There are 12 Common Tangents to Four Spheres

Problem Statement

We are given four spheres Sk, k=1..4, with centers sk and radii rk, and seek to determine all lines L that are tangent to those four spheres.  The problem arises, for example, in spatial geometric constraint solving, either as stated, or else when we seek to construct a line L that is to have a specific distance rk from known points  sk, where k=1...4.  We prove that there can be up to 12 such tangents in general, using the method of  and refer to a recent result  that shows that this bound is exact.

Basic Equation System

Assume that L has the coordinates (x,y,z;u,v,w), where (x,y,z) is the point of L closest to the origin and (u,v,w) is the line direction vector.  Note that the same line in the opposite direction has the coordinates (x,y,z;-u,-v,-w).  Now choose a coordinate system in which the first sphere has its center s1=(0,0,0) at the origin, the second sphere has s2=(a,0,0) on the x-axis, the center s3=(b,c,0) is in the xy-plane, and the center s4=(d,e,f) is in general position.  We obtain six equations that express the constraints on the line L, namely:

 x2+y2+z2-R0=0 (1) (x-a)2+y2+z2-(au)2-R1=0 (2) (x-b)2+(y-c)2+z2-(bu+cv)2-R2=0 (3) (x-d)2+(y-e)2+(z-f)2-(du+ev+fw)2-R3=0 (4) xu+yv+zw=0 (5) u2+v2+w2=1 (6)

where Rk = rk2.

The algebraic degree of that system is 64.  Furthermore, note that if (x,y,z;u,v,w) is a solution of the system, then so is (x,y,z;-u,-v,-w).

Simplifications

We can use equations (2), (3) and (4) to solve for x, y, and z.  We obtain the following expressions:

 x= (a2-a2u2-R1+R0)/(2a) y= (ab2-a2b+a2bu2+bR1-bR0+ac2-ab2u2-2abcuv-ac2v2-aR2+aR0)/(2ac) z= -(K0 + K1uv + 2acefvw + 2acdfuw + K2u2 + K3v2 + acf2w2) / (2fac) where K0= ace2+dcR0-dcR1+dca2 -acd2+aeR0-aeR2+aec2-ebR0+ebR0-eba2 +aeb2-acR0+acR3-acf2 K1= 2acde - 2abce K2= acd2 - dca2 - aeb2 + bea2 K3= ace2 - aec2

Substituting them into equations (1), (5) and (6) results in three equations in the unknowns u, v, and w, where the algebraic degrees are 4, 3, and 2, respectively.  Thus, the new system has a total degree of 24.

Note that in the expressions for x, y, and z the variables u, v, and w appear only quadratically.  Therefore, the new system also has the property that for each solution (x,y,z;u,v,w) there is a solution (x,y,z;-u,-v,-w).  Therefore, at most 12 geometrically distinct tangents can exist.

Exactness of Bound

In , the authors consider the case of equal radii and analyze both the general position and the degenerate positions.  They give an example of four unit spheres that are centered at the vertices of a tetrahedron and have exactly 12 real tangents.  Therefore, the bound of 12 tangents is exact.

References

1. C. M. Hoffmann and Bo Yuan, On Spatial Constraint Solving Approaches, presented at the Workshop for Automated Deductions in Geometry, ETH Zürich, September 2000.
2. I. G. Macdonald, J. Pach and T. Theobald, Common Tangents to Four Unit Balls in R3, to appear in J. of Discr. and Comp. Geometry.