If R = 4r in Exercise 1, show that the hypocycloid is parametrized by x = (R cos3 ϕ)i + (R...

If R = 4r in Exercise 1, show that the hypocycloid is parametrized by

 x = (R cos3 ϕ)i + (R sin3 ϕ)j.

(Hint: Use the identities cos 3θ = 4 cos3 θ − 3 cos θ and sin 3θ = 3 sin θ − 4 sin3 θ.) This is the hypocycloid of four cusps.

Exercise 1

In Figure 1.9.3 we can use the angle ϕ = ∠SOT as a parameter instead of the angle t. In this case show that a parametrization for the hypocycloid is