polepolar

Polepolar refers to the pole-polar relationship in geometry with respect to a conic. In a projective plane, given a nondegenerate conic C, to each point P not on C there corresponds a unique polar line p with respect to C, and to each line l not tangent to C there corresponds a unique pole P. The correspondence P ↔ p is a polarity, an incidence-reversing involution: the polar of P is p, and the pole of p is P. If P lies on C, its polar is the tangent to C at P; if l is tangent to C, its pole is the point of contact.

La Hire's theorem is a central property: if P lies on the polar of Q, then Q lies on the polar of P. Polarity provides a natural duality between points and lines, preserving incidence in reverse.

In the common case of a circle, the polar has concrete constructions. From an external point P, the two tangents to the circle touch it at A and B; the line AB is the polar of P. Conversely, the pole of a line l not meeting the circle is the intersection of the tangents at the endpoints of l’s chord of contact; the pole of a tangent line is its point of contact on the circle.

Analytically, with the unit circle x^2 + y^2 = 1, the polar of a point (a, b) is the line a x + b y = 1, and the pole of a line u x + v y = 1 is the point (u, v). Lines through the origin have poles at infinity in directions perpendicular to the line.

Polepolar underpins many geometric constructions and duality concepts in Euclidean and projective geometry, and serves as a foundational tool in problems involving tangency and conic sections.