[[Projective space]]
# Homogenous coördinates

**Homogenous coördinates** are a system of coördinates used to label (not uniquely) points in a [[Projective space]] $\mathrm{P}^n \mathbb{K}$.
A projective point is labelled by $n+1$ scalars that are not all zero, written as
$$
\begin{align*}
p = (p_{1}:p_{2}:\dots:p_{n+1})
\end{align*}
$$
where any nonzero multiple of the coördinates denotes the same point.
Viewing $\mathrm{P}^n \mathbb{K}$ as the [[Orbit space|quotient]] $\mathbb{K}^n \setminus \{ 0 \} / \mathbb{K}^\times$,
the point $(p_{1}:\dots:p_{n+1})$ denotes the equivalence class of $(p_{1},\dots,p_{n+1})$.

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