Hyperbola
A hyperbola is a set of points with the property that, given two foci, the difference in distance from the foci is a constant.
The gears in this diagram are shown in light grey. Grab onto the blue "Drag Me" point and crank it around the circle. Point P will trace the hyperbola. The dashed line becomes an asymptote as Point P whips off to infinity, and circles back to re-enter at the other side of the diagram.