Find all relevant intersections

Two radii at one angle

A ray turns about the origin, which is called the pole. The angle \(\theta\) gives the direction of the ray, and \(r\) is the distance from the pole along it.

The solid curve is the circle \(r = 2\cos\theta\), and the dashed curve is the circle \(r = 1\) centered at the pole.

Drag the angle slider. The ray meets each curve at an orange dot, and the orange segment from the pole to each dot shows r. The red segment between the dots shows the gap.

At \(\theta = \frac{\pi}{3}\), both curves have \(r = 1\), so the dots meet and the point lies on both curves. The dots meet again at \(\theta = -\frac{\pi}{3}\).

Any two curves

In general, the curves are \(r = f(\theta)\) and \(r = g(\theta)\). Where the gap is 0, \(f(\theta) = g(\theta)\) and the point lies on both curves.

A point has more than one polar name. The point \((r, \theta)\) is also \((-r, \theta + \pi)\), because a negative \(r\) puts the point on the opposite side of the pole.

Turn the ray to \(\frac{7\pi}{12}\). The circle's dot is also on the rose. The rose has a negative \(r\) here, so its dot is on the opposite side of the pole and the gap is not 0.

Press Use the other name. The rose reaches the same point at the angle \(\theta + \pi\), and the gap closes. The equation \(f(\theta) = g(\theta)\) compares the curves at one angle, so it misses this point.

Press Show the pole. Each circle reaches the pole, where \(r = 0\), at its own angle \(\theta_1\) or \(\theta_2\), so no single angle gives \(r = 0\) on both.

Notation

1. Set the two radii equal and solve for \(\theta\). Each solution is a point on both curves.

   \[f(\theta) = g(\theta)\]

   Give the angles in one turn, for example \(0 \le \theta < 2\pi\).

2. Check the other names. The point \((f(\theta), \theta)\) is also \((-f(\theta), \theta + \pi)\), so it lies on the second curve if \(-f(\theta) = g(\theta + \pi)\).

   \[f(\theta) = -g(\theta + \pi)\]

   If the second curve is a circle \(r = k\), the equation reads \(f(\theta) = -k\).

3. Check the pole, where \(r = 0\). Solve \(f(\theta) = 0\) and \(g(\theta) = 0\) separately. If both have a solution, the pole is a common point, which the red ring marks.

   \[f(\theta_1) = 0 \quad\text{and}\quad g(\theta_2) = 0\]

4. Compare your list with a drawing of both curves, which shows any common point that the equations missed.

When it fails

Solving \(1 + \sin\theta = 1 - \sin\theta\) gives \(\sin\theta = 0\), so \(\theta = 0\) or \(\theta = \pi\), and both points have \(r = 1\).

The graph shows a third common point, the pole. The first curve reaches it at \(\theta = \frac{3\pi}{2}\) and the second at \(\theta = \frac{\pi}{2}\), so solving at one angle cannot find it.

Example 1: a limaçon and a circle

Find the points where \(r = 4 + 2\cos\theta\) and \(r = 5\) meet, for \(0 \le \theta < 2\pi\).

1. Set the two radii equal.

   \[4 + 2\cos\theta = 5\]

2. Then \(\cos\theta = \tfrac12\). The orange rings mark the two angles in one turn.

   \[\cos\theta = \tfrac12 \;\Rightarrow\; \theta = \tfrac{\pi}{3},\; \tfrac{5\pi}{3}\]

3. At both angles \(r = 5\), the radius of the circle.

   \[\cy{(5,\ \tfrac{\pi}{3}) \quad (5,\ \tfrac{5\pi}{3})}\]

4. Check the other names. A point with \(r = -5\) is 5 from the pole, so it is also on the circle. The equation below gives \(\cos\theta = -\tfrac92\), which no angle satisfies.

   \[4 + 2\cos\theta = -5\]

5. Check the pole. The circle never has \(r = 0\), so the pole is not on it, and the curves meet at two points.

Example 2: a circle inside a cardioid

Find every point where \(r = 2\cos\theta\) and \(r = 1 + \cos\theta\) meet, for \(0 \le \theta < 2\pi\).

1. Set the two radii equal. Then \(\cos\theta = 1\).

   \[2\cos\theta = 1 + \cos\theta\]

2. In one turn, \(\cos\theta = 1\) only at \(\theta = 0\). There \(r = 2\), so the curves meet at \((2, 0)\). The other names give \(\theta = \pi\) with \(r = -2\), which is \((2, 0)\) again.

   \[\cy{(2,\ 0)}\]

3. A tempting wrong move: Stop here and conclude that the curves meet at one point.

   \[\text{one point: } (2,\ 0)\]

   Both curves pass through the pole, a second common point that the equation did not find.

4. Check the pole. The circle has \(r = 0\) at \(\theta = \frac{\pi}{2}\), and the cardioid has \(r = 0\) at \(\theta = \pi\), so the pole is on both curves.

   \[\begin{aligned} 2\cos\theta = 0 &\Rightarrow \theta = \tfrac{\pi}{2} \\ 1 + \cos\theta = 0 &\Rightarrow \theta = \pi \end{aligned}\]

5. The curves meet at two points: \((2, 0)\) and the pole.

Practice

Find every point where \(r = 2\sin 2\theta\) and \(r = \sqrt{3}\) meet, for \(0 \le \theta < 2\pi\).

The circle crosses each of the four petals of the rose twice, so your list needs as many points as the graph shows.

Questions

Questions 1 and 2 use the curve \(r = 1 - \cos\theta\) and the circle \(r = \cos\theta\).

1. How many points do the two curves have in common?

   • 1
   • 2
   • 3
   • 4
   Answer

   3

2. The curves meet at the pole. Why does solving \(\cos\theta = 1 - \cos\theta\) not give the pole?

   • The circle reaches the pole at \(\theta = \frac{\pi}{2}\), and the other curve reaches it at \(\theta = 0\).
   • The pole has no polar coordinates.
   • The pole is on only one of the curves.
   Answer

   The circle reaches the pole at \(\theta = \frac{\pi}{2}\), and the other curve reaches it at \(\theta = 0\).

3. Find the angles in \(-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\) where \(r = 8\cos\theta\) and \(r = 4\sqrt{3}\) meet.

   • \(\theta = \pm\frac{\pi}{6}\)
   • \(\theta = \pm\frac{\pi}{3}\)
   • \(\theta = \frac{\pi}{6}\) only
   • \(\theta = \pm\frac{\pi}{4}\)
   Answer

   \(\theta = \pm\frac{\pi}{6}\)

Summary

To find where two polar curves meet, you set the radii equal, then check the other names and the pole.

The checks are needed because a point on both curves can have a different angle on each curve.

You find these points before you choose the limits for an area between polar curves.

This is one of four skills in topic 9.9 of Which & Why: AP Calculus.