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, and some authors define a vertex to be more specifically a local extreme point of curvature. However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant.
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. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.
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. 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.
If a 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.
- Agoston (2005), p. 570; Gibson (2001), p. 126.
- Gibson (2001), p. 127.
- Fuks & Tabachnikov (2007), p. 141.
- Agoston (2005), p. 570; Gibson (2001), p. 127.
- Gibson (2001), p. 126.
- 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.
- 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
- Gibson, C. G. (2001), Elementary Geometry of Differentiable Curves: An Undergraduate Introduction, Cambridge University Press, ISBN 9780521011075.