A parameter lets us describe a curve by giving and separately as functions of a third variable, which we can think of as time.
Parameterization
Definition 16.1.1. A parametric equation relates and using an intermediary called a parameter. We accomplish this by writing and as functions of the parameter : and . Think of like time, and of and as the coordinates of a 2D particle on the Cartesian plane. The equations , are called parametric equations. Graphically, we plot all points for which there is a such that ,. If the possible values of are not specified, we assume all real values.
Example 16.1.2.
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Observe: at we get and . Therefore at “time” the particle is at .
Graphing parametric equations
Problem 16.1.3. Let ,.
(a) Compute many values for and plot them on a graph.
(b) Eliminate from the parametric equations to find a new equation in which only and are variables. Show that for every pair that satisfies our new equation, we can find a value of in the parametric equations such that .
(c) Suppose we limit to . How does this affect the graph?
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(a)
(b) From we get , so
Let : then .
Since and , saying is equivalent to saying
which shows that for all such , is defined (or: for all , and are defined?).
Problem 16.1.4. Graph the parametric equations ,.
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The points are . It is an upward-opening () parabola.
Problem 16.1.5. Find the graph of the parametric equations , for .
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Plotting the points for :
Guessing it’s supposed to be a circle.
Problem 16.1.6. Graph ,.
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Finding parametric equations
Problem 16.1.7. (a) Find parametric equations for such that the graph of the parametric equations is the same as the graph of .
(b) Is your answer unique? Are there any other parametric equations which produce the same graph?
(c) Suppose a particle starts at the point and moves at a constant speed clockwise around the origin along . If the particle completes a full revolution in seconds, find the parametric equations for the location of the particle in terms of , where is time.
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(a) Well, famously , so let ,.
(b) I mean, only if you consider things like , different equations, but they are the same as ,.
Genuinely different parameterizations do produce the same graph, though: for example , (clockwise from ), or ,,. So the answer is not unique.
(c) So we have covering one revolution, , so of rotation takes sec. Starting at the top, , and going clockwise, the angle at time is . Therefore we have
Problem 16.1.8. Find parametric equations for such that .
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Rewrite as , i.e. . This looks like .
First try: , giving ,. But that has and the wrong way round: it gives , the up–down (blue) hyperbola below. So instead take :
Check: .
Problem 16.1.9. I’m riding my bike at a constant speed of mph on flat land when I ride over a dot of fresh yellow paint, thereby getting a yellow spot on my front wheel, which is ft in diameter. I continue riding my bike in a straight line at mph. When seconds have passed since running over the dot, how many ft is the yellow spot on my tire from the yellow dot I ran over?
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Units first:
so the bike moves mi/sec ft/sec.
Let be the distance forward the spot has moved in seconds, and how far off the ground it is after sec. The distance is then
The wheel has radius ft. The spot starts at the bottom of the wheel (angle measured from the center), and as the wheel rolls forward it turns clockwise, so after turning through the spot is at angle from the center. A full rotation is radians, the bike covers ft each second, and since the wheel turns through radians in seconds.
So:
So we have
Expanding,
and pairing , the distance is
Exercises
Exercise 16.1.10. Slope of ,?
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.
Exercise 16.1.11. Graph ,.
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Exercise 16.1.12. Graph ,.
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An ellipse centered at : it reaches above and below the center (from to ) and has width .
Exercise 16.1.13. Graph ,.
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Let , which runs over . Then and , so the graph is the arc of a parabola from through to .
Exercise 16.1.14. Find parametric equations for .
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Exercise 16.1.15. A particle starts at and travels at a constant speed in a circle around the point . Find parametric equations to describe the location of the particle at sec when it moves counterclockwise and takes sec for a full revolution.
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(A quarter of the circle takes sec.)
Start with ,. Use , for the radius, then change the centering: ,. Next we need to do a phase shift, and , where and , since but . Finally we increase the period from to by using :
Exercise 16.1.16. Graph , parametrically.
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A circle of radius centered at , passing through .