Epicycloid--1-Cusped

A 1-cusped epicycloid has $b=a$, so $n=1$. The radius measured from the center of the large circle for a 1-cusped epicycloid is given by Epicycloid equation (9) with $n=1$ so

$$r^2 = {a^2\over n^2}\left[{(n^2+2n+2)-2(n+1)\cos(n\phi)}\right]$$

$$= a^2[(1^2+2\cdot 1+2)-2(1+1)\cos(1\cdot\phi)]$$

$$= a^2(5-4\cos\phi)$$ (1)

$$r=a\sqrt{5-4\cos\phi},$$ (2)

and

$$\tan\theta = {2\sin\phi-\sin(2\phi)\over2\cos\phi-\cos(2\phi)}.$$ (3)

The 1-cusped epicycloid is just an offset Cardioid.