Editing Hyperbola (section)

===Reciprocation of a circle===
The [[reciprocation (geometry)|reciprocation]] of a [[circle]] ''B'' in a circle ''C'' always yields a conic section such as a hyperbola. The process of "reciprocation in a circle ''C''" consists of replacing every line and point in a geometrical figure with their corresponding [[pole and polar]], respectively. The ''pole'' of a line is the [[inversive geometry#Circle inversion|inversion]] of its closest point to the circle ''C'', whereas the polar of a point is the converse, namely, a line whose closest point to ''C'' is the inversion of the point.

The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius ''r'' of reciprocation circle ''C''. If '''B''' and '''C''' represent the points at the centers of the corresponding circles, then

<math display="block">e = \frac{\overline{BC}}{r}.</math>

Since the eccentricity of a hyperbola is always greater than one, the center '''B''' must lie outside of the reciprocating circle ''C''.

This definition implies that the hyperbola is both the [[locus (mathematics)|locus]] of the poles of the tangent lines to the circle ''B'', as well as the [[envelope (mathematics)|envelope]] of the polar lines of the points on ''B''. Conversely, the circle ''B'' is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to ''B'' have no (finite) poles because they pass through the center '''C''' of the reciprocation circle ''C''; the polars of the corresponding tangent points on ''B'' are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle ''B'' that are separated by these tangent points.