Curvature - Curves On Surfaces

Curves On Surfaces

When a one dimensional curve lies on a two dimensional surface embedded in three dimensions R3, further measures of curvature are available, which take the surface's unit-normal vector, u into account. These are the normal curvature, geodesic curvature and geodesic torsion. Any non-singular curve on a smooth surface will have its tangent vector T lying in the tangent plane of the surface orthogonal to the normal vector. The normal curvature, kn, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, kg, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τr, measures the rate of change of the surface normal around the curve's tangent.

Let the curve be a unit speed curve and let t = u × T so that T, u, t form an orthonormal basis: the Darboux frame. The above quantities are related by:

\begin{pmatrix} \mathbf{T'}\\ \mathbf{t'}\\ \mathbf{u'} \end{pmatrix} = \begin{pmatrix} 0&\kappa_g&\kappa_n\\ -\kappa_g&0&\tau_r\\ -\kappa_n&-\tau_r&0 \end{pmatrix} \begin{pmatrix} \mathbf{T}\\ \mathbf{t}\\ \mathbf{u} \end{pmatrix}

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