OTHER REPRESENTATIONS OF GEOMETRY
Projective geometry need not have points and lines as its basic elements. For example circles through a fixed base point Z together with points may be used instead. Just as any two lines meet in one point, so any two circles through Z meet in just one other point. Dually just as any two points determine one line so any two points together with Z determine just one circle. We may then expect analogues of the basic theorems of projective geometry to apply to such a geometry of circles and points. The following diagram shows the construction of a "conic" in this geometry, where two projective ranges give rise to a set of "lines" (circles) joining them, enveloping a "conic" (lemniscate).
The construction is well known (e.g. Lockwood A Book of Curves) but this approach views it in another way.