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?

1 Answer
Nov 24, 2016

See 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


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]}