Question

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?

Answer 1

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 (theta-alpha)` 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 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]}