Multiples of Homogeneous Cartesian Coordinates represent Same Point
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Theorem
Let $\CC$ denote the Cartesian plane.
Let $P$ be an arbitrary point in $\CC$.
Let $P$ be expressed in homogeneous Cartesian coordinates as:
- $P = \tuple {X, Y, Z}$
Then $P$ can also be expressed as:
- $P = \tuple {\rho X, \rho Y, \rho Z}$
where $\rho \in \R$ is an arbitrary real number such that $\rho \ne 0$.
Proof
By definition of homogeneous Cartesian coordinates, $P$ can be expressed in conventional Cartesian coordinates as:
- $P = \tuple {x, y}$
where:
\(\ds x\) | \(=\) | \(\ds \dfrac X Z\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \dfrac Y Z\) |
for arbitrary $Z$.
We have that:
\(\ds \dfrac X Z\) | \(=\) | \(\ds \dfrac {\rho X} {\rho Z}\) | ||||||||||||
\(\ds \dfrac Y Z\) | \(=\) | \(\ds \dfrac {\rho Y} {\rho Z}\) |
The result follows.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity