J . Phys. Chem. 1987, 91, 4121-4122
4121
Volume of the Intersection of Three Spheres of Unequal Size. A Simplified Formula K. D. Gibson and H. A. Scheraga* Baker Laboratory of Chemistry, Cornell University, Ithaca. New York 14853-1 301 (Received: December 19, 1986)
By generalizing a formula of Powell, an expression is derived for the volume of intersection of three spheres of unequal size that meet in a pair of points. Expressions are also presented for other types of triple intersection.
Introduction In an accompanying paper, Kang et al. present an improved version of the hydration shell model for calculating free energies of hydration of organic and bioorganic molecules.’ An important feature of their method is that it includes an exact expression for the volume of the intersection of three spheres of unequal size. Kang et al. derived their expression by generalizing the equations of Rowlinson.2 Here, we present an alternative, and somewhat simpler, exact expression for the volume of the triple intersection, based upon the procedure of PowelL3 As Powell himself noted, although he presented the expression for the intersection of three spheres only of unit radius, his approach is simpler than Rowlimon’s and should generalize readily to several spheres of unequal size. However, in spite of the fact that Powell’s formula has been used quite frequently,- it does not yet seem to have been extended explicitly to cover even the case of three spheres of unequal size. In Rowlinson’s method,2 the triple intersection is divided into six disjoint spherical diangles,I0 whose volumes are calculated by integration. The expression for the volume of a spherical diangle has been simplified by Lustig,!O who has applied it to the computation of the volume of the intersection of three or four spheres of equal size.’OJ’ As shown by Lustig, this approach can readily be adapted to the computation of the surface areas of the intersections. In contrast, Powell’s method is based on the inclusion-exclusion principle for calculating the measure (volume) of a union of intersecting sets and cannot easily be adapted to the computation of surface areas. Both Rowlinson’s and Powell’s procedures fail if the surfaces of the three overlapping spheres have no common point; however, as will be seen below, the volume of the triple intersection can be computed quite simply under those circumstances. In the next section, we generalize Powell’s formula to cover three spheres of unequal size whose surfaces intersect in a pair of common points. In the final section, we present formulas for the volume of the triple intersection when there is no common point.
centered at A which is enclosed between the planes ABC, PAB,
0 1 w2 = (1/2) 1 1 1
1 0
1 y2
p2
1
1 a2
y2
0
a2
b2 =
p2
u2
a’
b2
0 c2
c2 0
Three Spheres Whose Surfaces Mutually Intersect Let A, B, and C be the centers of the spheres, and let P be one of the two points common to the surfaces of all three spheres. Powell’s procedure computes the volume of the triple intersection as twice the value of the following sum: (1) the volume of the tetrahedron PABC; (2) minus the volume of that part of the sphere (1) Kang, Y. K.; NCmethy, G.; Scheraga, H. A. J . Phys. Chem., first o f four papers in this issue. (2) Rowlinson, J. S. Mol. Phys. 1963, 6, 517. (3) Powell, M. J. D. Mol. Phys. 1964, 7 , 591. (4) Melnyck, T. W.; Rowlinson, J. S.;Sawford, B. L. Mol. Phys. 1972, 24, 809. ( 5 ) Stell, G.; Wu, K. C. J . Chem. Phys. 1975, 63, 491. (6) Perram, J. W.; White, L. R. Discuss. Faraday SOC.1975, 29. (7) Rushbrooke, G. S. Mol. Phys. 1979, 37, 761. (8) Torquato, S.; Stell, G. J . Chem. Phys. 1983, 79, 1505. (9) Emrich, K.; Zabolitzky, J. G. Phys. Reu. B: Condens. Matter 1984, 30, 2049. (10) Lustig, R. Mol. Phys. 1985, 55, 305. (11) Lustig, R. Mol. Phys. 1986, 59, 195.
0022-3654/87/2091-4121$01.50/0
Similar expressions are obtained for the volume contributions 3 and 4 by cyclically permuting the symbols a, b, c and a,@, y and the subscripts 1,2, 3. The dihedral angle between the planes ABC and PAB is tan-’ (2w/q3), and the volume contribution 5 is equal to (1/2n) V A B tan-! (2w/q3), where V A B is the volume of the intersection of the two spheres centered at A and B. The volume contributions 6 and 7 are found by cyclic permutation of the symbols A, B, C and the subscripts 1, 2, 3. In all of these expressions, the inverse tangents must be chosen to lie between 0 and T . 0 1987 American Chemical Society
4122
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
Gibson and Scheraga
C
Figure 1. Representation of the tetrahedron formed by the centers of spheres at A, B, and C and the point P, which is one of the two points common to the surfaces of all three spheres.
