---
tag: H7CXZYANX0
kind: concept
name: Analytic geometry
msc: [51N20, 51N10]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_geometry
same_as:
  wikidata: Q134787
  wikipedia: Analytic geometry
---

## Summary

Analytic geometry is geometry done with coordinates: points become tuples of numbers, and figures become sets of solutions of equations.
Through a [[Cartesian coordinate system|0RWKYX8339]], lines are given by linear equations, circles and [[conic sections|00HJCMC2CD]] by quadratic ones, and distances by the formula $\sqrt{(x_1 - y_1)^2 + \cdots + (x_n - y_n)^2}$.
Geometric questions then become algebraic computations, which can be carried out systematically.
It underlies much of calculus, linear algebra and algebraic geometry, and it is usually traced to Descartes and Fermat.

## Explanations

### L1 human

Put a grid on the plane, so that every point gets an address $(x, y)$.
Then a line is the set of points whose addresses satisfy an equation like $y = 2x + 1$, and a circle of radius $5$ around the origin is $x^2 + y^2 = 25$.
To find where the line meets the circle, one solves the two equations together, with no ruler or compass needed.
This turns pictures into algebra, and algebra into pictures.