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 |
Evolvent of a plane curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evolvent_of_a_plane_curve&oldid=42552