Poles and polars

The previous two posts have been about polar triangles. The first used a polar triangle to derive a theorem, and the second showed how to compute polars using linear algebra. But what exactly is a polar triangle and why is it called that? Poles and polars on a sphere Let’s start with a sphere S and a great circle G. Then G is the intersection of S with a plane through the center of S. The circle G has two poles, two points on opposite sides of S that are as far as possible (i.e. π/2 radians) from G. If you rotate the sphere so that G is the equator, the poles are the north and south pole. Conversely, if you start with a point P on the sphere S, the polar of P is the great circle for which P is a pole. So if you think of P as the north pole, the polar of P is the equator. The term equator would have been more intuitive than polar, but the terminology was established a couple centuries ago. In my opinion it would be clearer to call poles and polars both duals. The dual of a great circle…