Vertex (curve)

In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero.

[3] However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant.

On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form: it can be found by completing the square or by differentiation.

[5][6] In contrast, generic points on a curve typically only have 3-point contact with their osculating circle.

According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices.