Vertex (curve)

In the context of curves, a vertex is a point on the curve where the curvature attains a local maximum or minimum. Essentially, it represents a point where the curve is "bending" most sharply, either inward or outward, compared to its immediate surroundings.

More formally, a vertex is a stationary point of the curvature function, meaning the derivative of the curvature with respect to arc length is zero at a vertex. This implies that the rate of change of the curvature is zero at that point, signifying a local extremum.

Vertices are important features for analyzing the shape and properties of curves. They can be used for curve segmentation, shape recognition, and other geometric processing tasks. Not every curve possesses vertices; for instance, a straight line has constant zero curvature and therefore no vertices, while a circle has constant non-zero curvature and also lacks vertices. The presence and location of vertices provide key information about the non-uniformity and complexity of a curve's shape.