Let $S$ be the unit sphere in $\Bbb R^3$ with centre $(0, 0, 0)$
$\sigma(u, v) = (\cos v/\cosh u,\sin v/\cosh u,\tanh u)$
is a parametrization of $S$ minus the north and south poles.
Show that meridians and parallels on $S$ correspond under $\sigma$ to perpendicular straight lines in the plane with coordinates $(u, v)$.