  In this work we systematically derive the governing equations of supersonic
conical flow by projecting the 3D Euler equations onto the unit sphere. These
equations result from taking the assumption of conical invariance on the 3D
flow field. Under this assumption, the compressible Euler equations reduce to a
system defined on the surface of the unit sphere. This compressible flow
problem has been successfully used to study steady supersonic flow past cones
of arbitrary cross section by reducing the number of spatial dimensions from 3
down to 2 while still capturing many of the relevant 3D effects. In this paper
the powerful machinery of tensor calculus is utilized to avoid reference to any
particular coordinate system. With the flexibility to use any coordinate system
on the surface of a sphere, the equations can be more readily solved
numerically when a structured mesh is used by defining the mesh lines to be the
coordinate lines. The type of the system of partial differential equations
would be hyperbolic or elliptic based on whether the crossflow Mach number is
supersonic or subsonic.
