The cartesian equation of a parabola with directrix vertical is

The parametric equation of a parabola with directrix vertical is

The third parameter t is usually but not always considered a time element. For instance a moving object has position x and y at time t. However parametric equations can also draw a curve that has multiple values of y that do not pass the vertical line test, ie all the conic sections. It is easy to prove that the parametric equations above simplify to the cartesian equation for a parabola. Solve both of them for t, then set them equal to each other and simplify. Doing the reverse – given a cartestian equation find the parametric equations – is more difficult. Each cartesian equation (I read) has an infinite number of parametric equations. I will rely on published equations. One of my roadmap texts is “A Catalog of Special Plane Curves” J. Dennis Lawrence, 1972. “Parametric methods are most important for our purposes, and will be heavily relied on.” He also remarks that it is expedient to locate the coordinate system where it is convenient to simplify the algebra.

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