球冠
Richmond, Timothy J. (1984). “Solvent accessible surface area and excluded volume in proteins: Analytical equation for overlapping spheres and implications for the hydrophobic effect”. Journal of Molecular Biology 178 (1): 63–89. doi:10.1016/0022-2836(84)90231-6. PMID 6548264.
Lustig, Rolf (1986). “Geometry of four hard fused spheres in an arbitrary spatial configuration”. Molecular Physics 59 (2): 195–207. Bibcode: 1986MolPh..59..195L. doi:10.1080/00268978600102011.
Gibson, K. D.; Scheraga, Harold A. (1987).
半径 r 1 {\displaystyle r_{1}} と r 2 {\displaystyle r_{2}} の球が交差したときの和集合の体積は、
V = V ( 1 ) − V ( 2 ) , {\displaystyle V=V^{(1)}-V^{(2)}\,,}
ここで
V ( 1 ) = 4 π 3 r 1 3 + 4 π 3 r 2 3 {\displaystyle V^{(1)}={\frac {4\pi }{3}}r_{1}^{3}+{\frac {4\pi }{3}}r_{2}^{3}}
は2つの独立した球の体積の合計で
V ( 2 ) = π h 1 2 3 ( 3 r 1 − h 1 ) + π h 2 2 3 ( 3 r 2 − h 2 ) {\displaystyle V^{(2)}={\frac {\pi h_{1}^{2}}{3}}(3r_{1}-h_{1})+{\frac {\pi h_{2}^{2}}{3}}(3r_{2}-h_{2})}