A projective version

Recalling that the equation of a sphere is

$$\begin{eqnarray*} \alpha (x^2+y^2+z^2)+\beta x+\gamma y+\delta z + \epsilon&=&0, \end{eqnarray*}$$

regard the coefficients as a vector in a five dimensional space.

Because any non-zero multiple of the equation defines the same sphere, $\alpha$ may as well be chosen equal to 1 unless it, itself, is zero. Except for this degenerate case, substituting $f=\beta/\alpha$, $g=\gamma/\alpha$, and $h=\delta/\alpha$ gives a vector

$$\begin{eqnarray*} (f,g,h) & = & -2{\bf r}, \end{eqnarray*}$$

with r the center of the sphere. Similarly, set $k=\epsilon/\alpha$ to get

$$\begin{eqnarray*} k & = & r^2-R^2, \end{eqnarray*}$$

when $R$ is the radius of the sphere and $r$ the distance between its center and the origin.

Defining a metric matrix

$$\begin{eqnarray*} M & = & \left[ \begin{array}{ccccc} . & . & . & . & -1/2 \\ ... ...& . & . & 1 & . \\ -1/2 & . & . & . & . \end{array} \right]. \end{eqnarray*}$$

creates an inner product for which, given coefficient vectors $x_i^T=(\alpha_i, \beta_i,\gamma_i,\delta_i,\epsilon_i)$,

$$\begin{eqnarray*} x_1^T M x_2 & = & \beta_1\beta_2+\gamma_1\gamma_2+\delta_1\delta_2- \frac{1}{2}(\alpha_1\epsilon_2+\alpha_2\epsilon_1) \\ \end{eqnarray*}$$

If it were agreed to use normalized vectors $x_i^T=(1,f,g,h,k)$, we would have instead,

$$\begin{eqnarray*} x_1^T M x_2 & = & f_1 f_2 + g_1 g_2 + h_1 h_2 - \frac{1}{2}(k... ..._1^2+R_2^2-r_1^2-r_2^2) \\ & = & \frac{1}{2}(R_1^2+R_2^2-D^2), \end{eqnarray*}$$

The last two lines result from normalizing the vectors; the very last is a consequence of the cosine law and calling $D$ the distance between centers. According to this, the norm of a sphere is its radius, no matter where it sits.

Some further trigonometry produces the most interesting version of the formula,

$$\begin{eqnarray*} x_1^T M x_2 & = & R_1R_2\cos\theta \end{eqnarray*}$$

wherein $\theta$ is the angle at which the two spheres intersect, taken to be the angle between their tangent planes at the point of intersection. This is the same as the angle between the normals to the surface, which are themselves radii.

The remarkable conclusion is that orthogonal circles are those which intersect orthogonally, just as parallel or antiparallel circles are those which are tangent at their intersection. Parallel means internally tangent, antiparallel means externally tangent.