The Generalized Comrie Projection: A Closed-Form Transformation from the Fundamental Plane to Arbitrary Observer Altitude

This paper develops the Generalized Comrie Projection (GCP), a closed-form transformation from arbitrary points on the Besselian Fundamental Plane to geographic coordinates at arbitrary observer altitude. The method extends Leslie J. Comrie’s 1933 eclipse transformation by replacing the terrestrial reference ellipsoid with an altitude-dependent virtual ellipsoid while preserving the structure of Comrie’s auxiliary-sphere construction. A direct Cartesian Ellipsoid-Normal Intersection (ENI) formulation is developed for the same problem, and the GCP and ENI are proved algebraically equivalent for an arbitrary Fundamental-Plane point and common target ellipsoid. The GCP is also shown to be an exact right inverse of the corresponding forward observer transformation on the virtual ellipsoid, with the transformations mutually inverse on its Sun-facing hemisphere. The difference between the virtual ellipsoid and the exact surface of constant geodetic height is derived and quantified. Near the terrestrial surface, the leading discrepancy is fourth order in the Earth’s first eccentricity and is only centimeter-scale at ordinary aircraft and high-altitude observing altitudes. The maximum normal separation remains bounded as altitude increases. The formulation is applicable not only to eclipse centerlines but to arbitrary Fundamental-Plane points and loci, providing a closed-form basis for extending classical Besselian eclipse geometry to observers above or below the terrestrial reference ellipsoid. UPDATE 21 Sep 2026: v1.1 This revision clarifies that the GCP altitude parameter h is ellipsoidal height relative to the adopted reference ellipsoid rather than MSL altitude, and adds an appendix describing the required conversion from MSL/orthometric height using geoid undulation. It also adds practical guidance for GPS/GNSS and aircraft-altitude inputs, corrects several worked-example and cross-reference errors, removes an incorrect attribution of a 15 deg/h Earth-rotation rate to Meeus, and fixes the Appendix B numerical closure values for the 10-km example. The underlying GCP derivation, GCP–ENI equivalence proof, virtual-ellipsoid accuracy analysis, and principal conclusions are unchanged.