I love reading stuff like this. While most people (I'm guessing) are fascinated by the Higgs, Cosmology, Dark Energy, etc., I'm more of a sucker for "simple stuff" that really isn't that simple nor trivial. This is one such example.
The authors of this paper is investigating what they call the "comfortable roller coaster". It is a roller coaster that has a constant magnitude of the normal force acting on the rider. In other words, you don't get multiple g's pressing on you and don't get lifted off your seat.
Abstract: A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is, what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.
You can follow the derivation at your leisure. The resulting trajectories are shown in several figures, such as Fig. 1 and 2. Each of the trajectory corresponds to a particular ratio of N/mg. If you look at the trajectory of the loop-the-loop, you'll see that it is more of teardrop shape, rather than a circle or an oval. This shape is what you see at amusement parks and an important design to make sure that the riders are not subjected to unusually high g's during the ride.
A fun paper!
 A.B. Nordmark and H. Essén, Eur. J. Phys. v.31, p.1307 (2010). You may also obtain the paper for free during the first 30 days of online publication at this link.