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.
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
| (3.1) |
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
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
v2p2
- v2r2
= 0 .
(3.2)
for i=1, 2, ..., 2n-3.
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 3.10 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.