I am currently writing a mathematical essay concerning the close packing of various 3D shapes. One example I am investigating is the shape of an oblate spheroid (ellipsoid) with a minor axis of 5 mm and a major axis of 15 mm. I am aiming to find the maximum packing density of such a spheroid in a face-centered cubic lattice. In order to calculate this, I have taken a look at a cubic section from such an infinite lattice. This cube contains half a spheroid on each face and an additional 8 eighths of a spheroid in each corner. It was fairly simple calculating the dimensions of the top and bottom surfaces of the cube, but the sides have been more of an issue.
To make it short: I need to find the distance between the center points of two ellipses represented by the red arrow in the following graph. The graph shows the cube from one of the side faces.
The graph can be infinitely stacked in all direction to produce a 2D representation of the 3D lattice I am investigating. It is not to scale, nor are the ratios graphed correctly, but the numbers are correct. Any help is greatly appreciated.
Here's a picture of what happens. In any case, the distance between centers is half the rectangle diagonal. But the rectangle horizontal side $X$ and vertical side $Y$ must satisfy the following equation: $$ Y=10\sqrt{1-{X^2\over900}}. $$