GEOMETRY 



2.450 Cycloids and Trochoids. 



If a circle of radius a rolls on a straight line as base the extremity of any 

 radius, a, describes a cycloid. The rectangular equation of a cycloid is: 



x = a((j> sin </>), 

 y = a(i cos (/>), 



where the #-axis is the base with the origin at the initial point of contact. <j> is 

 the angle turned through by the moving circle. (Fig. 6.) 



Fic.6 



A = vertex of cycloid. 



C = center of generating circle, drawn tangent at A. 

 The tangent to the cycloid at P is parallel to the chord AQ. 



Arc AP = 2 x chord AQ. 

 The radius of curvature at P is parallel to the chord QD and equal to 2 x chord QD. 



PQ = circular arc AQ. 

 Length of cycloid: s = 8a; a = CA. 

 Area of cycloid: S = sira 2 . 



2.451 A point on the radius, b>a, describes a prolate trochoid. A point, 

 b<a, describes a curtate trochoid. The general equation of trochoids and 

 cycloids is 



x = a(f) (d + d} sin </>, 



y = (a + d) (i - cos <), 

 d = o Cycloid, 

 d>o Prolate trochoid, 

 d<o Curtate trochoid. 

 Radius of curvature: 



P = 



d 2 



