  As was first noted by Isaac Newton, the two most famous ellipses of classical
mechanics, arising out of the force laws F~r and F~1/r^2, can be mapped onto
each other by changing the location of center-of-force. What is perhaps less
well known is that this mapping can also be achieved by the complex
transformation, z -> z^2. We give a simple derivation of this result (and its
generalization) by writing the Gaussian curvature in its "covariant" form, and
then changing the \emph{metric} by a conformal transformation which "mimics"
this mapping of the curves. The final result also yields a relationship between
Newton's constant G, mass M of the central attracting body in Newton's law, the
energy E of the Hooke's law orbit, and the angular momenta of the two orbits.
We also indicate how the conserved Laplace-Runge-Lenz vector for the 1/r^2
force law transforms under this transformation, and compare it with the
corresponding quantities for the linear force law. Our main aim is to present
this duality in a geometric fashion, by introducing elementary notions from
differential geometry.
