Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates
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Theorem
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.
Proof
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.
Hence:
\(\ds l X + m Y + n Z\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds l \dfrac X Z + m \dfrac Y Z + n\) | \(=\) | \(\ds 0\) | dividing by $Z$ | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds l x + m y + n\) | \(=\) | \(\ds 0\) |
Hence the result.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity