Question

lines are polar lines with respect to the sphere

Answer 1

The pole of a line L in a circle C is a point P that is the inversion in C of the point Q on L that is closest to the center of the circle. Conversely, the polar line (or polar) of a point P in a circle C is the line L such that its closest point Q to the center of the circle is the inversion of P in C.

If a point A lies on the polar line q of another point Q, then Q lies on the polar line a of A. More generally, the polars of all the points on the line qmust pass through its pole Q.

The relationship between poles and polars is reciprocal. Thus, if a point A lies on the polar line q of a point Q, then the point Q must lie on the polar line aof the point A. The two polar lines a and q need not be parallel.

There is another description of the polar line of a point P in the case that it lies outside the circle C. In this case, there are two lines through P which are tangent to the circle, and the polar of P is the line joining the two points of tangency (not shown here). This shows that pole and polar line are concepts in the projective geometry of the plane and generalize with any nonsingular conic in the place of the circle C.

Reciprocation and projective duality