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Section 14.1 Graphs in polar coordinates

Functions expressed in polar coordinates can be graphed. For the function \(r(\phi)\text{,}\) for each value of \(\phi\text{,}\) an angle measured counterclockwise from the \(x\)-axis, plot the distance \(r(\phi)\text{,}\) measured outward from the origin.

For example, Keplerian orbits can be described in the form

\begin{equation*} r(\phi) = \frac{\alpha}{1+\varepsilon\cos(\phi-\delta)} \end{equation*}

as shown in Figure 14.1.1.

Figure 14.1.1. The polar plot of conic sections.

Activity 14.1.1.

Use the applet above to explore how the shape of a Keplerian orbit is influenced by the parameters \(\alpha\text{,}\) \(\epsilon\text{,}\) and \(\delta\text{.}\)