A 2D parametric curve is represented by a function p(t) = [f(t), g(t)]. In this context we call the variable t the parameter where it belongs to some domain D. The parametric curve is also expressed by a set of equations, so-called parametric equations:
u = f(t) v = g(t)A set of parametric equations can be graphed in either xy-Cartesian coordinate system or rθ-polar coordinate system. They can accordingly be expressed as follows:
x = f(t) y = g(t) or r = f(t) θ = g(t)Alternatively, they can be represented by:
p(t) = [x(t), y(t)] or p(t) = [r(t), θ(t)]The graph of the parametric equations is simply the set of points (ordered pairs) as defined above for t in D.
Note that the graph of parametric equations is NOT the graph of the function p(t), which is a three-dimensional curve, but the graph of the range of p(t).The following figures, created by our parametric curve graphing calculator, show the graphs of some parametric curves drawn in the Cartesian and polar coordinate systems.
The graphs of the same parametric equations shown above are also drawn with axes rotated in each coordinate system.
