Reading the Shape of a Limaçon
A small change in a polar equation can produce a conspicuous change in its graph. In the family \(r=a+b\cos\theta\), the relative sizes of \(a\) and \(b\) determine whether the curve turns back through the pole, develops a dimple, or has a convex outline. Recognizing these cases makes a sketch more reliable than plotting a few isolated points.
We will consider the standard limaçon equations with \(a>0\) and \(b>0\). The same shape classifications apply to the sine form \(r=a+b\sin\theta\); its graph is a rotation of the corresponding cosine graph. The focus here is the role of the ratio \(a/b\), together with the key values and crossings that make each shape visible.
Key Values and Symmetry
For \(r=a+b\cos\theta\), the cosine reaches its maximum at \(\theta=0\) and minimum at \(\theta=\pi\). Thus the signed radius at those angles is \(a+b\) and \(a-b\), respectively. At \(\theta=0\), the point is on the positive horizontal axis. At \(\theta=\pi\), it is on the negative horizontal axis if its radius is positive; if its radius is negative, the plotted point lies on the opposite ray.
The cosine equation is symmetric about the horizontal axis: replacing \(\theta\) with \(-\theta\) leaves \(\cos\theta\), and therefore \(r\), unchanged. The points at angles \(\theta\) and \(-\theta\) are reflections across that axis. For the sine form, the key values are \(r(\pi/2)=a+b\) and \(r(3\pi/2)=a-b\); its symmetry axis is vertical.
The two forms are rotations of one another, not different shape families. In fact, setting \(\phi=\theta-\pi/2\) gives \(\cos\phi=\sin\theta\), so \(r=a+b\sin\theta\) has the same radial rule as the cosine form with its angle shifted by a quarter-turn. This observation lets us classify the sine form using the same ratio.
How the Ratio Determines the Shape
The key question is whether the minimum signed radius \(a-b\) is negative, zero, or positive. If it is negative, the curve crosses the pole and includes a segment traced with negative radius. If it is positive, the radius stays positive, and curvature distinguishes a dimpled outline from a convex one.
| Parameter ratio | Radius and shape |
|---|---|
| \(0<a/b<1\) | The radius is negative for part of the cycle. The curve crosses the pole and has an inner loop. |
| \(a/b=1\) | The minimum radius is zero. This is the boundary case, a cardioid, treated in more detail separately. |
| \(1<a/b<2\) | The radius stays positive, but part of the outline bends inward, making a dimpled limaçon. |
| \(a/b\geq 2\) | The outline is convex. At the boundary \(a/b=2\), it has zero curvature at one point but does not turn inward. |
For the inner-loop case, the radius is negative exactly when \(a+b\cos\theta<0\), or \(\cos\theta<-a/b\). Since \(0<a/b<1\), this inequality holds over an interval around \(\theta=\pi\). At either endpoint of that interval, \(r=0\), so the curve reaches the pole. Between the crossings, the negative radius places points on rays opposite to the angles being used. This return through the pole creates the inner loop.
When \(a>b\), the minimum radius \(a-b\) is positive, so there is no pole crossing. That alone does not guarantee a convex outline: for \(b<a<2b\), the curve bends inward near its minimum-radius side. The boundary between this dimpled case and the convex case can be justified by checking the curve’s signed curvature.
Why the Dimple Boundary Is at \(a/b=2\)
For a polar curve with radius \(r(\theta)\), its Cartesian parametrization is \(x=r\cos\theta\), \(y=r\sin\theta\). At a regular point, the sign of \(x'y''-y'x''\) determines the direction of bending as the parameter increases. For a positively oriented, simple closed curve, a change to negative signed curvature indicates an inward-bending portion. If the signed curvature is nonnegative all the way around, the outline is convex; a zero at an isolated point can mark a flat transition without creating a dimple.
Here is the calculation behind that result. Differentiating \(x=r\cos\theta\) and \(y=r\sin\theta\) gives \(x'=r'\cos\theta-r\sin\theta\) and \(y'=r'\sin\theta+r\cos\theta\). Differentiating again and substituting into the determinant yields the general polar expression below. In the limaçon equation, \(r'=-b\sin\theta\) and \(r''=-b\cos\theta\).
