ringsandarcs

Ringsandarcs is a term used in plane geometry to describe configurations formed from rings and circular arcs. A ring denotes the annulus between two concentric circles. An arc is a connected portion of a circle’s circumference, defined by its endpoints or by its intersections with other elements.

An arrangement of rings and arcs consists of a finite set of rings and arcs embedded in the plane, with arcs allowed to meet rings and other arcs. The arrangement induces a planar graph: vertices are arc endpoints and intersection points; edges are the arc and ring segments between consecutive vertices. The primary objects of study are the combinatorial and topological properties of these graphs, such as the number of connected components, the number of regions, and the structure of the boundaries.

Typical assumptions include transversal intersections and simplicity (no overlapping arcs along the same curve) unless stated otherwise. Ringsandarcs may be constrained to lie on ring boundaries, to cross rings, or to connect different rings via endpoints.

Analytical and computational methods from the theory of circle arrangements and planar graphs apply. Construction algorithms, such as plane-sweep approaches, enable the explicit rendering and region counting of ringsandarcs configurations. Invariants like the Euler characteristic and the genus (in extended contexts) provide classification tools.

Applications occur in computer-aided design, graphic ornamentation, and motion-planning problems involving curved annular regions. The concept also serves as a link to related topics, including arrangements of circles, annuli, and curved networks, and is subject to ongoing formalization in specialized literature.