# Conjugate isothermal coordinates

Coordinates on a surface in which the second fundamental form is written as

\$\$ \tag{* } \textrm{ II } = - \Lambda ( u, v) ( du ^ {2} + dv ^ {2} ). \$\$

Conjugate isothermal coordinates can always be introduced in a sufficiently small neighbourhood of an elliptic point of a regular surface. In a sufficiently small neighbourhood of a hyperbolic point of a regular surface one can introduce coordinates in which

\$\$ \textrm{ II } = \Lambda ( u, v) ( du ^ {2} - dv ^ {2} ), \$\$

but in this case one often prefers the so-called asymptotic coordinates \$ \widetilde{u} , \widetilde{v} \$ for which

\$\$ \textrm{ II } = \widetilde \Lambda ( \widetilde{u} , \widetilde{v} ) \ d \widetilde{u} d \widetilde{v} . \$\$