Why Pole Crossings Matter
A polar equation \(r=f(\theta)\) describes points by pairing a directed radius with an angle. When \(r=0\), the curve is at the pole, regardless of the value of \(\theta\). These pole crossings often mark the endpoints of a loop, so finding them is a useful first step when sketching a polar curve.
The sign of \(r\) matters just as much as whether it is zero. A negative radius does not mean the curve disappears or that the point is invalid. It means the point is plotted in the direction opposite the angle. That change in direction can create an inner loop, but it can also make an interval retrace points already traced elsewhere.
Negative Radius and Equivalent Points
The Cartesian coordinates of a point with polar coordinates \((r,\theta)\) are \((r\cos\theta,r\sin\theta)\). If \(a>0\), then
Therefore, the polar representations \((-a,\theta)\) and \((a,\theta+\pi)\) give the same point. A negative radius plots the point a distance \(|r|\) from the pole, along the ray opposite the one at angle \(\theta\).
This equivalence explains why a sign change can alter the curve’s tracing in a substantial way. As \(\theta\) varies through an interval where \(r<0\), the plotted point lies at angle \(\theta+\pi\) with positive distance \(-r\). To sketch the curve, track both the changing distance and this reversed direction.
A Reliable Tracing Process
For a continuous polar function on the interval being examined, its sign cannot change without passing through zero. Thus, solving \(r=0\) divides the angle interval into pieces on which the radius keeps one sign. A sign test at one angle in each piece determines that sign throughout the piece.
Use a convenient interval that covers the curve of interest, such as an interval of length \(2\pi\), but remain alert to repeated tracing.
Solve \(r=0\) on the chosen interval and list the resulting angles in order.
Choose a test angle in each interval. A positive radius follows angle \(\theta\); a negative radius follows the opposite direction \(\theta+\pi\).
Intervals with pole endpoints may trace loops, but check the geometry and look for repeated points before claiming that each piece is a distinct loop.
Worked Example: Finding an Inner Loop of a Limaçon
Worked Example: Finding an Inner Loop of a Limaçon
Analyze the pole crossings and the negative-radius interval for \(r=1+2\cos\theta\) on \(0\leq\theta\leq2\pi\).
Step 1: Solve for the pole crossings. Set \(r=0\):
In the stated interval, the solutions are \(\theta=\frac{2\pi}{3}\) and \(\theta=\frac{4\pi}{3}\). At each one, \(r=0\), so the curve reaches the pole.
Step 2: Test the signs. Before the first crossing, use \(\theta=0\):
Between the crossings, use \(\theta=\pi\):
After the second crossing, use \(\theta=2\pi\):
Thus, the radius is negative on \(\frac{2\pi}{3}<\theta<\frac{4\pi}{3}\), and positive on the two remaining pieces within the chosen interval.
Step 3: Interpret the negative-radius piece. On the negative interval, the plotted direction is \(\theta+\pi\), and the positive distance is \(-r=-1-2\cos\theta\). As \(\theta\) runs from \(\frac{2\pi}{3}\) to \(\frac{4\pi}{3}\), this plotted direction runs from \(\frac{5\pi}{3}\) to \(\frac{7\pi}{3}\), a total angle change of \(\frac{2\pi}{3}\). The distance is zero at both ends and reaches \(1\) at \(\theta=\pi\). In particular, at the midpoint the plotted point is
The interval therefore leaves the pole, reaches a point one unit to the right, and returns to the pole. It traces the limaçon’s inner loop. The plotted direction changes by less than \(2\pi\), so the interval does not make a full turn around the pole and retrace the entire loop.
Check: Substituting either crossing angle gives \(\cos\theta=-\frac12\), so \(r=1+2(-\frac12)=0\), as required. The sign test at \(\pi\) confirms that the loop’s interval has negative radius.
A Sign Change Is Not the Whole Story
Consecutive pole crossings and a sign change are useful clues, not a guarantee that the corresponding arc is a distinct simple loop. Different parameter intervals can trace the same geometric points. Also, a pole crossing need not involve a sign change: the radius may be zero at an angle while remaining nonnegative on both sides. Such a crossing may produce a cusp or a touch at the pole rather than an inner loop.
