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

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