# Vertex (curve)

In the geometry of planar 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 extreme point 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.

## Examples

A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry.^{[2]} On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.^{[4]}

For a circle, which has constant curvature, every point is a vertex.

## Cusps and osculation

Vertices are points where the curve has 4-point contact with the osculating circle at that point.^{[5]}^{[6]} In contrast, generic points on a curve typically only have 3-point contact with their osculating circle. The evolute of a curve will generically have a cusp when the curve has a vertex;^{[6]} other, more degenerate and non-stable singularities may occur at higher-order vertices, at which the osculating circle has contact of higher order than four.^{[5]} Although a single generic curve will not have any higher-order vertices, they will generically occur within a one-parameter family of curves, at the curve in the family for which two ordinary vertices coalesce to form a higher vertex and then annihilate.

The symmetry set of a curve 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.

## Other properties

According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices.^{[7]} A more general fact is that every simple closed space curve which lies on the boundary of a convex body, or even bounds a locally convex disk, must have four vertices.^{[8]}

If a planar 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.

## Notes

- ↑ Agoston (2005), p. 570; Gibson (2001), p. 126.
- 1 2 Gibson (2001), p. 127.
- ↑ Fuks & Tabachnikov (2007), p. 141.
- ↑ Agoston (2005), p. 570; Gibson (2001), p. 127.
- 1 2 Gibson (2001), p. 126.
- 1 2 Fuks & Tabachnikov (2007), p. 142.
- ↑ Agoston (2005), Theorem 9.3.9, p. 570; Gibson (2001), Section 9.3, "The Four Vertex Theorem", pp. 133–136; Fuks & Tabachnikov (2007), Theorem 10.3, p. 149.
- ↑ Sedykh (1994); Ghomi (2015)

## References

- Agoston, Max K. (2005),
*Computer Graphics and Geometric Modelling: Mathematics*, Springer, ISBN 9781852338176. - Fuks, D. B.; Tabachnikov, Serge (2007),
*Mathematical Omnibus: Thirty Lectures on Classic Mathematics*, American Mathematical Society, ISBN 9780821843161 - Ghomi, Mohammad (2015),
*Boundary torsion and convex caps of locally convex surfaces*, arXiv:1501.07626 - Gibson, C. G. (2001),
*Elementary Geometry of Differentiable Curves: An Undergraduate Introduction*, Cambridge University Press, ISBN 9780521011075. - Sedykh, V.D. (1994), "Four vertices of a convex space curve",
*Bull. London Math. Soc.*,**26**(2): 177–180