Isometry of a cone
= Isometry of a cone
{wiki}
Every <isometry> of a complete singular <flat cone>, of <cone angle> $\theta\ne2\pi$, fixes its vertex: this is the unique <point> without a locally Euclidean neighbourhood. It preserves <distance> from that vertex and has the form
$$
(r,\varphi)\mapsto(r,\varphi+c)
\quad\hbox{or}\quad
(r,\varphi)\mapsto(r,-\varphi+c).
$$
These are angular rotations and reflections. The <stabilizer> of a nonvertex <point> consists of the identity and its radial reflection. The vertex <stabilizer> is the entire <isometry group>, including all rotations: an unspecified fixed <point> does not give the two-element conclusion.
When $\theta=2\pi$, the complete cone is the <Euclidean plane>, allowing translations and every rotation about a fixed <point>. This smooth case is excluded from the singular-cone statement.