Coordinates of Pole of Given Polar/Homogeneous Coordinates

Theorem
Let $\CC$ be a circle of radius $r$ whose center is at the origin of a Cartesian plane.

Let $\LL$ be a straight line whose equation is given as:
 * $l x + m y + n = 0$

The pole $P$ of $\LL$ with respect to $\CC$, given in homogeneous Cartesian coordinates is:
 * $P = \tuple {l, m, -\dfrac n {r^2} }$

Proof
From Coordinates of Pole of Given Polar, $P$ can be expressed in conventional Cartesian coordinates as:
 * $P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2}$

Hence in homogeneous Cartesian coordinates:


 * $P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2, 1}$

From Multiples of Homogeneous Cartesian Coordinates represent Same Point, we multiply each coordinate by $-\dfrac n {r^2}$:
 * $P = \tuple {l, m, -\dfrac n {r^2} }$