Simulators use an interactive, physical device to simulate reality. For example, space shuttle simulators were used to help astronauts understand the intricacies and various scenarios associated with flying the shuttle; people can even improve their driving skills using driving simulators. But how do simulators work in terms of mathematics?

One way of seeing how this works is with an airplane flight simulator. Although there are different types—ranging from full cockpit to full motion simulators—they have the same purpose: to make the person “flying” the simulator feel as if he or she is really controlling an airplane. When a person sits in the cockpit, he or she sees a virtual world projected onto a screen—what it would look like outside a real plane’s cockpit. As the pilot “flies” by moving the instruments, the actions get translated to a computer hooked to the simulator; from there, each component of the process is translated using physics-based equations that describe the relationship between the input and output of the actions. Not only do the instruments respond in kind, but often, if the simulator is hooked to hydraulic mounts, movements will also be fed by computer into the simulator. Thus, the pilot will truly feel as if he or she is flying—all thanks to mathematical equations translated by the computer and into the simulator.