^3}+ \frac{1/3 \pi h^2 (3 R – h)}{4/3 \pi r^3}+ \frac{1/3 \pi h^2 (3 r – h)}{4/3 \pi R^3}$$
R and r are the radii. If I read the examples correctly, h is the same for both spheres, x is the distance from the center to line h, and d is the distance between centers. The resulting formula adds up the four energy percentages. The percentage of energy in cap 1 from sphere 1, the percentage of energy in cap 2 from sphere 2, the percentage of energy in cap 1 from sphere 2 and the percentage of energy in cap 2 from sphere 1 that occupy the lens. But the formula is ugly. I think it can it be simplified because the term (1/3 pi h^2) is in each piece but I can’t make it work when I try to simplify it.
I’m an accountant and good at math when it comes to a checkbook or a balance sheet. But can someone help me put some thoughts together mathematically to deal with the intersection of two spheres?
Refer to these diagrams: Spherical Cap, Sphere-sphere intersection, and circle-circle intersection.
The volume V of a Spherical cap = Vcap = 1/3 pi h^2 (3 R – h) and where two spheres intersect they form an overlap in the shape of a 3-D lens. The math for the volume of the lens uses the above formula for each cap and adds the volumes together.
My question adds an assumption that the spheres are spheres of energy density and that the overlap is a combination of the energy density of the two spheres. Using that premise, if the two spheres were equal in size and energy density, then the 3-D lens would have twice the energy density as the individual spheres.
But if the spheres were different sizes and have equal total energy but different energy density I wanted a formula that would work.
Since there are two spherical caps, one from each sphere, and each cap contains overlapping or combined energy from each sphere I thought that by using the volume of the cap as a percentage of the sphere and adding the percentages together would work. Since I am assuming different size spheres and different energy density but equal total energy in each sphere and since the lens is made up of energy from each sphere, I came up with this for the calculation of the combined percentage of energy in the 3-D lens shaped intersection:
$$\frac{V_{cap1}}{V_1} + \frac{V_{cap2}}{V_2} + \frac{V_{cap1}}{V_2} + \frac{V_{cap2}}{V_1} = \frac{1/3 \pi h^2 (3 R – h)}{4/3 \pi R^3} + \frac{1/3 \pi h^2 (3 r – h)}{4/3 \pi r^3}+ \frac{1/3 \pi h^2 (3 R – h)}{4/3 \pi r^3}+ \frac{1/3 \pi h^2 (3 r – h)}{4/3 \pi R^3}$$
R and r are the radii. If I read the examples correctly, h is the same for both spheres, x is the distance from the center to line h, and d is the distance between centers. The resulting formula adds up the four energy percentages. The percentage of energy in cap 1 from sphere 1, the percentage of energy in cap 2 from sphere 2, the percentage of energy in cap 1 from sphere 2 and the percentage of energy in cap 2 from sphere 1 that occupy the lens. But the formula is ugly. I think it can it be simplified because the term (1/3 pi h^2) is in each piece but I can’t make it work when I try to simplify it.
Last edited:
