Parametric curves in the plane

Purpose

The purpose of this lab is to give you practice with parametrizing curves in the plane and in visualizing parametric curves as representing motion of particles.

Background

A parametric curve in the plane can be defined as an ordered pair, , of functions, with representing the x coordinate and the y coordinate. Parametric curves arise naturally as the solutions of differential equations and often represent the motion of a particle or other mechanical systems. They also often arise in studying oscillations in electrical circuits. For this lab, however, we want to think of parametric curves as representing motion of a particle in space.

For example, neglecting air resistance, the position of a projectile fired from the origin at an initial speed of and angle of inclination is given by the parametric equations

where t is time and g is the acceleration due to gravity.

Graphically, a parametric curve can be represented several ways. One simple way is to plot the component functions, and , individually versus the independent variable t. Another way is to plot the set of points . This gives you the curve along which the particle moves, but information on how it moves has been lost. On the other hand, plotting the component functions individually makes it hard to see how the particle is actually moving.

To help you to visualize parametric curves as representing motion, a Maple routine called ParamPlot has been written. It uses the Maple animate command to actually show the particle moving along its trajectory. To use the ParamPlot command, you will first have to load the CalcP package with the command.

  > with(CalcP):

The syntax for ParamPlot is very simple, as shown by the following examples. Try them! They use the Maple animate command, which has controls similar to those on a VCR, allowing you to slow down or speed up the playback, or even to put it into a continuous loop mode. The last example shows how to animate more than one curve at a time.

  > ParamPlot([t,t^2],t=-2..2);

  > ParamPlot([-t, t^2],t=0..2*Pi);

  > ParamPlot([t,t^2],t=-1..2);

  > ParamPlot([cos(t), sin(t)],t=0..2*Pi);

  > ParamPlot([cos(t), sin(2*t)],t=0..2*Pi);

  > ParamPlot([t^2, t^3-t], t=-2..2);

  > ParamPlot([t,t^2],[-t,2*t],t=-2..2);

Different parametric descriptions of a curve in the plane

Given a curve defined by the graph of a function , there are an infinite number of ways of representing this curve parametrically, corresponding to different motions on the curve. About the simplest way of parametrizing is with the pair , which traverses the curve from left to right as t increases. One can reverse the direction of motion by changing to the parametrization .

By restricting t to an interval , you can get a parametric description of a portion of the curve. For example, the right half of the parabola would result from for .

More complicated parametrizations of can be obtained with parametric curves of the form . By choosing appropriately, for example, you can make the particle stop and turn back on the curve. For example, suppose that the curve to be parametrized is the graph of the function y=2x. The following examples give three different parametrizations of this curve.

  > ParamPlot([sin(t),2*sin(t)], t=0..4*Pi);

  > ParamPlot([t-1,2*(t-1)],t=0..2);

  > ParamPlot([(1-t)^2,2*(1-t)^2],t=0..3);

Exercises

1. Suppose that a cannon can fire a projectile at an initial velocity of 400 feet per second. Neglecting air resistance, what angle of elevation should be used to hit a target at the same elevation 500 yards away? Explain how your result was obtained.

2. Lissajous figures. Investigate the family of parametric curves

   

   where n and m are positive integers. Explain what you see. ( Due to limits on ParamPlot, you will probably have to keep n and m smaller than about 5.)

3. The last Maple example, concerning the curve y=2x, provided three different parametrizations. Match the parametrizations with the descriptions below. Justify your choices.
   1. Starts at the point . Moves to the left initially, but reverses direction once
   2. Starts at the origin, then oscillates between the points and .
   3. Starts at the point and then moves to the right along the line.

4. Given a function , find conditions on that are necessary to make the parametric description do the following. Give conditions on and its derivatives, as well as examples, and make sure that you explain your reasoning.
   1. Traverse the curve from left to right as t increases.
   2. Reverse direction once.

5. Suppose that at time zero, flight 12 is at the point at an altitude of 30,000 feet and traveling Northeast at 420 mph and that flight 33 is at the same altitude, but is traveling due west at a speed of 388 mph. At time zero, flight 33 is at the point .
   1. What is the minimum distance between flights 12 and 33, and at what time does it occur?
   2. Suppose that FAA regulations require that the closest approach of any two aircraft must be greater than 2000 feet. Can you change the speed of flight 33 so that this regulation is satisfied? What is your new speed, and what is the closest approach distance?
   (Hint - set up parametric descriptions of the positions of the two flights and use the distance formula. Make sure that you explain each step of your procedure.)