9.5.2

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Special cases arise when the both circles are set to the same radius value (known as Pascal’s curve).
If, in addition, point P is on the radius of the outer circle (a=r), the result is known as a Cardioid. (See
the illustrations below.)
R = 2, r = 2, a = 1. R = 2, r = 2, a = 3. R = 2, r = 2, a = 2
(cardioid).
With the basic Hypocycloid, the observed point P (that forms the curve) is on the radius of the inner
circle (a = r). With the shortened Hypocycloid, it is located inside the inner circle (a < r), and with
the extended Hypocycloid it is outside (a > r). If the radius of the outer circle is exactly four times the
size of the inner circle, the result is an Astroid. The examples below illustrate some of the curves that
are possible.
R = 6, r = 2, a = 2. R = 8, r = 1, a = 1. R = 8, r = 2, a = 2
(astroid).
R = 8, r = 2, a = 1. R = 12, r = 3, a = 6.