subarcs

A subarc is a portion of an arc, typically studied in geometry and topology. An arc is a simple curve that is homeomorphic to a closed interval, or more generally the image of an embedding of [0,1] into a space. A subarc is then a contiguous segment of that arc.

Formally, if c: [0,1] → X is an embedding whose image is an arc, then for any 0 ≤ a ≤ b ≤ 1, the image c([a,b]) is a subarc of c([0,1]). The endpoints of the subarc are c(a) and c(b), and the subarc itself is an arc. The degenerate case a = b yields a single point on the arc.

In the common setting of a circle or a planar curve, a subarc of an arc with endpoints P and Q is any arc whose endpoints lie on the original arc and whose entire interior lies within it. Equivalently, if an arc is parameterized along a circle or curve, subarcs correspond to taking a subinterval of the parameter domain.

Key properties include that subarcs are connected and compact whenever the original arc is; they inherit the order along the parent arc and reflect the same local geometric behavior. Different parameterizations can yield the same subarc, but as a subset of the ambient space, a subarc is well-defined.

Subarcs are used to study the structure of curves, to analyze segments of curves under limits or deformations, and to express local-to-global properties of arcs within larger spaces. Related notions include simple arcs and the broader concept of subspaces induced by restricting a parameterization.