MATH 1113 PreCalculus

Day #35, Parametric Equations

Parametric Equations define the location of a point on the plane (x, y) in terms of a parameter, usually thought of as "time," and given the variable, "t."

Example: x = 2*t, y = -3*t.  For any time, t (assume it's in seconds) the location of the point, (x, y) is given by x = 2*t and y = -3*t.  So, at time t = 0, the point is at (2*0, -3*0) = (0, 0).  After 1 second, the point is at (2*1, -3*1) = (2, -3).  You get the idea.  The path traced out by the point as t ranges through some continuous domain of values determines a curve in the plane. In this case it doesn't "curve," it's a straight line segment or straight line through the origin with slope = -3/2, and so it has the standard equation y = -3/2*x.  (It's quite often not possible, or at least not practical, to express the curve explicitly in terms of x and y.)

Many interesting examples of parametric equations involve trig functions, since the periodicity allows the same x or y coordinates to be visited repeatedly.  Take a look at the family of parametric curves in this Geogebra applet.

Applet:

Notice that the family looks like:

x = A*cos(t) + B

y = C*sin(t) + D

If B = D = 0, and A = C, it is "obvious" (think about WHY this really is "obvious") you will get a circle centered at the origin, with radius = A (= C).

Adjusting the values of "B" and "D" translate the function in a very logical way horizontally and vertically -- since both of these are in the "output," there's none of this "subtracting moves it to the right" stuff like we had in the trig translations.  So, you get that:

x = 2*cos(t) -3, y = 2*sin(t) + 5 would be a circle with radius = 2, centered at (-3, 5).

Also, very logically, adjusting the constants "A" and "C" separately will squeeze or stretch the curve in the x or y direction, creating ellipses, when "A" and "C" are not equal.  Play around with the applet until you get a better feeling for this.

A second interesting family can be formed by changing the frequencies of the cosine and sine functions. 

The general family looks like:

x = A*cos(Bt), y = C*sin(Dt)

Most often, we restrict "B" and "D" to non-negative integers, although that's certainly not necessary; it just makes the shapes simpler to understand.

This family is called the family of Lissajous figures, named after Jules Antoine Lissajous.  They are also sometimes called "Bowditch curves," after Nathaniel Bowditch.  They are also sometimes defined with both "x" and "y" equations in terms of sine, and then allowing a phase shift in the x-curve, which is more general than the definition I've given (since cos(t) = sin(t +pi/2), remember?)

Check out this Geogebra applet:

Applet:

Notice, that if "B" = "D", you get a 2*A x 2*C ellipse.

This points out a very important fact about parametric equations: the representation is by no means unique.

In fact, if the domain for t is all real numbers, then it is always the case that the curves traced out by:

x = f(t), y = g(t) and x = f(u(t)), y = g(u(t)) will always be the same, as long as the range of u(t)  is also all real numbers.  However, the motion of the particle will be different.

Example:  Think about x = cos(3t - 2) and y = sin(3t - 2).  This will still trace out a unit circle; it will just trace it out three times as fast as the usual parameterization, and will not start at (1, 0) for t = 0.

To see what's going on, look at the DivX videos on the next page.

Page Last Modified: 2 April, 2007

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