Vertex (curve)

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 and in a quadratic of the form:

:$ax^2\; +\; bx\; +\; c\backslash ,\backslash !$

it can be found by completing the square or by differentiation.{{harvtxt|Gibson|2001}}, p. 127. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.{{harvtxt|Agoston|2005}}, p. 570; {{harvtxt|Gibson|2001}}, p. 127.

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

Vertices are points where the curve has 4-point contact with the osculating circle at that point.{{harvtxt|Gibson|2001}}, p. 126.{{harvtxt|Fuchs|Tabachnikov|2007}}, p. 142. 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; 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. 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.

According to the classical four-vertex theorem, every simple closed planar smooth curve must have at least four vertices.{{harvtxt|Agoston|2005}}, Theorem 9.3.9, p. 570; {{harvtxt|Gibson|2001}}, Section 9.3, "The Four Vertex Theorem", pp. 133–136; {{harvtxt|Fuchs|Tabachnikov|2007}}, Theorem 10.3, p. 149. 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.{{harvtxt|Sedykh|1994}}; {{harvtxt|Ghomi|2015}} Every curve of constant width must have at least six vertices.{{harvtxt|Martinez-Maure|1996}}; {{harvtxt|Craizer|Teixeira|Balestro|2018}}

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.

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Category:Curves