Because \(\cos\theta\) ranges from \(-1\) to \(1\), the minimum occurs at \(\theta=\pi\). Factoring that minimum gives
In the dimpled range \(b<a<2b\), the first factor is positive and the second negative, so the determinant is negative near \(\theta=\pi\). In the range \(a>2b\), both factors are positive, and the determinant remains positive everywhere. At \(a=2b\), the minimum is zero only at \(\theta=\pi\); it does not become negative. This establishes the stated boundary and explains why a curve can be convex even though its curvature vanishes at one point.
Worked Examples
Worked Example: Identifying an Inner Loop
Classify and sketch the main features of \(r=1+2\cos\theta\).
Here \(a=1\) and \(b=2\), so the ratio is \(a/b=1/2\). Since \(0<1/2<1\), this is an inner-loop limaçon. The key radii are
To find where the graph crosses the pole, set \(r=0\):
Between these angles the radius is negative; for example, \(r(\pi)=1-2=-1\). The point at \(\theta=\pi\) is therefore plotted one unit along the positive horizontal axis, opposite the angle-\(\pi\) ray. The graph is symmetric about the horizontal axis. As a check, substituting the pole-crossing angles gives \(r=1+2(-1/2)=0\) at each, and substituting \(0\) and \(\pi\) gives the stated maximum and negative minimum signed radii.
Worked Example: Recognizing a Dimpled Limaçon
Classify \(r=3+2\cos\theta\) and verify that it has an inward-bending portion.
The ratio is \(a/b=3/2\), which lies between \(1\) and \(2\). The radius stays positive because its minimum is \(3-2=1\), but the classification predicts a dimple. The signed-curvature numerator has minimum
Thus it is negative at \(\theta=\pi\), where \(\cos\theta=-1\). Direct substitution confirms the value: \(a^2+2b^2+3ab\cos\pi=9+8+18(-1)=-1\). The key radii are \(r(0)=3+2=5\) and \(r(\pi)=3-2=1\), so the graph never reaches the pole. These positive radii, together with the negative curvature near the minimum-radius side, check that this is a dimpled limaçon rather than an inner-loop curve.
Worked Example: Classifying a Convex Limaçon
Classify \(r=5+2\sin\theta\). State the key radii and use the corresponding cosine curve to check convexity.
The ratio is \(5/2=2.5\), which exceeds \(2\), so the graph is convex. Its maximum and minimum radii occur where \(\sin\theta\) is \(1\) and \(-1\):
To check the curvature classification, write the sine curve as a quarter-turn of \(r=5+2\cos\phi\). For that cosine curve, the minimum signed-curvature numerator is
It has positive curvature throughout, and the rotated sine curve is therefore convex as well. The radius check agrees: its minimum is \(3>0\), so there is no inner loop or pole crossing. Substitution at the two extremal angles gives \(7\) and \(3\), as claimed.
A Practical Sketching Method
For a standard limaçon with positive parameters, use the following sequence. The ratio classifies the shape; key values and symmetry then place the important parts of the curve.
Identify \(a\) and \(b\), then compare \(a/b\) with \(1\) and \(2\).
For the cosine form it is \(a-b\); for the sine form it is also \(a-b\). A negative minimum signals an inner loop, while a positive minimum rules out a pole crossing.
Use \(\theta=0,\pi\) for the cosine form and \(\theta=\pi/2,3\pi/2\) for the sine form.
Reflect across the horizontal axis for the cosine form or the vertical axis for the sine form. Interpret every negative radius on the opposite ray.
A Common Mistake
A common error is to reverse the ratio. For \(r=3+2\cos\theta\), one might calculate \(b/a=2/3\) and wrongly conclude that the curve has an inner loop. But the classification uses \(a/b\), not \(b/a\): here \(a/b=3/2\), so the curve is dimpled. Checking the minimum radius catches the error: \(3-2=1>0\), so the radius never becomes negative and the curve does not cross the pole.
Another important point is that a negative radius is not an instruction to discard a point. It means the point is plotted in the direction opposite to the angle. Ignoring this convention removes the inner loop from the sketch and changes the curve being represented.
Check Your Understanding
Use the ratio, key radii, and pole-crossing condition to answer the following.
- Classify \(r=2+5\cos\theta\). For which angles between \(0\) and \(2\pi\) does it cross the pole?
- Is \(r=7+4\sin\theta\) inner-loop, dimpled, or convex? What are its maximum and minimum radii?
- Explain why \(r=3+2\cos\theta\) has no inner loop even though it is not convex.
- What happens to the signed-curvature numerator at the minimum-radius side when \(a=2b\)?
- How is the graph of \(r=a+b\sin\theta\) related to the graph of \(r=a+b\cos\theta\)?
