What was the old system of using right circular cones to solve problems about circles in the plane?

Question

[I asked this originally at the Math Stack Exchange, and they suggested I also ask about it here.]

I heard about this from a college professor but haven't ever been able to find any other mention of it. I remember it being called "La Guerre" geometry, but I could be wrong there.

The story was that a French(?) mathematician was taken as a POW in some war (no idea which one, but pre-WWII anyway and possibly hundreds of years before that). He was bored and passed the time doing geometry in the dirt on the floor of his cell. He noticed that if you have circle problems in the Euclidean plane you can sometimes solve them more easily by turning the problem into a 3-space problem with cones instead.

The general idea is that each circle is the base of a right circular cone (h=r). Then the vertex of each cone (r units above the center of the circle) acts as a stand-in for the whole initial circle.

Something like that anyway.

There is, confusingly, also a mathematician named Laguerre, but what I can find on him doesn't seem to be quite what I'm talking about.

In any case, I'd love to know more about it all. I remember we did some very cool proofs using this method and I just haven't been able to coax Google into finding what I need. Hopefully I'll have more luck here.

Answer 1

Laguerre geometry is the geometry of Laguerre plane, and it was introduced by Laguerre, who defined it as the geometry of the oriented lines and circles in the real Euclidean plane (modern definition is as an incidence geometry of parabolas and lines). It was studied classically using the so-called cyclographic mapping, which can indeed be defined via projections involving intersections of right angled cones with a plane. Computational Line Geometry by Pottmann and Wallner, p.366ff gives a description of the method, and references to classical books of Coolidge and Muller-Krames, and to modern applications in computer assisted geometric design.