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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.

The orbifold fundamental group is the group of deck transformations of the universal orbifold cover. It retains the local finite group data that the ordinary fundamental group of the underlying topological space forgets.^[ref]

For an oriented closed topological surface of genus $g$ with cone points of orders $m_1,\dots ,m_r$, the group presentation isEach local cone relation says $c_i^{m_i}=1$; it does not guarantee that $c_i$ has order exactly $m_i$ in the resulting global group. For the teardrop orbifold, a two-sphere with one cone point of order $m>1$, the remaining relation is $c_1=1$, so its orbifold fundamental group is trivial despite nontrivial local isotropy.^[ref]

For a complete hyperbolic surface orbifold, the universal cover is the hyperbolic plane. A torsion-free finite-index subgroup then acts freely and gives a smooth surface cover. The hyperbolic covering model is what identifies local cone stabilizers with actual finite subgroups in this case.