How to graph a limaçon and identify its shape

A limaçon can have a loop, a dimple, or a smooth convex outline. The ratio of the constant to the trig coefficient tells us which shape to expect. Let’s use that ratio and a radius table to graph an inner-loop example.

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Compare the two positive coefficients

For $r=a+b\cos \theta$ or $r=a+b\sin \theta$ with positive $a$ and $b$, compare $a/b$. The same classification works with a minus sign before the trig term.

Limaçon shapes from the coefficient ratio

Ratio	Shape
$0<\frac{a}{b}<1$	Inner loop
$\frac{a}{b}=1$	Cardioid
$1<\frac{a}{b}<2$	Dimpled
$\frac{a}{b}\ge 2$	Convex

Find the key points for an inner-loop limaçon

For $r=3+4\cos \theta$, the ratio is $3/4$, so the graph has an inner loop. The cosine makes it symmetric about the horizontal axis.

Key radius values

Angle	Radius
$0$	$7$
$\frac{\pi }{2}$	$3$
$\pi$	$-1$
$\frac{3\pi }{2}$	$3$

At $\theta =\pi$, the negative radius puts the point at $(1,0)$. To find where the curve reaches the pole, solve $3+4\cos \theta =0$, giving $\cos \theta =-3/4$. The negative-radius interval between those two angles traces the inner loop.

The small loop lies inside the larger loop on the right of the pole.

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