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The Four-Vertex Theorem

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This shows the result of the four-vertex theorem: A simple closed curve has at least four vertices. You can transform the closed curve by dragging the locator. If the caustic extends beyond the window, you can reduce its size.

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Let be a smooth plane curve parametrized by arc length , that is, for all . The number is called the curvature of at . A vertex of is a point where . A vertex corresponds to a cusp of the caustic generated by the curve. The theorem implies that the caustic of a general simple closed curce has at least four cusps (for a caustic, see Caustics on Spline Curves).

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Contributed by: Takaharu Tsukada (January 2012)
Open content licensed under CC BY-NC-SA

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Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

Ian R. Porteous, Geometric Differentiation: For the Intelligence of Curves and Surfaces, Cambridge University Press, 1994.

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