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3.v. Common tangent lines to Spheres in
R^{n}
Consider the following.
Question 3.9
How many common tangent lines are there to 2n  2 spheres in
R^{n}?
For example, when n = 3, how many common tangent lines are there to four
spheres in R^{3}?
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
2
n  2 general spheres in B>R
^{n}
(
n at least 3) have 3 2
^{n1} complex common
tangent lines, and there are 2
n  2 such spheres with all
common tangent lines real.
Figure 9 displays a configuration of 4 spheres in
R^{3} 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 R^{n} and is necessary for there to be
finitely many common tangents.
Represent a line in R^{n} by a point p on the
line and its direction vector v in P^{n1}.
Imposing the condition
<p,v> := 

p_{i} v_{i}
= 0 , 

(3.1) 
makes this representation unique.
Here, <p,v> is the usual Euclidean dot product.
Write v^{2} for <v,v>.
A line represented by the pair (p,v) is tangent to the sphere
with center c and radius r when
v^{2}p^{2}
 2v^{2}<p,c>
+ v^{2}c^{2}
 <v,c>^{2}
 v^{2}r^{2}
= 0 .
Suppose we have 2n  2 spheres with centers
c_{1}, c_{2}, ...,
c_{2n2} and
corresponding radii r_{1}, r_{2}, ...,
r_{2n2}.
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
v^{2}p^{2}
 v^{2}r^{2}
= 0 . 
(3.2) 
Subtracting this from the equations for the other
spheres, we obtain the equations
2v^{2}<p, c_{i}>
= v^{2}c_{i}^{2}
 <v, c_{i}>^{2}
 v^{2}(r_{i}^{2}r^{2})

(3.3) 
for i=1, 2, ..., 2n3.
These last equations are linear in p.
If the centers c_{1}, c_{2}, ...,
c_{n} are linearly independent (which they are, by
our assumption on generality), then we use the
corresponding equations to solve for v^{2}p
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
v^{2}p
into (3.3) for
i=n+1, n+2, ..., 2n3 gives
n3 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^{n3} = 3 2^{n1}.
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
c_{1} := ( 1, 1, 1, 
0, ..., 0) 
c_{2} := ( 1,1,1, 
0, ..., 0) 
c_{3} := (1, 1,1, 
0, ..., 0) 
c_{4} := (1,1, 1, 
0, ..., 0) 
and subsequent centers are at the points ae_{j}
and ae_{j}, for j = 4, 5, ..., n,
where
e_{1}, e_{1}, ..., e_{n} is the
standard basis for R^{n}.
Let g :=
a^{2}(n1)/(a^{2}+n3), which is
positive.
Theorem 3.11 ([
STh,
Theorem 5]) When
a r (r^{2}  3) (3  g) (a^{2} 
2) (r^{2}  g) ( (3  g)^{2} + 4g
 4r^{2})
does not vanish, there are exactly 3 2
^{n1}
complex lines tangent to the spheres.
If we have
 (a)
 (3  g)^{2}/4 + g > r^{2}
> g and
 (b)
 (n  1)/(n  4) + 2 >
a^{2} > 2,
then all the 3 2
^{n1} 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
R^{2}.
This case of n=2 does generalize, though.
Theorem 3.12 (Megyesi [
Me])
Four unit spheres in
R^{3} 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: 4. The Schubert Calculus
Up: 3. Enumerative Real Algebraic Geometry:
Table of Contents
Previous: 3.iv. Real rational cubics through
8 points in R^{2}