next up prev
Next: 4. The Schubert Calculus
Up: 3. Enumerative Real Algebraic Geometry: Table of Contents
Previous: 3.iv. Real rational cubics through 8 points in R2

3.v. Common tangent lines to Spheres in Rn

Consider the following.

Question 3.9   How many common tangent lines are there to 2n - 2 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].

Theorem 3.10   2n - 2 general spheres in B>Rn (n at least 3) have 3 2n-1 complex common tangent lines, and there are 2n - 2 such spheres with all common tangent lines real.

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

Figure 9: Four spheres with 12 common tangents

The number 2n - 2 is the dimension of the space of lines in Rn and is necessary for there to be finitely many common tangents.

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

<p,v> := pi vi   =   0 ,

makes this representation unique. Here, <p,v> is the usual Euclidean dot product. Write v2 for <v,v>. A line represented by the pair (p,v) is tangent to the sphere with center c and radius r when

v2p2 - 2v2<p,c> + v2c2 - <v,c>2 - v2r2   =   0 .

Suppose we have 2n - 2 spheres with centers c1, c2, ..., c2n-2 and corresponding radii r1, r2, ..., r2n-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

v2p2 - v2r2   =   0 . (3.2)

Subtracting this from the equations for the other spheres, we obtain the equations
2v2<p, ci>   =   v2ci2 - <v, ci>2 - v2(ri2-r2) (3.3)

for i=1, 2, ..., 2n-3.

These last equations are linear in p. If the centers c1, c2, ..., cn 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 2n-3 = 3 2n-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

c1 := (  1,  1,  1, 0, ..., 0)
c2 := (  1,-1,-1, 0, ..., 0)
c3 := (-1,  1,-1, 0, ..., 0)
c4 := (-1,-1,  1, 0, ..., 0)

and subsequent centers are at the points aej and -aej, for j = 4, 5, ..., n, where e1, e1, ..., en is the standard basis for Rn. Let g := a2(n-1)/(a2+n-3), which is positive.

Theorem 3.11 ([STh, Theorem 5])   When

a r (r2 - 3) (3 - g) (a2 - 2) (r2 - g) ( (3 - g)2 + 4g - 4r2)

does not vanish, there are exactly 3 2n-1 complex lines tangent to the spheres. If we have
(3 - g)2/4 + g   >   r2   >   g     and
(n - 1)/(n - 4) + 2   >   a2   >   2,
then all the 3 2n-1 lines are in fact real.

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.

Theorem 3.12 (Megyesi [Me])   Four unit spheres in R3 whose centers are coplanar but otherwise general have 12 common complex tangents. At most 8 of these 12 are real.

Remark 3.13   This problem of common tangents to 4 spheres with equal radii and coplanar centers gives an example of an enumerative geometric problem that is not fully real. We do not feel this contradicts the observation that there are no enumerative problems not known to be fully real, as the spheres are not sufficiently general.

§The question of the maximal number of (real) common tangents to 4 balls was first formulated by David Larman [Lar] at DIMACS in 1990.
This was solved during the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

next up prev
Next: 4. The Schubert Calculus
Up: 3. Enumerative Real Algebraic Geometry: Table of Contents
Previous: 3.iv. Real rational cubics through 8 points in R2