Definition of Vertex
A vertex on a planar curve of differentiability class [latex]C^2[/latex] is a point at which the curvature function has a local maximum or minimum value.
This definition appears in standard expositions of the differential geometry of plane curves [1].
If the curvature is constant over an arc of the curve, then every point of the arc may be regarded as a vertex.
Aside
This definition applies to plane curves and does not carry over to curves in
3-space. For a [latex]C^3[/latex] curve in 3-space, a vertex is a point at which the torsion is 0. Torsion may be regarded as a scalar measure of the rate of change with respect to arc length of the binormal vector. The binormal is a unit vector perpendicular to the osculating plane, the plane that contains the tangent vector and the principal normal. Therefore, torsion may be regarded as the rate at which the osculating plane twists.
