Dupin indicatrix
Encyclopedia
The Dupin indicatrix is a method for characterising the local shape of a surface
. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature
. The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was invented by Charles Dupin
.
For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse
aligned with the principal direction
s.
For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola
. Two different hyperbola will be formed on either side of the tangent plane. These hyperbola share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic directions.
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
. The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was invented by Charles Dupin
Charles Dupin
Pierre Charles François Dupin was a French Catholic mathematician.He studied geometry with Monge at the École Polytechnique and then became a naval engineer. In 1819 he was appointed professor at the Conservatoire des Arts et Métiers; he kept this post until 1854...
.
For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
aligned with the principal direction
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...
s.
For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
. Two different hyperbola will be formed on either side of the tangent plane. These hyperbola share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic directions.