when does a limacon have an inner loop
Limaçon As   b  {\displaystyle b}  is decreased relative to   a  {\displaystyle a} , the indentation becomes more pronounced until, at   b = a  {\displaystyle b=a} , the cardioid, it becomes a cusp. For   0 < b < a  {\displaystyle 0<b<a} , the cusp expands to an inner loop and the curve crosses itself at the origin. As   b  {\displaystyle b}  approaches 0 the loop fills up the outer curve and, in the limit, the limaÃ§on becomes a circle traversed twice.