3.v. Common tangent lines to Spheres in Rⁿ

For example, when n = 3, how many common tangent lines are there to four spheres in R³? Despite its simplicity, this question does not seem to have been asked classically, but rather arose in discrete and computational geometry^§. The case n = 3 was solved by Macdonald, Pach, and Theobald [MPT] and the general case more recently [STh].

Figure 9 displays a configuration of 4 spheres in R³ with 12 real common tangents.

Figure 9: Four spheres with 12 common tangents

Represent a line in Rⁿ by a point p on the line and its direction vector v in P^n-1. Imposing the condition

<p,v> := pi vi   =   0 ,

(3.1)

Suppose we have 2n - 2 spheres with centers c₁, c₂, ..., c_2n-2 and corresponding radii r₁, r₂, ..., r_2n-2. Without any loss of generality, we may assume that the last sphere is centered at the origin and has radius r. Then its equation is

These last equations are linear in p. If the centers c₁, c₂, ..., c_n are linearly independent (which they are, by our assumption on generality), then we use the corresponding equations to solve for v²p as a homogeneous quadratic in v. Substituting this into the equations (3.1) and (3.2) gives a cubic and a quartic in v, and substituting the expression for v²p into (3.3) for i=n+1, n+2, ..., 2n-3 gives n-3 quadratics in v. By Bézout's Theorem, if there are finitely many complex solutions to these equations, their number is bounded by 3 4 2^n-3 = 3 2^n-1.

This upper bound is attained with all solutions real. Suppose that the spheres all have the same radius, r, and the first four have centers

Theorem 3.10 is false when n=2. There are 4 lines tangent to 2 general circles in the plane, and all may be real. The argument given for Theorem 3.10 fails because the centers of the spheres do not affinely span R². This case of n=2 does generalize, though.