As was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising from the force laws [formula omitted] and [formula omitted], can be mapped onto each other by changing the location of the center of force. Less well known is that this mapping can also be achieved by the complex transformation, [formula omitted]. We derive this result and its generalization by writing the Gaussian curvature in its covariant form, and then changing the metric by a conformal transformation which mimics this mapping of the curves. We indicate how the conserved Laplace–Runge–Lenz vector for the [formula omitted] 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 by introducing concepts from differential geometry. © 2011, American Association of Physics Teachers. All rights reserved.