Section 7.1 Parametric Equations
Introduction.
Many interesting curves in the plane cannot be described as the graph of a function \(y = f(x)\text{;}\) the circle \(x^2 + y^2 = 1\) is a very familiar example.
Relatedly, many curves describe the position of a moving object as a function of time; an object moving around the above circle might have coordinates at time \(t\) given by \(x(t)=\cos t, y(t)=\sin t\text{.}\) This is an example of a parametric description of a curve, with the new, “auxilliary” variable \(t\) called a parameter.
Note that this parametric form can convey more information that the equation \(x^2 + y^2 = 1\) for the curve, because of the information about time and place—this is important for example for computing the velocity of a moving object.
Parametric Equations and Their Graphs.
Let us introduce the main new concepts in this chapter:
Definition 7.1.1.
For two functions \(F\) and \(G\) defined on a common interval \(I\text{,}\) the pair of equations
\begin{equation}
x = x(t) = F(t),\, y = y(t) = G(t)\tag{7.1.1}
\end{equation}
are the parametric equations of a curve.
The set of all points \(x(t), y(t)\) for all \(t \in I\) is called the graph of these equations; also known as a parametric curve or plane curve, and typically denoted \(C\text{.}\)
If the domain is a closed interval \(I = [a, b]\) then the curve has initial point \((F(a), G(a))\) and terminal point \((F(b), G(b))\text{;}\) these are are collectively called the endpoints of the curve.
Note however that that the interval could be open or semi-open and so lack one or both endpoints: it can even be infinite, like \((-\infty, \infty)\) or \([0, \infty)\text{.}\)
The parameter \(t\) often has the physical meaning of time, and will informally be referred to as “time” here.
Example 7.1.2. Circles.
For any constant \(R>0\) and any point \((c,d)\) in the plane
\begin{equation*}
x = c + R \cos t, \, y=d + R\sin t, \, 0 \leq t \leq 2\pi
\end{equation*}
describes a circle of radius \(R\text{,}\) center \((c,d)\text{.}\)
Checkpoint 7.1.3. Graphing this circle with Desmos.
Graph the above parametric curve for the case of radius
\(R=1\text{,}\) center
\((2, 3)\text{,}\) using the
Desmos online graphing calculator 2 : input the parametric equations as
(2 + cos(t), 3 + sin(t))
and then edit the limits of the t values (the value \(\pi\) can be entered by typing “pi”).
Practice using the mouse/trackpad/finger to move around the graph and to zoom in and out.
Note that in this case, the initial and terminal points are the same; the right-most extremity, \((c+R,d)\text{:}\)
Definition 7.1.4.
A closed curve is one whose initial and final points are the same.
Example 7.1.5. A spiral.
The curve
\begin{equation*}
x=t \cos t, \, y=t\sin t, \, 0 \leq t < \infty
\end{equation*}
describes one kind of spiral; at time \(t\text{,}\) the point is at distance \(t\) from the origin and as the parameter increases, the position rotates around the origin infinitely often. Its initial point is the origin, but it has no terminal point.
Checkpoint 7.1.6. Graph this spiral with Desmos.
One catch is that
Desmos 3 cannot handle an infinite interval of
\(t\) values, and anyway the whole spiral is infinitely large; thus, experiment with graphing a couple of turns; say
\(0 \le t \le 4\pi\text{.}\)
Example 7.1.7. An exponential spiral.
The curves
\begin{equation*}
x=e^{a t} \cos t, \, y = e^{a t} \sin t, \, -\infty < t < \infty
\end{equation*}
describe another kind spiral; this time the point is at distance \(e^{a t}\) from the origin at time \(t\text{.}\)
This has no initial or terminal point; however it makes sense to say that (for \(a > 0\))
\begin{equation*}
\lim_{t \to -\infty}(x(t), y(t)) = (0, 0)
\end{equation*}
so informally, it starts at the origin.
Checkpoint 7.1.8. Visualizing exponential spirals.
The above exponential spiral grows rather fast so to visualize, it is best to keep the parameter
\(a\) small. Thus, start by look at a case like
\(x=e^{(t/4)} \cos t, \, y = e^{(t/4)} \sin t\text{;}\) Desmos 4 input can be done as (exp(t/4) cos(t), exp(t/4) sin(t))
(You can also experiment with inputing exponents, to get the notation \(e^{t/4}\text{.}\))
Again the infinite interval has to be reduced; start with one turn on either side \(t=0\) with \(-2\pi \le t \le 2\pi\text{.}\) Then larger \(t\) intervals can be visualized with the help of zooming in and out.
