---
tag: HPW5F0GQ4B
kind: concept
name: Distance geometry
msc: [51K05, 51K99]
sources:
  - url: https://en.wikipedia.org/wiki/Distance_geometry
same_as:
  wikidata: Q2088417
  wikipedia: Distance geometry
---

## Summary

Distance geometry studies sets of points through the distances between them alone.
Its central question asks when given distances between some pairs of points can be realized by actual points in a Euclidean space.
Tools include the [[Cayley–Menger determinant|KGC5CYDF53]], [[Euclidean distance matrices|1BCJ7A73KS]] and the theory of [[rigid frameworks|V5YS9BZX0N]].
It has applications to molecular shape, sensor networks and the design of structures.

## Explanations

### L1 human

Suppose you know how far apart some cities are, but you have no map.
Distance geometry asks whether such a map exists and whether it is unique.
For example, three distances $3$, $4$ and $5$ fit a right triangle in the plane.
The distances $1$, $1$ and $3$ fit nothing, because $3 > 1 + 1$ breaks the [[triangle inequality|01M6CW9VM6]].