an excellent question!!
First, try plotting some of the multi theta curves , like r = 4 +cos(6 theta ) on a graphing calculator
It is a circle , almost , with a wiggling cos curve running up and down on the circle .
Cosine goes from +1 to - 1 so the outer/inner range of this sort of function goes from
4+1 =5 to 4-1 =3
now slowly change the first letter , in this case the 4 down to 3 to 2 to 1, and watch the curve .
Its still a circle, with a wiggling cos (theta )
Eventually the outer / inner range will go from 1 + 1 = 2 and 1-1 = 0
at this point the inner wiggle will touch the origin.
Now change the 6theta down to 5 and to 4 and to3 .. and the number of wiggles will slowly decrease , until at 1 theta, it is almost invisible, as it goes "up and down" just once per revolution .
Now with a limacon, like r = 1 + 2 cos(theta)
The circle is still at radius 1, but the 2cos(theta) causes the circle to have an
outer/ inner range of 1+2 =3 and 1-2 = -1
This is the important point : the outer range of the parent circle becomes +3 at some values of r
and at some values , it drops to "-1"
an r of +3 means the point is at a distance of 3 along the radius vector r
and now :
An r of "-1" means that the radius vector is a distance of -1 from the origin or reversed in the opposite direction .
So these values, where r becomes NEGATIVE in length, are the places where limacons
have inside out loops .
If you look at limacons , with r = a +bcos(theta )
--->if a is big enough, the wiggles of the bcos(theta) part will just cause a dimple to appear
--->If the a is smaller, like 1, and the b is also 1, the wiggles will dip into the origin
and finally ---> if the b is bigger than the a, as in r = 2 +4cos(theta) the wiggles will cause the r to become negative, or reversed, and for these curves, you will get an inside out loop.