---
tag: 579PT4D349
kind: definition
name: Analytic geometry
msc: [51N20, 51N10]
defines: H7CXZYANX0
uses: [0RWKYX8339, 0XPR1E85W0, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_geometry
---

## Definition

Let $E$ be a [[Euclidean space|0XPR1E85W0]] of dimension $n$, and fix a [[Cartesian coordinate system|0RWKYX8339]] on $E$.
It is a bijection $P \mapsto x(P) \in \R^n$ given by an origin and $n$ mutually perpendicular unit axes.
*Analytic geometry* is the study of figures in $E$ through their coordinates:

1. a figure $F \subseteq E$ is described by equations and inequalities, as the set of points $P$ whose coordinates satisfy them;
2. the [[distance|0NPPGHQNXQ]] between $P$ and $Q$ is computed as $\abs{x(P) - x(Q)} = \sqrt{\sum_{i=1}^n (x_i(P) - x_i(Q))^2}$, and angles through the dot product of coordinate vectors;
3. geometric statements are proved by algebraic computation on these equations.

## Remarks

Example: in the plane, the line $y = x$ and the circle $x^2 + y^2 = 2$ meet where $2x^2 = 2$, that is at $(1, 1)$ and $(-1, -1)$.
The distance between these points is $\sqrt{2^2 + 2^2} = 2\sqrt{2}$, the diameter of the circle, as it should be for a line through the centre.
Non-example: with oblique axes at an angle of $60^\circ$ and unit lengths on both axes, the formula of clause 2 fails.
The points with coordinates $(1, 0)$ and $(0, 1)$ are then at distance $1$, since with the origin they form an equilateral triangle, and not $\sqrt{2}$.
Affine notions such as lines, midpoints and parallelism can be studied with any affine coordinates; metric notions need Cartesian ones.
In the plane, a line is $\set{(x, y) : ax + by = c}$ with $(a, b) \ne (0, 0)$, and the circle with centre $(a, b)$ and radius $r$ is $\set{(x, y) : (x - a)^2 + (y - b)^2 = r^2}$.