Vertex (curve)

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For other uses, see Vertex.

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 have a cusp when the curve has a vertex. 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.

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