---
tag: XYJ6T5CKR6
kind: concept
name: Umbilical point
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/Umbilical_point
same_as:
  wikidata: Q3351935
  wikipedia: Umbilical point
---

## Summary

An umbilical point, or umbilic, of a surface is a point where the two [[principal curvatures|FA8QE1SGP4]] are equal.
There the surface bends by the same amount in every direction, like a sphere or a plane.
Every point of a sphere or a plane is umbilical.
Conversely, a connected surface made only of umbilical points is a piece of a sphere or of a plane.

## Explanations

### L1 human

At most points of a surface there is a direction of strongest bending and a direction of weakest bending.
At an umbilical point there is no preferred direction: the surface looks the same in every direction, up to second order.
On a ball every point is like this.
An egg-shaped ellipsoid with three different axes has exactly four such special points.