The Dupin indicatrix is a method for characterising the local shape of a surface. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. 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. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was invented by Charles Dupin. Pierre Charles François Dupin ( October 6, 1784 in Varzy, France – January 18, 1873 in Paris, France
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 directions. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Differential geometry, the two principal curvatures at a given point of a Surface measure how the surface bends by different amounts in different directions
For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions 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. In the Differential geometry of surfaces, an asymptotic curve is a Curve always Tangent to an asymptotic direction of the surface (where they exist
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