# Evolvent of a plane curve

A curve assigned to the plane curve such that is the evolute of . If (where is the arc length parameter of ) is the equation of , then the equation of its evolvent has the form

where is an arbitrary constant and the unit tangent vector to . The figures show the construction of the evolvent in two typical cases: a) if for any the curvature of does not vanish (the evolvent is a regular curve); and b) if vanishes only for and (the point corresponding to on the evolvent is a cusp of the second kind).

Figure: e036720a

Figure: e036720b

About the evolvent of a surface, see Evolute (surface).

#### Comments

The evolvent is often called the involute of the curve. Involvents play a part in the construction of gears.

For references see also Evolute.

#### References

[a1] | K. Strubecker, "Differential geometry" , I , de Gruyter (1964) |

[a2] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. 305ff (Translated from French) |

[a3] | J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959) pp. 195 |

[a4] | H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60 |

[a5] | M. Berger, "Geometry" , I , Springer (1987) pp. 253–254 |

**How to Cite This Entry:**

Evolvent of a plane curve.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Evolvent_of_a_plane_curve&oldid=15581