For this project, I used parametric equations to model real-life roller coasters. I modeled Kingda Ka from Six Flags, NJ, Blue Streak from Cedar Point, OH, as well as a generic 100ft high ferris wheel. I simulated the surfaces as well as the path of the coasters in three dimensions (x,y,z). The equations were formed through the actual dimensions of the roller coasters and through estimation.
Parametric equations relate the path of the coaster to time. Instead of relating time to the amount of seconds the coaster lasted, I related it to a circle (0<t<2π). This allowed me to plot the track (x[t],y[t]) as an ellipse. The height/drops of the coaster (z[t]) was then mapped using piecewise functions. When inputted into Mathematica, all three equations model out a coaster.