# 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

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

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

The ones below are for

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]}