eccentricity

Eccentricity is a parameter that measures deviation of a geometric figure from a circle or, more broadly, from a reference symmetry. In geometry, it is a constant associated with a conic section that classifies the curve as circle, ellipse, parabola, or hyperbola. In graph theory, eccentricity of a vertex is the greatest distance to any other vertex, and graph radius and diameter are derived from these.

In conic sections, eccentricity e is defined as the ratio of the distance from a point on the conic to a fixed focus and the distance from that point to a fixed directrix. For a circle, e = 0; for an ellipse, 0 < e < 1; for a parabola, e = 1; for a hyperbola, e > 1. The eccentricity is the same for all points on a given conic.

In celestial mechanics, the orbital eccentricity characterizes the shape of an orbit. Planets orbit with e between 0 and about 0.2; comets often have e near or above 1. The energy and angular momentum of the orbit determine e via relationships in the two-body problem.

In graph theory, a vertex's eccentricity is the greatest distance to any other vertex; the radius is the minimum of these values, and the diameter is the maximum. The concept is defined for connected graphs, reflecting how spread out a graph’s vertices are from a given point.