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