# The general form for a limacon is r=a+bcostheta or r=a+bsintheta. When will a limacon neither have an inner loop nor a dimple?

Nov 24, 2016

See explanation.

#### Explanation:

graph{x^2+y^2-sqrt(x^2+y^2)-2y=0 [-10, 10, -5, 5]}

If you agree that the polar coordinate r is length (modulus ) of a

vector and accordingly r = the positive $\sqrt{{x}^{2} + {y}^{2}}$, limacons

$r = a + b \cos \left(\theta - \alpha\right)$ have dimples but not inner loops.

For examples, r=2+cos theta and r = 1-sin theta#

have nodes ( points with two distinct tangent directions ) that create

dimples

Unless you allow negative values for r,

the question of having inner loop is ruled out.

The first limacon is for $r = 1 - \sin \theta$

The ones below are for $r = 2 + \cos \theta \mathmr{and} r = 1 - 1.5 \sin \theta$.

Note that all are cardioid-like in shape but the depths of the dents at

the dimples differ. The dimple is irremovable. The dimple

disappears, when a or b = 0 and the limacon becomes a circle.

graph{x^2+y^2-2 sqrt(x^2+y^2)-x=0 [-10, 10, -5, 5]}
graph{x^2+y^2-sqrt(x^2+y^2)+1.3y=0 [-5, 5, -2.5, 2.5]}