Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates

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A straight line $\LL$ is the set of all points $P$ in $\R^2$, where $P$ is described in homogeneous Cartesian coordinates as:

$l X + m Y + n Z = 0$

where $l, m, n \in \R$ are given, and not both $l$ and $m$ are zero.


Let $P = \tuple {X, Y, Z}$ be a point on $L$ defined in homogeneous Cartesian coordinates.

Then by definition:

\(\ds x\) \(=\) \(\ds \dfrac X Z\)
\(\ds y\) \(=\) \(\ds \dfrac Y Z\)

where $P = \tuple {x, y}$ described in conventional Cartesian coordinates.


\(\ds l X + m Y + n Z\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds l \dfrac X Z + m \dfrac Y Z + n\) \(=\) \(\ds 0\) dividing by $Z$
\(\ds \leadstoandfrom \ \ \) \(\ds l x + m y + n\) \(=\) \(\ds 0\)

Hence the result.