---
tag: NVBD7Y86ES
kind: concept
name: Screw theory
msc: [70B10, 53A17]
sources:
  - url: https://en.wikipedia.org/wiki/Screw_theory
same_as:
  wikidata: Q7891441
  wikipedia: Screw theory
---

## Summary

Screw theory describes the velocities of a rigid body and the forces on it by six-dimensional vectors called screws.
Twists describe motions, and wrenches describe forces and moments.
Its geometric basis is Chasles' theorem.
It says that every orientation-preserving rigid motion of space is a screw motion.
That is, it is a rotation about a line combined with a translation along the same line.
A screw has an axis, a line in space, and a pitch, the ratio of translation to rotation.
It is used in the kinematics and statics of mechanisms and robots.
There the power of a wrench on a twist is given by a simple pairing.

## Explanations

### L1 human

Turn a screw with a screwdriver: it rotates and at the same time moves forward along its axis.
Screw theory says that every motion of a rigid object is like this at each instant: a turn about some axis plus a slide along that axis.
The axis and the amount of slide per turn, the pitch, describe the motion completely, up to its speed.
A pure rotation is a screw of pitch $0$, and a pure translation is a screw of infinite pitch.
Forces work the same way: any system of forces on a body acts like one force along a line together with a twisting moment about that line.