Vertex (curve)

For other uses, see Vertex (disambiguation).

An ellipse (red) and its evolute (blue). The dots are the vertices of the curve, each corresponding to a cusp on the evolute.

In the geometry of curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For a circle which has constant curvature, every point is a vertex.

The four-vertex theorem states that every closed curve must have at least four vertices.

Vertices are points where the curve has 4-point contact with the osculating circle at that point. The evolute of a curve will generically have a cusp when the curve has a vertex. Other, more degenerate and non-stable singularities occur at higher vertices. Higher vertices generically occur in a one-parameter family of curves when two ordinary vertices coalesce to form a higher vertex; after which they annihilate.

The symmetry set has endpoints at the cusps corresponding to the vertices, and the medial axis, a subset of the symmetry set, also has its endpoints in the cusps.

If a curve is bilaterally symmetric, it will have a vertex at the point or points where the axis of symmetry crosses the curve. Thus, the notion of a vertex for a curve is closely related to that of an optical vertex, the point where an optical axis crosses a lens surface.

Vertices of a conic section

A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.