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The Four-Vertex Theorem

Graph of the Curvature

This demo simply takes a curve as input and displays the evolute to that curve and then, in another window, draws the graph of kg(t).

The purpose of this demo and the following exercise is to illustrate what is called the four-vertex theorem and will be explicitly stated below.

    Compare the numbers of cusps of the evolute curve for the ellipse, the epicycloid, and the curves in the cardioid family

      Xc(t )=((c+cos( t))cos(t ),(c+cos(t))sin(t))
    for various values of c .

Hopefully, the exercise demonstrated the following theorem. The four-vertex theorem states that for a closed convex curve kg(t) has at least two minima and at least two maxima. Furthermore, this implies that the evolute to any closed convex curve has a least four cusps . The result does not say anything about curves that are self-intersecting. Such curves may have just two cusps on their evolute curves.


Next: Winding Numbers of Plane Curves