Parametrizing Planar Curves

with(CalcP): with(plots):     

The parametrization of a curve is a description of a curve in terms of coordinate functions. In Cartesian two-space, the goal is to find functions x(t) and y(t) so that the object is located at (x(t), y(t)) for each value of the parameter t.

If a curve is the graph of a function f, then a parametrization is [t, f(t)]. For example part of the line y=3x + 2 can be parametrized by (t,3t + 2) for -4< t < 4 To see the parametrization, use Maple command
ParamPlot([t, 3*t + 2], t= -4..4);

The "buttons" above the graph work like buttons on a Walkman or CD player. Click on the question mark box on the far right to show the labels of each button. We have provided further explanation on how the buttons work. The same curve has have many parametrizations. For example, our line t -> (t, 3t + 4) could also be parametrized by t -> (2t, 6t + 4) for -2 < t < 2 . No matter how we parametrize this line, the functions x(t) and y(t) satisfy the equation
y(t) = 3 x(t) + 4
for every value of t.

Question #1 (Review! : Do not hand in this question.)

1. Parametrize the line segment connecting the points (2,1) to (4,9). Over what interval does your parameter vary?
2. Parametrize the line segment so that your parameter varies from 0 to 1.
3. Parametrize the line segment so that as your parameter increases from 0 to 1, the segment travels from right to left.
4. Parametrize the curve so that it is traced with speed 3.