Point at Infinity of Intersection of Parallel Lines
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Theorem
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$ such that $\LL_1$ and $\LL_2$ are parallel.
By Condition for Straight Lines in Plane to be Parallel, $\LL_1$ and $\LL_2$ can be expressed as the general equations:
\(\ds \LL_1: \ \ \) | \(\ds l x + m y + n_1\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \LL_2: \ \ \) | \(\ds l x + m y + n_2\) | \(=\) | \(\ds 0\) |
The point at infinity of $\LL_1$ and $\LL_2$ can thence be expressed in homogeneous Cartesian coordinates as $\tuple {-m, l, 0}$.
Proof
Let $\LL_1$ be expressed in the form:
- $l x + m y + n = 0$
Hence let $\LL_2$ be expressed in the form:
- $l x + m y + k n = 0$
where $k \ne 1$.
Let their point of intersection be expressed in homogeneous Cartesian coordinates as $\tuple {X, Y, Z}$
Then:
\(\ds \tuple {X, Y, Z}\) | \(=\) | \(\ds \tuple {m n \paren {k - 1}, n l \paren {1 - k}, 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {-m, l, 0}\) |
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity