Objectives
- To formulate the parametric equation of an ellipse
- To understand how transformations to a parametric equation alters the shape of the ellipse including stretching and translation
Section 1.3 Parametric Equations of Ellipses
Subsection 1.3.1 Ellipse Parametric Equation
In Subsection 1.1.2 we learned that to parametrize the implicit equation of a circle with center \((h,k)\) and radius \(r > 0\) given by
\begin{equation*}
(x-h)^2 + (y-k)^2 = r^2
\end{equation*}
we can use the relationship between \(\sin\) and \(\cos\text{.}\) We found a parametric equation for the circle can be expressed by
\begin{equation*}
x(t) =r \cos(\theta) + h \qquad y(t) = r \sin(\theta) + k.
\end{equation*}
The conic section most closely related to the circle is the ellipse. We have been reminded in class that the general equation of an ellipse is given by
\begin{equation*}
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.
\end{equation*}
Below is an ellipse that you can "play" around with:
We can continue to make use of the relationship between \(\sin\) and \(\cos\) to discover parametric equations for an ellipse. In fact, without the \(a\) and \(b\) in the equation things would work perfectly. To remedy this (that is, get rid of \(a,b\)) we simply multiply by \(\sin\) and \(\cos\) by these values. Consider the equations
\begin{equation*}
x(t) = a \cos(t) \qquad y(t) = b \sin(t).
\end{equation*}
Activity 1.3.1.
(a)
Substitute the equations for \(x(t), y(t)\) above into
\begin{equation*}
\frac{x^2}{a^2} + \frac{y^2}{b^2}
\end{equation*}
and simplify.
(b)
Consider the ellipse given by \(\frac{x^2}{9} + \frac{y^2}{4} = 1.\) What are the parametric equations for this ellipse? Graph them below to ensure you obtain the exact same graph.
(c)
In Figure 1.3.1, update the equation in \(x\) and \(y\) as well as the parametric equations for the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1.\)
(d)
Write a few sentences about the orientation of the parametric equations. In particular, you should talk about the following: at what point do you "start", i.e. what point corresponds to \(t=0\text{?}\) How do you move around the ellipse (clockwise or counterclockwise)?
(e)
Update the equation in \(x\) and \(y\) as well as the parametric equations so that the ellipse has a center of \((-2,1)\text{.}\)
(f)
Since we know that
\begin{equation*}
(\sin(t))^2 +(\cos(t))^2=1 \quad \text{for every }t\in\mathbb{R}
\end{equation*}
and the form of a translated ellipse is given by
\begin{equation*}
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,
\end{equation*}
set
\begin{equation*}
\frac{(x-h)^2}{a^2}=(\sin(t))^2
\end{equation*}
and solve for x.
(g)
Set up a similar equation involving \(y\) and \(\cos(t)\) then solve for \(y\) to get a general set of parametric equations for the translated ellipse.
Activity 1.3.2. Parameterizing a Translated Hyperbola.
Another conic section that will be useful to quickly parameterize is the hyperbola. The translated hyperbola is given by
\begin{equation}
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\tag{1.3.1}
\end{equation}
for horizontal orientation and
\begin{equation}
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\tag{1.3.2}
\end{equation}
for vertical orientation.
(a)
Find a trig identity of the form
\begin{equation*}
(\underline{\hspace{0.5in}})^2-(\underline{\hspace{0.5in}})^2=1
\end{equation*}
Hint.
Look for one involving secant and tangent.
(b)
Set the first blank element of your trig identity from the previous part equal to the following expression, and then solve for \(x\text{.}\)
\begin{equation*}
\frac{(x-h)^2}{a^2}
\end{equation*}
(c)
Set up a similar equation involving \(y\) and the trig function from the second blank of Task 1.3.2.a then solve for \(y\) to get a general set of parametric equations for the translated hyperbola.
(d)
Subsitute in your parametric equations for the translated hyperbola into the Desmos Interactive below to check that your equations trace the same graph as the translated hyperbola. You should move the sliders for \(h\text{,}\) \(k\text{,}\) \(a\text{,}\) and \(b\) to convince yourself that this set of parametric equations works for all cases (of the horizontal orientation.)
