Can the center of a circle be considered a pole with a polar line?

Question

Can the center of a circle be considered a pole with a polar line?

I read few web pages. Some of them said that a pole can be any point in the plane of circle, while some say it cannot be the center of the circle.

Please clarify for me.

Answer 1

It depends on the plane your circle lives in. In a standard Euclidean plane, the center has no polar line. In the projective plane, the polar of the center is the line at infinity.

You can observe that as a point approaches the center, it's polar moves increasingly further away from the circle. In projective geometry the limit of this process had its own well-defined representation.