52 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2.452 Epi- and Hypocycloids. An epicycloid is described by a point on a 

 circle of radius a that rolls on the convex side o a fixed circle of radius b. An 

 hypocycloid is described by a point on a circle of radius a that rolls on the con- 

 cave side of a fixed circle of radius b. 

 Equations of epi- and hypocycloids. 



Upper sign: Epicycloid, 

 Lower sign: Hypocycloid. 



/L \ & =t a ^ 



x = (b 0) cos 9 =F cos - <p, 



f , \ JL . , 



y = (b o) sin 9 - a sin - 9. 



The origin is at the center of the fixed circle. The #-axis is the line joining the 

 centers of the two circles in the initial position and <j) is the angle turned through 

 by the moving circle. 



. Radius of curvature: 



2a(b a) . a , 



P = r - " sm ~r $ 



b 2a 2b 



2.453 In the epicycloid put b = a. The curve becomes a Cardioid: 



(3? + /) 2 - 6a 2 (r> + f) + 80 s * = $a\ 



2.454 Catenary. The equation may be written: 



2. y = a cosh - 



- 



The radius of curvature, which is equal to the length of the normal, is: 



ro x 



p = a cosh 2 

 a 



2.455 Spiral of Archimedes. A point moving uniformly along a line which 

 rotates uniformly about a fixed point describes a spiral of Archimedes. The 

 equation is: 



r = aO, 

 or 



V* 2 + / = a tan' 1 ^ 



The polar subtangent = polar subnormal = a. 

 Radius of curvature: 



_ r( T + ff)3 ^ (r* + 02)1 



= 0(2 + 2 ) r 2 + 2a 2 ' 



2.456 Hyperbolic spiral: 



