Conservation Equation along Great Circles (G.C.) and Great Ellipses (G.E.). Modified Clairaut's Relation of a G.E.

Two tools of Differential Geometry (diffeomorphisms between two-manifolds, -surfaces-, pushforwardof related vector fields) will allow us to write, in closed form, the relationships for thedetermination of: 1) azimuth of points along Great Ellipse (G.E.) on ellipsoid of revolution E2, as afunction of both latitude and longitude, known the initial and final points of the route; 2) coordinatesof the vertices and the expression of longitude as a function of latitude and its inverse (G.E. equation).We will able to write the above relations by demostrating the existence of a Conservation Equation(C.Eq.), for Great Circle (G.C.) on sphere S2, and for G.E. on E2, that which will allow us animmediate derivation of Clairaut Relation on G.C. in a completely innovative form, with the ClairautConstant, C l , determined directly to the constant of motion of C.Eq., C1,2. It will also be possible,again using the Conservation Equation, to write the constant C l as a measure of area, (definiteintegral of the square of the cosine of the latitude). Finally, as the G.E. is not, in general, a geodesicof the ellipsoid of revolution, will be given a "modified Clairaut Relation" for G.E.