Homogeneous coordinates

Builds on Projective space Equivalence class

Definition

Let $k$ be a field and let $n \ge 0$ . A point $p$ of the projective space $\mathbb{P}^n(k)$ is an equivalence class of nonzero vectors of $k^{n+1}$ . Two vectors are equivalent when one is a nonzero scalar multiple of the other. A tuple $(x_0, \dots, x_n) \in k^{n+1} \setminus \set{0}$ in this class is a tuple of homogeneous coordinates of $p$ , and one writes $p = [x_0 : \dots : x_n]$ .

Remarks

Two tuples are homogeneous coordinates of the same point exactly when they differ by a nonzero factor $\lambda \in k$ .

Example: in $\mathbb{P}^2(\R)$ , $[1 : 2 : 3] = [-2 : -4 : -6]$ . Check: $(-2, -4, -6) = -2 \cdot (1, 2, 3)$ .

Non-example: $(0, 0, 0)$ is not a tuple of homogeneous coordinates of any point, since the zero vector is excluded.

The affine point $(x_1, \dots, x_n)$ of $k^n$ corresponds to $[x_1 : \dots : x_n : 1]$ ; the points with last coordinate $0$ form the hyperplane at infinity. For a homogeneous polynomial $F$ , the condition $F(x_0, \dots, x_n) = 0$ does not depend on the chosen coordinates of a point, since $F(\lambda x) = \lambda^d F(x)$ for $F$ of degree $d$ .

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Tag	`Q2V8JVPB5E`
URI	`m2://Q2V8JVPB5E`
Kind	definition
Writing	W0 stub
MSC	51N15, 14N05
Defines	Homogeneous coordinates
Uses	Projective space, Equivalence class

Homogeneous coordinates

Definition

Remarks

Reading path

Further reading

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