Parametic Equations

Parametic Equations

by: Al Byrnes

Parameterizing a curve, or defining a curve in terms of one variable, can be extremely helpful in many situations in math. Let's take a closer look at the following parameterization:

In Graphing Calculator 4.0, we can see how this parameterization will look for values of t along the domain 0 ≤ t ≤ 2π. See below:

This curve can be conceptualized as all of order pairs, whose x and y coordinate values are in thr form (cos(t), sin(t)) for values of t defined by the domain. The unit circle is displayed again below, but the two curves x = cos(t) and y = sin(t), displayed in red and blue, respectively, highlight the fact that the circle is the trace of the point at which the two functions are coincident as the value of t varies:

Let's investigate this parametric equation further. Firstly, what will happen if we multiply both functions x = cos(t) and y = sin(t) by the same constant? Let n be a real number, 1 < n ≤ 5. See the animation below for an illustration of how effects the graph of the following parametric equation:

It appears that as n varies, so does the size of the radius of the unit circle. As a check, we have graphed the Cartesian standard form of a circle centered at the origin with a radius of 5 next to the parametric equation for the circle centered at the origin and we see that these two circles are similar:

Through the comprison made above, we can make a speculation about how the Cartesian standard form for a circle and the parameterization of the curve using the sin(t) and cos(t) funtions are related. Specifically, the coefficients of the trigonometric function corresponds to the radius of the circle, which corresponds to the squared constant term on the right hand side of the equation in its Cartesian standard form.

In the previous example, we analyzed an example in which the coefficients a and b in the parametic equation shown below were equal:

What if the coefficents a and b in the parametric equation shown above are not equal? Above, we made a connection between the values of a and b with regards to the radius of the circle corresponding to the parametric function (x, y) = (5cos(t), 5sin(t)). How could a circle have two different "radii"? Is there a shape that comes to mind, whose parametric is of the following form:

Examine the following movie to get a sense for how the the graph of our parametric equation changes when the value of coefficient b is held at constant value of 5 and the value for a is defined on the domain 0 < a ≤ 9

Now we see what has happened! When the value of the coefficient a = 5 = b we have a circle. When the coefficient a < 5 (eg: a < b), we have an ellipse with a minor axis of length 2(a) and major axis of length 2(b) = 10. Likewise when the value of coefficient a > 5 (eg: a > b), then we have an ellipse with major axis of length 2a and minor axis of length 2(b).

How is the parametic equation for an ellipse centered at the origin related to the equation for the same ellipse expressed in Cartesian standard form? The Cartesian standard form and the parametric equations for the ellipse with major axis 2a = 18 and major axis length 2b = 10 are shown below along with their corresponding graphs:

For circles and ellipses centered at the origin, we have a better understanding of the parametric and Cartesian standard equation forms. As an extension, use what we have learned about the parametric equations for circles and ellipses to devise parametric equations for the circle with radius 5 centered at (1, 1) in the plane. How about the parametric equation for an ellipse with a major axis of length 18 (along the x-axis) and a minor axis of length 10 (along the y-axis) centered at (-1, 3)? Can you devise a general form for the parametric equations of a circle and an ellipse centered at any point (c, d); c, d are elements of R²?

Parameterization has its advantages. As we saw in our investigation above, it gives us another tool with which we can graph function, whose Cartesian standard forms might be slightly more difficult to interpret. For instance graphing the parametric equation (x, y) = (cos(t), sin(t)) for 0 < t ≤ 2π seems less threatening than finding solutions to y = √(1 - x²) and y = -√(1 - x²).

There is another type of curve, which can be graphed quite easily using parameterized functions. If we continue to play around with the initial parametric equation, let's change the function defining the y-coordinate of the parametric equation to match the function defining the x- coordinate:

Original parametic equation:

New parametic equation:

See the the animation below to see the trace of the parametric function in the plane for the values of t: 0 < t ≤ π:

What have we done here? We have the graph for the line segment y = x, -1 ≤ x ≤ 1! So we can use parameterization to graph line and curve segments. Admittedly, there is a more straight forward way to parameterize the line segment y = x, -1 ≤ x ≤ 1 (eg: (x, y) = (t, t), -1 ≤ t ≤ 1), however, I wanted to show how I was able to make this connection between parameterizing circles and ellipses and parameterizing curve segments by manipulating our original parametric equation, which was composed of trigonometric functions. Can you use parameterization to graph some other line and curve segements? Consider the following parametric equations and the curve segments that they produce. For both examples 0 < t < π:

A segment of the sine curve:

In conclusion, in this module we explored using parameterized function to gain insight into graphing circles, ellipses, and line and curve segments in the plane.