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 exactlyis a polar triangle and why is it called that?

Poles and polars on a sphere

Let's start with a sphereS and a great circleG. ThenG is the intersection ofS with a plane through the center ofS. The circleG has two poles, two points on opposite sides ofS that are as far as possible (i.e. /2 radians) fromG. If you rotate the sphere so thatG is the equator, the poles are the north and south pole.

Conversely, if you start with a pointP on the sphereS, thepolar ofP is the great circle for whichP is a pole. So if you think ofP as the north pole, the polar ofP 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 bothduals. The dual of a great circle is a point, and the dual of a point is a great circle.

Now let's look at a spherical triangle ABCsides a,b, andc. The vertices of the polar triangle of ABC are the poles of the sides a,b, andc. This definition is essentially correct, but it needs a little refinement.

The pole A' of the side a is one of the two poles of the great circle containinga, the one that that lies in the same hemisphere asA. Define B' and C' similarly, and the polar triangle is the spherical triangle with vertices A'B'C'.

Poles and polars in the plane

The termspole and polar were coined for spherical geometry, then generalized to plane (projective) geometry. If you think of great circles as the lines of spherical geometry, then you can carry the idea of poles and polars over to plane geometry where the pole (dual) of a line is a point, and the polar (dual) of a point is a line.

However, in the plane you can't define the pole of a line without more context. You have to start with a circle (or more generally a conic section) and define the pole of a linewith respect to the circle and the define the polar of a pointwith respect to the same circle.

Given a circleS with center O, to find the polar of a point P O, letQ be the inversion ofP inS, and let be the line throughQ perpendicular to the lineOP.

Conversely, the pole of a line with respect to a circleS is the inversion ofQ inS whereQ is the intersection of withOQ. Here we assume the line does not pass through the center of the circleO because otherwiseO =Q and it doesn't make sense to speak of the lineOQ.

Projective geometry

The definitions above have a couple holes that can be filled in by extending the Euclidean plane to the projective plane.

We defined the pole of a line except when passes through the center of the circle. If passes throughO then the pole of is the point at infinity in the direction of . Similarly, we defined the polar of all points exceptO. In projective geometry, we can define the polar ofO to the be the line at infinity, the line containing all the points at infinity.

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