Cone point of an orbifold ($2\pi /m$)

A cone point of an oriented two-dimensional orbifold is a point with a chart modeled on a disk modulo a nontrivial finite cyclic rotation group. For a group of order $m$, its local quotient metric has total angle $2\pi /m$, and its stabilizer is cyclic of order $m$.

For example, identifying the edges of a planar sector of angle $2\pi /3$ gives the local order-three model. The underlying quotient neighborhood is homeomorphic to a disk, so the cone order is not detectable from topology alone. A point with order one is an ordinary nonsingular point.

A cone point is distinct from a mirror boundary point, whose local group acts by a reflection. It also differs from a general metric cone singularity: an orbifold cone angle must be of the form $2\pi /m$ with positive integer $m$. A teardrop orbifold has exactly one such exceptional point on an underlying sphere.