Question

Hypocycloid When a circle rolls on the inside of a fixed circle, any point P on the circumference of the rolling circle describes a hypocycloid. Let the fixed circle be x2 + y2 = a", let the radius of the rolling circle be b, and let the initial position of the tracing point P be A(a, 0). Find parametric equations for the hypocycloid, using as the parameter the angle θ from the positive x-axis to the line joining the circles' centers. In particular, if b a/4, as in the accompanying figure, show that the hypocycloid is the astroid x = a cos3 θ, y = a sin3 θ bA(a, 0)

Answer 1

The to-ord^nales «f the cmaller mouina circle's centre 、C, ae : e «maler moUin (a-b) sin e Now, the to- «rclnales point P rel ahive. , to the cửeles renhe . C are : Hence to the orgin , are :

L- O θ is mea-ured (ou nkrclockwis<e i s measure clockvise ω.γ.t. As the Phe cirum ference the arcs in contact must be e Smaller crle is movin, o n Hxed bioger circle , ot crcles hat cam e equal lengths

- Sin alA 34- sin θ - a sin 30 χ= a a cos θ