Dupin indicatrix

Under an indicatrix is understood in the differential geometry of curved surfaces in space a plane conic section that describes the local curvature behavior of the surface at a certain point. The term was introduced by Charles Dupin at the beginning of the 19th century and therefore is also called Dupin indicatrix.

Geometric description

In a sufficiently small neighborhood of a point a of a surface ( given approximately by z = f (x, y) with f twice continuously differentiable ) can be the face by a quadric, ie by a surface second order of the form z = g ( x, y), approach with arbitrary precision. This Schmiegequadrik will cut the infinitesimally shifted in the direction of the surface normal or her opposite direction tangential plane. Here, four cases can occur:

  • The intersections are always empty; the Schmiegequadrik degenerates to the tangent plane. Nevertheless is called a parabolic point (because the determinant of the second fundamental form vanishes ).
  • The intersection consists of two parallel lines on one side of the surface and of the empty set on the other ( as in the case of a cylinder ); it is called a point of the parabolic surface. The Schmiegequadrik is a parabolic cylinder (see web link below)
  • The intersection is with a shift in the normal direction of an ellipse and displacement in the opposite direction is empty ( as in the case of a spherical surface ); we say that a point of an elliptic surface. The Schmiegequadrik is an elliptic paraboloid.
  • The intersection results depending on the direction of displacement of one or the other hyperbola of a conjugated Hyperbelpaars ( as in the case of a saddle surface; see chart at right); then we say that a point of a hyperbolic surface. The Schmiegequadrik is a hyperbolic paraboloid.

The two main curvatures

These four cases are now commonly distinguished on the two principal curvatures of the surface. For these apply:

  • Both principal curvatures are zero when the Schmiegequadrik degenerates to the tangent plane.
  • Exactly one of the two is zero in the case of a parabolic point with non-planar Schmiegequadrik.
  • Both have the same sign in the case of an elliptic point.
  • Both have different signs in the case of a hyperbolic point.

The product of the main curvatures, the so-called Gaussian curvature, that is positive in the case of an elliptical spot in the case of a hyperbolic negative point; otherwise it is zero.

Formal Description

Each passing through the point of a straight line corresponding to a tangential plane of the cam piece on the surface; This has a certain normal curvature κ in a. If κ is not zero, the radius of curvature in the circle A is given by the reciprocal value of | κ |. Then the two are at a distance from a nearby points of the output straight to the indicatrix of a

Applications

  • The index ellipsoid is an indicatrix that is used for calculation of birefringence.
  • With the Tissotschen indicatrix distortion characteristics are checked by card network designs.
Tissot's indicatrix
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