Y'
Multiplying by 2, combining all these terms, and simplifying give the final formula for the volume of the triple intersection VAW
-
x'
\
I+ (0 5 tan-' 5 r) ( 5 )
As a numerical example, we consider spheres of radii 1.0, 2.0, and 3.0 A centered at A, B, and C, respectively, and set the distances AB, BC, and C A equal to 2.0, 4.0, and 3.0 A, respectively. Then the volume of the triple intersection calculated from eq 5 is 0.5736 A3,in exact agreement with the result obtained by Kang et al.' (This result was obtained independently by one of the referees, using a numerical method.)
Three Spheres Whose Surfaces Have No Common Point If w2 > 0, the three spheres intersect in two points, and VABc can be found from eq 5. If n? = 0, the spheres intersect in a single point and V A= ~0. If w2 C 0, the spheres have no common point and the procedures of Powell and Rowlinson fail. There are then three possibilities, which can be distinguished by considering the projections of the three circles of pairwise intersection onto the plane of the centers of the spheres, as illustrated in Figure 2. Let the end points of these projections be X, X' for the intersection of the spheres centered at B and C; Y , Y' for the intersection of the spheres centered at C and A; and Z, Z' for the intersection of the spheres centered at A and B (Figure 2). Define the quantities t and top?
+ b + c)(-a + b + C ) ( U - b + C ) ( U + b - C)
(6)
+ P + 7N-a + P + r)(a - P + r)(a + P - Y)
(7)
f2 = (a
= (a
together with two similar quantities tbya and tCapobtained by cyclically permuting the symbols a, b, c and cy, P, y in eq 7. Set
p1 = [(b2- 'C p2 = [(b' - 'C
+ P' - 7')' + ( t - t,p,)']/4a2 + P2 - Y ' ) ~+ ( t + t,p,)2]/4a2
- a2 (8) -
cy2
(9)
In eq 8, the first term on the right-hand side is the square of the distance XA, while the first term on the right-hand side of eq 9 is the square of the distance X'A (Figure 2). Analogous quantities p 3 and p4 are defined by cyclically permuting the symbols a, b, c and a , /3, y in eq 8 and 9, respectively, and a further cyclic
Figure 2. Intersections of three spheres when the surfaces of the spheres have no point in common, projected onto the plane of the centers. A, B, and C are the centers of the spheres; XX', YY', and ZZ' are the projections of the circles of pairwise intersection. (a) No triple intersection; (b) and (c) two different types of triple intersection. Tests for distinguishing the three cases are described in the text. permutation of the symbols leads to two more quantities p 5 and p6*
If all pi> 0, each circle of pairwise intersection lies outside the third sphere, and there is no triple intersection (Figure 2a). If p i < 0 and p 2 C 0 but all other pi > 0, the sphere centered at C contains the pairwise intersection of the other two spheres (Figure 2b), and therefore VABC=
J'AB
(10)
where VABis the volume of the pairwise intersection of the spheres centered at A and B. Similarly, if p 3 C 0 and p4 < 0 or if p 5 < 0 and p6 C 0, all other p i s being positive, one sphere contains the pairwise intersection of the other two. If p 1 > 0 and p 2 > 0 but p , , p4,p 5 , and P6 are negative, there is a triple intersection even though the surfaces of the spheres have no common point (Figure 2c). In this case VABC= VAB+ VAC- VA (1 1) where VAis the volume of the sphere centered at A. Analogous formulas apply if p 3 > 0 and p4 > 0 or if p s > 0 and p6 > 0, while the remaining p i s are negative. This type of triple intersection cannot occur if the spheres are of equal size.
Acknowledgment. This work was supported by research grants from the National Science Foundation (DMB84-01811) and from the National Institute of General Medical Sciences (GM-14312) of the National Institutes of Health, U S . Public Health Service, and by a grant from Hoffmann-La Roche, Inc. We thank the referee for helpful comments.
