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 four vertices.
Vertices are points where the curve has 4-point contact with the osculating circle at that point. The evolute of 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.
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.
References
 |
This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2007) |
 |
This geometry-related article is a stub. You can help Wikipedia by expanding it. |
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
