---
tag: J2KM2VYA3J
kind: concept
name: Steiner conic
msc: ["51A30", "51N15"]
sources:
  - url: https://en.wikipedia.org/wiki/Steiner_conic
same_as:
  wikidata: Q15844090
  wikipedia: Steiner conic
---

## Summary

A Steiner conic is a conic built synthetically from two pencils of lines.
Take two points $U$ and $V$ and a projective map from the lines through $U$ to the lines through $V$.
The points where each line meets its image form the Steiner conic.
In a projective plane over a field, Steiner conics are exactly the non-degenerate [[projective conics|R3TMVHTP3A]], so this gives a definition of conics with no equations.
The construction goes back to Jakob Steiner.

## Explanations

### L1 human

Pin one ruler at a point $U$ and another at a point $V$.
Turn them together by a fixed rule, so that each position of the first ruler decides the position of the second.
Mark where the two rulers cross.
When the rule is a projective map, and the line $UV$ is not sent to itself, the marks trace out a conic through $U$ and $V$.