Coordinates of Pole of Given Polar/Homogeneous Coordinates
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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 $-\dfrac n {r^2}$:
- $P = \tuple {l, m, -\dfrac n {r^2} }$
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $7$. To find the pole of the line $\ldots$ with respect to the circle $\ldots$