indicatrix

Indicatrix is a term used in differential geometry and related areas to denote a curve, surface, or bundle that encodes directional information about a geometric object. The most common senses are grouped around two themes: unit spheres defined by a norm, and the tracing of unit directions along a curve.

In normed or Finsler geometry, the indicatrix at a point x of a manifold M is the set I_x = { v in the tangent space T_xM : F(x, v) = 1 }, where F is a norm or a Minkowski norm on the tangent spaces. This unit sphere or unit circle (in the plane) represents directions of unit length according to the given metric. The shape of the indicatrix reflects anisotropy of the metric: for the standard Euclidean metric it is the ordinary unit sphere, while in a general Finsler space it may be a non-spherical convex curve or surface. The collection of all I_x across x in M forms the indicatrix bundle, a foundational object in Finsler geometry that influences curvature notions such as flag curvature.

A related usage arises in the study of space curves. The tangent indicatrix of a space curve γ(s) is the map s ↦ T(s), where T(s) is the unit tangent vector, viewed as a curve on the unit sphere. Similarly, one may consider the normal indicatrix s ↦ N(s) and the binormal indicatrix s ↦ B(s). In the plane, the tangent indicatrix lies on the unit circle and encodes how the direction of the curve changes.

Etymology and usage: indicatrix comes from Latin indicare, “to indicate.” See also unit sphere, Minkowski norm, and tangent indicatrix.