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
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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