Show that the Cartesian equation for the curve in Exercise 1 is
x2/3 + y2/3 = R2/3
Exercise 1
If R = 4r in Exercise 2, 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 2
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
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