Vertex (curve)
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For other uses, see Vertex.
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 4 vertices.
Vertices are points where the curve has 4-point contact with a circle. The evolute 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.
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