Dupin indicatrix
curvature indicatrix
A plane curve illustrating the normal curvatures of a surface at a point of this surface. The Dupin indicatrix lies in the tangent plane to the surface at the point
and there it is described by the radius vector
of length
, where
is the normal curvature of
at
in the direction
. Let
be a parametrization of
in a neighbourhood of
. One introduces a coordinate system on the tangent plane to
at
, taking
as the coordinate origin, and the vectors
and
as the basis vectors of this coordinate system. The equation of the Dupin indicatrix will then be
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where and
are the coordinates of a point on the Dupin indicatrix, and
,
and
are the coefficients of the second fundamental form of
calculated at
. The Dupin indicatrix is: a) an ellipse if
is an elliptic point (a circle if
is an umbilical point); b) a pair of conjugate hyperbolas if
is a hyperbolic point; and c) a pair of parallel straight lines if
is a parabolic point. The curve is named after Ch. Dupin (1813), who was the first to use this curve in the study of surfaces.
Figure: d034180a
References
[1] | V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) (In Russian) |
Comments
The Dupin indicatrix does not exist at a flat point.
The Dupin indicatrix at can be obtained as the limit of suitably normalized (inter)sections of the surface with planes parallel to the tangent plane of
at
which are approaching this plane, see [a1], p. 370; [a2], p. 363-365.
References
[a1] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |
[a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) |
[a3] | C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4 |
Dupin indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dupin_indicatrix&oldid=13694