Wikipedia's "Limaçon" entry gives the following information about the polar curve $r=b+a\cos(\theta)$:
When $b>a$, the limaçon is a simple closed curve. However, the origin satisfies [the Cartesian translation of the equation], so the graph of this equation has an acnode or isolated point.
When $b>2a$, the area bounded by the curve is convex, and when ${\displaystyle a<b<2a}$, the curve has an indentation bounded by two inflection points. At ${\displaystyle b=2a}$, the point ${\displaystyle (-a,0)}$ is a point of 0 curvature.
As $b$ is decreased relative to $a$, the indentation becomes more pronounced until, at $b=a$, the curve becomes a cardioid, and the indentation becomes a cusp. For $0<b<a$, the cusp expands to an inner loop, and the curve crosses itself at the origin. As $b$ approaches $0$, the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.
I'm familiar with polar curves and some of its special cases,however these results about Limaçon is new to me and I don't know hoe to prove them (notice that I'm not asking about the formula of a Limaçon itself).