As a further experiment with Desmos, include the parameter \(k\) with the form
(exp(k t) cos(t), exp(k t) sin(t)),
and see how Desmos allows setting up sliders for parameters.
Eliminating the Parameter.
Sometimes the parameter can be eliminated, getting back to an equation of the form \(y = F(x)\) or \(x = G(y)\text{,}\) or just a more general equation form \(F(x, y) = 0\) like the equation for a circle. However, we will soon see that this is not always possible, and even when it is, some useful information can be lost.
Example 7.1.9. Circles again.
Consider the parametric equations
\begin{equation*}
x = \cos t, \, y = \sin t, \, 0 \leq t \leq 3\pi/2
\end{equation*}
with initial point \((1, 0)\) and terminal point \((0, -1)\text{.}\) We can use a very familiar trig. identity to get
\begin{equation*}
x^2 + y^2 = 1,
\end{equation*}
which looks like the equation of a circle.
However, three things are lost here:
Information about where the point is at a give time \(t\text{,}\)
the fact that this only covers three-quarters of the circle, due to the limits on the parameter values, and
the “function” form that will allow us to do calculus with curves in the next section, like computing their slopes.
Also, we needed a bit of luck here, with the trig. identity; this strategy often fails, as seen with the example of cycloids below.
Example 7.1.10. Part of a parabola.
For the parametric curve
\begin{equation*}
x = \cos^2(t) + 2, \, y = \cos t, \, 0 \leq t \leq 2\pi
\end{equation*}
we can to some extent do better than above, getting a function describing this curve: substituting the second equation into the first gives
\begin{equation*}
x = y^2 + 2
\end{equation*}
This equation describes a side-ways parabola, but it hides two facts:
The y values are only in the interval \(-1 \le y \le 1\)
The curve both starts and ends at the point \((1, 3)\text{,}\) in between traveling to \((-1, 3)\) and then back-tracking.
Checkpoint 7.1.11. Graph this parametric curve \(x = \cos^2(t) + 2, \, y = \cos t\).
Note that “
\(\cos^2 t\)” can be typed into
Desmos 5 as either “cos^2(t)” or “cos(t)^2”.
Cycloids and Other Parametric Curves.
A very useful example of a parametric curve are the cycloids
\begin{equation}
x = a(t - \sin t),\, y = a(1 - \cos t)\tag{7.1.2}
\end{equation}
because this cannot be written as the graph of a function in any useful way, and yet we can answer all kinds of questions about it, like computing the slope at a point on it, the length along the curve between points on this curve, and related areas under the curve.
The origin of this curve is that it describes the trajectory of a point on a wheel of radius \(a\) as that wheel rolls along, starting on the ground at point \((0, 0)\) at time \(t=0\text{.}\)
Note that no domain for \(t\) is specified above; this curve can be consider as defined for all time. However it is also use to resrict to a single rotation
\begin{equation}
x = a(t - \sin t),\, y = a(1 - \cos t), \; 0 \leq t \leq 2\pi\tag{7.1.3}
\end{equation}
which goes from initial point \((0, 0)\) to terminal point \((2 \pi a, 0)\) with \(y>0\) in between, looking like an “arch”; for times before and after that, the curve “repeats” with copies of that arch shifted left and righ by multiples of \(2 \pi a\text{.}\)
Checkpoint 7.1.12. Graph two arches of a cycloid.
Set the scale as
\(a=1\text{,}\) so the
Desmos 6 input can be (t-sin(t), 1-cos(t)); use interval
\(0 \leq t \leq 4\pi\text{.}\)
Note the special behavior at the points where \(y=0\text{,}\) and zoom in on the point \((2\pi,0)\) given by \(t=2\pi\text{.}\)
If you wish to explore further the capablities of Desmos, use the full form
(a(t-sin(t)), a(1-cos(t))) with parameter \(a\text{,}\)
and see how it allows setting up a slider for it.
Checkpoint 7.1.13. Another Desmos experiment: Prolate and Curtoid Cycloids.
If you instead look at a point on the edge of the overhanging flange of a train wheel, so that the flange has radius \(b > a\text{,}\) one gets a prolate cycloid
\begin{equation}
x = a t - b \sin t,\, y = a - b \cos t\tag{7.1.4}
\end{equation}
If instead \(b < a\text{,}\) this describes the motion of a point on the wheel that is closer to the center, and is called a curtate cycloid.
Look at these with
Desmos 7 , using input (a t - b sin(t), a - b cos(t)) and setup sliders for both parameters.
Note the special behavior at the points where \(t=0, 2\pi\text{,}\) etc.
openstax.org/books/calculus-volume-2/pages/7-1-parametric-equations
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