Frame Fields: Anisotropic and Non-Orthogonal Cross Fields Additional material

If we consider the quotient space of each tangent plane with respect to this equivalence relation, and the related quotient tangent bundle TS/∼1, a field FD : S −→ TS/∼1 maps each point on the surface to a direction. It is customary to take unit-length vectors as representatives of their equivalence classes, so that a direction field can be regarded as a vector field where the length of all vectors is 1. This is equivalent to identifying the quotient space of the tangent plane at p to the unit circle centered at p. Note that if we take a vector field F and we define its corresponding directional field F/∼1 by mapping each vector in the image ofF to its representative direction, then F/∼1 is undefined at points where F vanishes. Vanishing points of vector fields, as well as isolated points where directional fields are undefined, are called singularities.