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Parametric Equations

Examples of surfaces generated parametrically

In physics, as well as many other branches of science, graphs of functions are simply inadequate to the task of modelling all of the phenomena which arise. In particular, to describe the motion of a particle in the xy-plane, it is necessary to use expressions that give the position (x,y) of the particle at any time t.

As a simple example, consider the trajectory of a ball that is thrown horizontally at a speed of 35 feet per second from a height of 5 feet above the ground by a person standing at the origin of coordinates. In this coordinate system, y points in the vertical direction and x in the horizontal direction. If there is no air resistance, then there is nothing to impede or increase the horizontal velocity of the ball. Therefore, t seconds after the ball is thrown, we expect the x-coordinate of the ball to be 35t feet. The y coordinate will be governed by the constant acceleration model discussed in an earlier section, and t seconds after the ball is thrown we expect its height above the floor to be 5 - 16t² feet. Thus, the path of the ball is well-expressed by the pair of equations:

or by the equation (x(t), y(t)) = (35t, 5 - 16t² ), which displays the same information in the form of ordered pairs.

Do exercise 1-32 to learn more about parametric curves using ordered pairs.

An especially important parametrically defined curve is the unit circle (cos(t), sin(t)). To change the radius of the circle, we need only multiply the x and y coordinate functions by a constant: (a cos(t), a sin(t)). For various values of a, we get circles of different radii all centered at the origin. To move the center to the point (c₁,c₂), of course, we just add c₁ to the x-coordinate and c₂ to the y-coordinate. Contrast this method of drawing circles with that done earlier in the week using functions.

Do exercise 1-33. to learn more about parametric curves using ordered pairs.

We would be remiss not to mention the close relative of the circle, the ellipse, whose parametric equations are (a cos(t), b sin(t)) for . You might do some experiments with Maple to see how interesting ellipses can get. Ellipses found use in the mid 1500's when Johannes Kepler observed that the orbits of planets around the Sun were ellipses and not perfect circles as the Greeks had imagined.

Parametrically defined curves can be pretty exotic. Nature clearly uses the idea of parameters which are repeated in fixed relation to one another in many of its designs. Hence, its possible to mimic a seashell with a simple parametric description:

tubeplot([sin(t),cos(t),t],t=-3*Pi..4*Pi,radius=1.+t/10.,numpoints=80);

Some other interesting and useful parametric curves are provided by the Lissajous figures. Their parametric equations look like (sin(at), sin(bt)) for various values of a and b. An example is shown in the applet below.

For a much better display of Lissajous figures (and as an inspiration to the formation of your own figures using Maple) see the excellent Lissajous display at Edward R. Hobbs's site.