Ringspanning

Ringspanning is a term used in geometry and related disciplines to describe the angular extent required on a circular ring to cover a set of points that lie on, or near, the ring. It provides a measure of dispersion for circular layouts and can be used to compare how spread out different point sets are around a circle.

Formally, suppose each point is mapped to an angle θi in [0, 2π). After sorting these angles, compute the circular gaps between consecutive points, including the wrap-around gap Gmax = max{θi+1 − θi, with θn+1 = θ1 + 2π}. The ringspan is defined as 2π − Gmax. This is the smallest arc length on the circle that contains all the given points. The ringspan thus ranges from 0 (all points coincide) to 2π (points are spread across the entire circle).

Variants and applications exist where ringspan is used as a design metric. In network design and circular layouts, it can quantify how evenly nodes are distributed around a ring topology, which has implications for latency, load balancing, and redundancy. In data visualization or geospatial analysis, ringspan helps assess how well a set of observations covers a circular region or boundary.

Related concepts include arc length, angular dispersion, and circular statistics. Ringspanning is often described as an informal or context-dependent notion, with precise definitions adapted to the specific application, such as exact tolerance for points near the ring or extensions to three-dimensional toroidal geometries.