Indicatrix

from Wikipedia, the free encyclopedia

In the differential geometry of curved surfaces in space, an indicatrix is understood to be 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 also bears the name Dupin's indicatrix .

Geometric description

Hyperbolic point of a surface (brown) with a tangent plane from the English Wikipedia

In a sufficiently small neighborhood of a point a of a surface (given for example by z = f (x, y) with f twice continuously differentiable) the surface can be divided into a quadric , i.e. a second order surface of the form z = g ( x, y), approximate as precisely as you want. This osculating quadric will intersect the infinitesimally displaced tangential plane in the direction of the surface normal or in the opposite direction. There are four possible cases:

  • The intersections are always empty; the osculating quadric has degenerated to the tangential plane. Nevertheless, a is called a parabolic point (because the determinant of the second fundamental form vanishes).
  • The intersection consists of two parallel straight lines on one side of the surface and the empty set on the other (for example in the case of a cylinder); one speaks of a parabolic point of the surface. The osculating square is a parabolic cylinder (see web link below)
  • The intersection is an ellipse when moved in a normal direction and empty when moved in the opposite direction (for example in the case of a spherical surface); a is called an elliptical point on the surface. The osculating square is an elliptical paraboloid .
  • Depending on the direction of the shift, the intersection results in one or the other hyperbola of a conjugated hyperbola pair (for example in the case of a saddle surface; see graphic on the right); a is then called a hyperbolic point of the surface. The osculating quadric is a hyperbolic paraboloid .

The two main curves

These four cases are usually differentiated today by the two main curvatures of the surface. For these apply:

  • Both main curvatures are zero if the osculating quadric degenerates to the tangential plane.
  • Exactly one of the two is zero in the case of a parabolic point with a non-planar osculating quadric.
  • Both have the same sign in the case of an elliptical point.
  • Both have different signs in the case of a hyperbolic point.

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

Formal description

Every straight line of the tangential plane running through point a corresponds to a curve segment on the surface; this has a certain normal curvature κ in a. If κ is not zero, the radius of the circle of curvature in a is given by the reciprocal of | κ |. Then the two points of the starting line at a distance from a belong to the indicatrix of a.

Applications

literature

Web links

  • Images of quadrics Only the areas designated there as parabolic quadrics appear as osculating quadrics . Retrieved August 13, 2009