Evolute
of a plane curve
The set 
of the centres of curvature of the given curve 
. If 
(where 
is the arc length parameter of 
) is the equation of 
, then the equation of its evolute has the form
 |
where 
is the curvature and 
the unit normal to 
. The figures shows the construction of the evolute in three typical cases:
a) if along the entire curve 
has a fixed sign and 
does not vanish;
Figure: e036670a
Figure: e036670b
b) if along the entire curve 
has a fixed sign and 
vanishes for 
; and
c) if 
for 
; 
for 
; 
, and 
does not vanish (the point of the evolute corresponding to 
is a cusp).
Figure: e036670c
The length of the arc of the evolute corresponding to the segment 
of 
is
 |
The evolute is the envelope of the normals to 
. The curve 
is called the evolvent of its evolute (cf. Evolvent of a plane curve).
Comments
The evolvent is also called the involute; thus, if 
is the evolute of 
, then 
is the involute of 
, cf. Evolvent of a plane curve.
References
| [a1] | H.-R. Müller, "Kinematik" , de Gruyter (1963) |
| [a2] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. 305 (Translated from French) |
| [a3] | J.L. Coolidge, "Algebraic plane curves" , Dover, reprint (1959) pp. 195 |
| [a4] | M. Berger, "Geometry" , I , Springer (1987) pp. 253–254 |
| [a5] | H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60 |
Evolute. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evolute&oldid=46863