Parametric Doodling
William T Webber
In this project you are asked the find the parametric equations for some given graphs. In order to ease your task we first give a brief introduction to parametric equations and then show a couple of examples.
Parametric equations of a curve in the plane are a pair of equations. The first equation describes the x-coordinate of a point as a function of some parameter. The second equation describes the y-coordinate of the point as a function of the same parameter. A good example of this comes from trigonometry. The unit circle can be described by the parametric functions
x = cos(t)
y = sin(t).
The parameter t can be thought of in two ways. If we think of t as an angle with vertex at the origin and initial ray on the positive x-axis, then the point (x, y) is the point where the terminal ray intersects the unit circle. As the angle t increases the point (x, y) moves around the circle. The second way to view the parameter t is to view it as time. In this way we see the point (x, y) moving around the plane as time increases. In this example the point moves around the unit circle counterclockwise.
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The example to the right shows x = sin(t), y = sin(2t). If we think of a bug that starts at the origin and flies into the first quadrant, then the x-coordinate starts at 0, increases to 1, comes back to 0, then drops down to –1, and finally returns to zero. At the same time, the y-coordinate goes from 0 to 1 to 0 to –1 to 0 to 1 to 0 to –1 to zero. These graphs of x and y as functions of t are given below. |
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| x as a function of t | y as a function of t |
As you can see, the graph of the x-coordinate is the graph of x = sin(t), and the graph of the y-coordinate is the graph of y = sin(2t).
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Now we will analyze the graph to the right. Again if we start our bug at the origin and let it fly into the first quadrant the x-coordinate goes from 0 to 1 to –1 to 0 three times before the bug flies the complete loop. This would indicate that the function for the x-coordinate might involve sin(3t). Likewise, the y-coordinate goes from 0 to 1 to –1 to 0 twice in the course of flying around the whole path. So we would expect the function for the y-coordinate to involve sin(2t). The graphs of the x and y coordinates are shown below. |
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| x = sin(3t) | y = sin(2t) |
Your job is to find the parametric equations for each of the graphs shown below. Some of the involve combinations of sines and cosines t, others involve polynomials.
For each graph your report should include
This project is to be done in groups of 2 or three people. The group will submit 1 report with all names on it.
Enjoy.
For a printable page containing just the 12 curves click here.
If you can not see the LiveMath below try this link.
