Vertex (curve)

Not to be confused with Vertex (geometry).
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 plane curves, a vertex is a point of where the first derivative of curvature is zero.[1] This is typically a local maximum or minimum of curvature,[2] and some authors define a vertex to be more specifically a local extremum of curvature.[3] However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For space curves, on the other hand, a vertex is a point where the torsion vanishes.

  1. ^ Agoston (2005), p. 570; Gibson (2001), p. 126.
  2. ^ Cite error: The named reference g01-127 was invoked but never defined (see the help page).
  3. ^ Fuchs & Tabachnikov (2007), p. 141.