---
tag: 6FFVXT5KMH
kind: concept
name: Arc in a finite plane
msc: ["51E21", "51E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Arc_(projective_geometry)
same_as:
  wikidata: Q4785004
  wikipedia: Arc (projective geometry)
---

## Summary

A $k$-arc in a finite [[projective plane|DANFNN8C4Y]] is a set of $k$ points with no three on a common line.
In a plane of order $n$ a $k$-arc has at most $n + 2$ points, and at most $n + 1$ points when $n$ is odd.
Arcs of maximal size, such as conics, are central objects of finite geometry and are linked to maximum distance separable codes.

## Explanations

### L1 human

Pick points in a finite plane so that no line passes through three of them.
Such a set is an arc, a discrete version of a curve that bends everywhere.
In the plane over the field with $q$ elements, the points $(1 : t : t^2)$ for $t$ in the field, together with $(0 : 0 : 1)$, form an arc with $q + 1$ points.
It is the finite analogue of a parabola together with its point at infinity.