Worked Example: A Pole Crossing Without a Negative-Radius Interval
Worked Example: A Pole Crossing Without a Negative-Radius Interval
Determine whether \(r=1+\cos\theta\) has a negative-radius interval on \(0\leq\theta\leq2\pi\).
Step 1: Rewrite the radius. Using \(1+\cos\theta=2\cos^2(\theta/2)\),
The radius is never negative. It equals zero when \(\cos(\theta/2)=0\), which in the given interval occurs at \(\theta=\pi\).
Step 2: Check the signs on either side. For example,
The curve reaches the pole at \(\pi\), but the radius is positive on both sides. There is no interval that must be plotted in the opposite direction, so this pole crossing does not create an inner loop by a negative-radius interval.
Check: At the crossing, \(r(\pi)=1+\cos\pi=1-1=0\). The squared-cosine form proves \(r\geq0\) throughout the full interval, confirming that no negative-radius piece was missed.
Worked Example: Detecting a Retraced Curve
Worked Example: Detecting a Retraced Curve
For \(r=2\sin\theta\), determine whether tracing from \(0\) to \(2\pi\) produces new points throughout.
Step 1: Locate the pole crossings and signs. Solving \(2\sin\theta=0\) gives \(\theta=0,\pi,2\pi\) on the interval. The radius is positive on \(0<\theta<\pi\) and negative on \(\pi<\theta<2\pi\).
Step 2: Write the Cartesian coordinates. For any \(\theta\),
Consequently,
Every plotted point lies on the circle with center \((0,1)\) and radius \(1\).
Step 3: Check for repeated tracing. Replacing \(\theta\) by \(\theta+\pi\) leaves both coordinates unchanged:
Thus, each point obtained for \(0\leq\theta\leq\pi\) is traced again when the parameter is increased by \(\pi\). The first interval already traces the full circle once: \(2\theta\) ranges from \(0\) to \(2\pi\), making \((x,y)=(\sin(2\theta),1-\cos(2\theta))\) complete one revolution. The second interval repeats those points.
Check: At \(\theta=0\) and \(\theta=\pi\), the coordinates are both \((0,0)\), the pole. The coordinate identities show directly that the second half repeats the first, so using all of \(0\leq\theta\leq2\pi\) would trace the circle twice.
Common Mistake: Treating Negative Radius as No Point
A tempting but incorrect interpretation is that the graph stops wherever \(r<0\), because a distance cannot be negative. For the limaçon above, this would discard the entire interval \(\frac{2\pi}{3}<\theta<\frac{4\pi}{3}\) and omit its inner loop.
The correction is to use the opposite ray. Since \((-1,\pi)\) is equivalent to \((1,0)\), the point exists and is one unit to the right of the pole:
A related error is to assume that every sign interval produces a new part of the graph. The example \(r=2\sin\theta\) shows why this fails: its negative-radius interval retraces the circle traced by the positive-radius interval. Checking coordinates or comparing equivalent parameter values can reveal the repetition.
How to Use Pole Crossings in a Sketch
Before plotting many sample points, list the pole-crossing angles and organize them in order. Then record the sign of \(r\) on each interval. For a negative interval, mentally add \(\pi\) to the plotted direction and use \(|r|\) as the distance. Finally, compare the resulting arc with other parts of the parameter range to see whether it is a new loop or a retracing.
These checks make the sketch more reliable: a loop boundary is supported by pole crossings, the radius sign explains which way the curve is plotted, and a comparison of parameter intervals prevents counting the same points twice. They also prepare you to choose meaningful angle bounds when studying areas of polar regions.
Check Your Understanding
Use pole crossings, sign tests, and the negative-radius interpretation to answer the following.
- For \(r=3\cos\theta\), find the pole crossings on \(0\leq\theta\leq2\pi\) and state the sign of \(r\) between consecutive crossings.
- Explain why \((-2,\frac{\pi}{6})\) and \((2,\frac{7\pi}{6})\) represent the same point.
- For \(r=2+\cos\theta\), can the radius be negative? Justify your answer using a bound or identity.
- For \(r=2\sin\theta\), why is it unnecessary to use both intervals \(0\leq\theta\leq\pi\) and \(\pi\leq\theta\leq2\pi\) to trace the circle once?
- Why does finding two pole crossings with a negative radius between them suggest a loop without proving that the interval traces a distinct simple loop?
