indicatrices

Indicatrices is the plural form of indicatrix, a term used in differential geometry and related areas to describe a locus that encodes directional curvature or norm information at a point.

In surface theory, the Dupin indicatrix is a conic in the tangent plane at a point p of a smooth surface S in Euclidean space. It is defined by the equation IIp(v, v) = 1, where IIp is the second fundamental form and v lies in the tangent plane TpS. If the principal curvatures at p are κ1 and κ2, then IIp in the principal directions can be written as IIp(x, y) = κ1 x^2 + κ2 y^2. The locus κ1 x^2 + κ2 y^2 = 1 is an ellipse when κ1κ2 > 0 (Gaussian curvature K = κ1κ2 > 0), a hyperbola when κ1κ2 < 0 (K < 0), or two lines when κ1κ2 = 0 (K = 0). The indicatrix thus encodes the local shape of the surface and the directions of principal curvature, with its semiaxes related to the reciprocals of the principal curvatures.

Beyond surfaces, in normed or Finsler geometry, the indicatrix at a point of a manifold is the unit sphere in the tangent space according to the given norm. It is the boundary of the unit ball in TpM and can be a non-Euclidean, strictly convex hypersurface. Its shape reflects anisotropy of the norm and may vary with location on the manifold.

Thus indicatrices serve as a compact way to summarize local curvature behavior (Dupin indicatrix) or local norm geometry (unit indicatrix) in geometric spaces.