A sphere tangent to four others

One of the interesting results from the geometry of antiquity is the construction by Appolonius of Perga of a fourth circle tangent to three arbitrary circles. In general there are eight solutions to the problem, according to whether the three tangencies are internal or external. The construction clearly generalizes to spheres and hyperspheres, leading us to expect sixteen different spheres tangent to each of four arbitrarily placed spheres.

The system of equations which has to be solved is slightly more complicated than when finding the orthogonal sphere, because the basic equation is

$$\begin{eqnarray*} f_i f +g_i g + h_i h -\frac{1}{2}(k_i+k) & = & R_iR. \end{eqnarray*}$$

In fact, we could write a similar system of equations,

$$\begin{eqnarray*} \left[ \begin{array}{ccccc} f_1 & g_1 & h_1 & -1/2 \\ f_2 ... ... \begin{array}{c} R_1 \\ R_2 \\ R_3 \\ R_4 \end{array} \right], \end{eqnarray*}$$

Unfortunately $R$, the radius of the tangent sphere, is dependent upon $k$, so that this is not a system of linear equations; this no doubt reflects the fact that for certain configurations of spheres, not all of the tangencies may be real. Thus another approach is needed, one of which is to assume a value for $R$ and try to arrive at a self-consistent value by iteration.

There is no reason that each sphere should not be given its own angle of intersection, or that a common angle different from $0^{o}$, $90^{o}$ or $180^{o}$ could not to be chosen.