Math Insight

An introduction to parametrized curves

 

A simple way to visualize a scalar-valued function of one or two variables is through their graphs. In a graph, you plot the domain and range of the function on the same set of axes, so the value of the function for a value of its input can be immediately read off the graph. Whether you plot the function $f(x)$ by the curve of points $(x,f(x))$ or the function $f(x,y)$ by the surface of points $(x,y,f(x,y))$, these graphs give a fairly comprehensive view of the behavior of the function.

If we have a two-dimensional vector-valued function of a single variable, $\vc{f}: \R \to \R^2$ (confused?), it is still possible to display its graph. If the function is $\vc{f}(t)=(f_1(t),f_2(t))$, we could plot the set of points $(t,f_1(t),f_2(t))$ and obtain a curve in three-dimensional space.

For example, the graph of the function $\dllp(t)=(3\cos t)\vc{i} + (2 \sin t) \vc{j}$ is a spiral, or helix, as shown here.

Applet: Graph of a function that parametrizes an ellipse

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Graph of a function that parametrizes an ellipse

Graph of a function that parametrizes an ellipse. The green curve is the graph of the vector-valued function $\dllp(t) = (3\cos t, 2\sin t)$. This function parametrizes an ellipse. Its graph, however, is the set of points $(t,3\cos t, 2\sin t)$, which forms a spiral.

More information about applet.

Graphing a vector valued function is possible for functions $\vc{f}: \R \to \R^2$ because the graph required only three dimensions. However, we are unable to plot the graph with an higher-dimensional input or output space, as these graphs would require more than three dimensions. We need a new method to visualize these functions.

One way to visualize vector-valued functions is by choosing a set in their domain and viewing how the function maps this set into its range. This procedure is particularly effective for vector-valued functions of a single variable. We pick an interval in their domain, and these functions will map that interval into a curve. If the function is two or three-dimensional, we can easily plot these curves to visualize the behavior of the function.

Returning to the example function $\dllp(t)=(3\cos t)\vc{i} + (2 \sin t) \vc{j}$, you can see in the below applet how it maps the interval $[0,2\pi]$ into an ellipse. As $t$ sweeps through the interval $[0,2\pi]$, the vector $\dllp(t)$ sweeps out the ellipse. We refer to $t$ as a free parameter, and say that $\dllp$ parametrizes the ellipse. (You can see the applet's page for details on how to show that $\dllp$ indeed sweeps out a ellipse.)

Applet: Parametrized ellipse

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Parametrized ellipse

Parametrized ellipse. The vector-valued function $\dllp(t)=(3\cos t)\vc{i} + (2 \sin t) \vc{j}$ parametrizes an ellipse, shown in green. This ellipse is the image of the interval $[0,2\pi]$ (shown in red) under the mapping of $\dllp$. For each value of $t$, the blue vector is $\dllp(t)$. As you change $t$ by moving the blue point along the interval $[0,2\pi]$, the head of the arrow traces out the ellipse.

More information about applet.

The ellipse is the image of the $[0,2\pi]$ under the mapping $\dllp$. If we think of $[0,2\pi]$ being being made of rubber, the function $\dllp$ stretches the rubber and curves it into the ellipse. Viewing $\dllp(t)$ as parametrizing ellipse hides some of the information, as the dependence on the parameter is less obvious. But, we still retain knowledge of which values of the parameter $t$ correspond to which points of the ellipse. To emphasize that we have more than just a curve of points, we refer to such curves as parametrized curves.

Often we think of the variable $t$ as representing time and $\dllp(t)$ as representing position as a function of time. For example, in the introduction to the chain rule, we used a parametrized curve to represent position of a hiker climbing a mountain. We retained the representation of time $t$ so that we could calculate how fast the climber ascended.

A single image curve, such as the ellipse, could have many parametrizations. For example, we could parametrize the ellipse by the function $\adllp(t)=(3\cos\frac{t^2}{2\pi})\vc{i} + (2 \sin \frac{t^2}{2\pi}) \vc{j}$, as shown in the below applet. This function also maps the interval $[0, 2\pi]$ onto the ellipse. In this case, the function stretches the interval unevenly as it curves $[0, 2\pi]$ into the ellipse; the right end of the interval is greatly stretched while the left end of the interval is compressed. If you change $t$ at a constant speed from 0 to $2\pi$, the vector $\adllp(t)$ moves slowly at first and picks up speed as $t$ increases.

Applet: Parametrized ellipse

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Parametrized ellipse

Parametrized ellipse. The vector-valued function $\adllp(t)=(3\cos \frac{t^2}{2\pi})\vc{i} + (2 \sin \frac{t^2}{2\pi}) \vc{j}$ parametrizes an ellipse, shown in green. This ellipse is the image of the interval $[0,2\pi]$ (shown in red) under the mapping of $\adllp$. For each value of $t$, the blue vector is $\adllp(t)$. As you change $t$ by moving the blue point along the interval $[0,2\pi]$, the head of the arrow traces out the ellipse.

More information about applet.

The differences between $\dllp(t)$ and $\adllp(t)$ are not obvious when looking at the curves they parametrize. We could make the difference more obvious by labeling points along the curve by their corresponding values of $t$. The difference is much more evident from their graphs; compare the following graph $\adllp$ with the graph of $\dllp$ at the beginning of this page.

Applet: Graph of a function that parametrizes an ellipse

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Graph of a function that parametrizes an ellipse

Graph of a function that parametrizes an ellipse. Graph of the vector-valued function $\adllp(t)=(3\cos \frac{t^2}{2\pi})\vc{i} + (2 \sin \frac{t^2}{2\pi}) \vc{j}$. This function parametrizes an ellipse. Its graph, however, is the set of points $(t,3\cos \frac{t^2}{2\pi},2 \sin \frac{t^2}{2\pi})$, which forms a curve that spirals more quickly with increasing $t$.

More information about applet.

Three-dimensional vector-valued functions can parametrize curves embedded in three-dimensions. The function $\sadllp(t) = (3\cos t)\vc{i} + (2 \sin t) \vc{j} + t\vc{k}$ parametrizes a helix, as shown below. This helix looks like the graph of $\dllp$, above. But, in this case, we have a mapping from the interval $[0,2\pi]$ onto the helix, so this is a parametrized helix. If we wanted to graph $\sadllp$, we would need four dimensions.

Applet: Parametrized elliptical helix

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Parametrized elliptical helix

Parametrized elliptical helix. The vector-valued function $\sadllp(t)=(3\cos \frac{t^2}{2\pi})\vc{i} + (2 \sin \frac{t^2}{2\pi}) \vc{j}+ t \vc{k}$ parametrizes an elliptical helix, shown in red. This helix is the image of the interval $[0,2\pi]$ (shown in magenta) under the mapping of $\sadllp$. For each value of $t$, the cyan point represents the vector $\sadllp(t)$. As you change $t$ by moving the blue point along the interval $[0,2\pi]$, the cyan point traces out the helix.

More information about applet.

A special case of a parametrized curve is a parametrized line. In this page, we introduce a simple parametrization of a line. But after seeing how we could change the vector-valued function and still parametrize the same curve, can you think of other types of vector-valued functions that would parametrize a line?

The derivative of a parametrized curve also has special meaning.