In the mathematical field of differential geometry a Liouville surface[1] (named after Joseph Liouville) is a type of surface which in local coordinates may be written as a graph in R3 such that the first fundamental form is of the form Sometimes a metric of this form is called a Liouville metric.
Darboux[2] gives a general treatment of such surfaces considering a two-dimensional space
is a constant related to the direction of the geodesic by where
is the angle of the geodesic measured from a line of constant
In this way, the solution of geodesics on Liouville surfaces is reduced to quadrature.
This was first demonstrated by Jacobi for the case of geodesics on a triaxial ellipsoid,[3] a special case of a Liouville surface.
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