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Section 1.2 Some Examples of Parametrizing Curves
Example 1.10 . Parametrizing Curves.
Find the parametric equations corresponding to the following curves.
The line segment connecting the two points
\((1,0)\) and
\((0,1)\text{.}\)
The ellipse
\(\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\text{.}\)
The circle of radius
\(2\) centered at
\((2,0)\text{.}\)
Solution.
Part A. The segment lies on the line \(y = 1 - x\text{,}\) so we may take
\begin{equation}
x = t, \qquad y = 1 - t, \qquad 0 \le t \le 1.\tag{1.4}
\end{equation}
At
\(t = 0\) we are at the point
\((0,1)\text{,}\) and at
\(t = 1\) we arrive at the point
\((1,0)\text{.}\) See
Figure 1.11 and
Figure 1.12 .
Figure 1.11. The line segment from \((0,1)\) to \((1,0)\text{,}\) traced by \(x = t\text{,}\) \(y = 1-t\) as \(t\) increases from \(0\) to \(1\text{.}\)
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Figure 1.12. Part A: the segment \(x = t\text{,}\) \(y = 1-t\text{,}\) \(0 \le t \le 1\text{,}\) which starts at \((0,1)\) when \(t=0\) and ends at \((1,0)\) when \(t=1\text{.}\)
Part B. Guided by the unit circle, we take
\begin{equation}
x = 2\cos(t), \qquad y = 3\sin(t), \qquad 0 \le t \lt 2\pi.\tag{1.5}
\end{equation}
To verify, note that
\begin{equation*}
\frac{(2\cos t)^2}{4} + \frac{(3\sin t)^2}{9}
= \cos^2 t + \sin^2 t = 1.
\end{equation*}
Figure 1.13. The ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1\text{,}\) traced counterclockwise by \(x = 2\cos(t)\text{,}\) \(y = 3\sin(t)\text{,}\) with the points at \(t = 0\text{,}\) \(\pi/2\text{,}\) \(\pi\text{,}\) and \(3\pi/2\) marked.
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Figure 1.14. Part B: the ellipse \(x = 2\cos(t)\text{,}\) \(y = 3\sin(t)\text{,}\) \(0 \le t \lt 2\pi\text{.}\)
Part C. The equation of such a circle in Cartesian coordinates is \((x-2)^2 + y^2 = 4\text{,}\) so we shift the standard parametrization of a circle of radius \(2\) by \(2\) units in the \(x\) -direction:
\begin{equation}
x = 2\cos(t) + 2, \qquad y = 2\sin(t), \qquad 0 \le t \lt 2\pi.\tag{1.6}
\end{equation}
To verify, note that
\begin{equation*}
\bigl((2\cos t + 2) - 2\bigr)^2 + (2\sin t)^2
= 4\cos^2 t + 4\sin^2 t = 4.
\end{equation*}
Figure 1.15. The circle of radius \(2\) centered at \((2,0)\text{,}\) traced by \(x = 2\cos(t) + 2\text{,}\) \(y = 2\sin(t)\text{.}\)
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Figure 1.16. Part C: the circle \(x = 2\cos(t)+2\text{,}\) \(y = 2\sin(t)\text{,}\) \(0 \le t \lt 2\pi\text{,}\) of radius \(2\) centered at \((2,0)\text{.}\) You have attempted
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