This paper explores orbits in extended mass distributions and develops an analytic approximation scheme based on epicycloids (spirograph patterns). We focus on the Hernquist potential ¼ 1/(1 þ ), which provides a good model for many astrophysical systems, including elliptical galaxies (with an R 1/4 law), dark matter halos (where N-body simulations indicate a nearly universal density profile), and young embedded star clusters (with gas density $ À1 ). For a given potential, one can readily calculate orbital solutions as a function of energy and angular momentum using numerical methods. In contrast, this paper presents a number of analytic results for the Hernquist potential and proves a series of general constraints showing that orbits have similar properties for any extended mass distribution (including, e.g., the NFW profile). We discuss circular orbits, radial orbits, zero-energy orbits, different definitions of eccentricity, analogs of Kepler's law, the definition of orbital elements, and the relation of these orbits to spirograph patterns (epicycloids). Over a large portion of parameter space, the orbits can be adequately described (with accuracy better than 10%) using the parametric equations of epicycloids, thereby providing an analytic description of the orbits. As an application of this formal development, we find a solution for the orbit of the Large Magellanic Cloud in the potential of our Galaxy.