This animation showcases the shape diversity of trochoids and cycloids, specifically epicycloids, pericycloids, and hypocycloids.

Animation and Variation of a cycloid or a trochoid with interactive sliders

Epitrochoid (or rather with cusps Epicycloid)
A moveable (yellow) wheel runs around a fixed (gray) wheel

Hypotrochoid (or rather with cusps Hypocycloid)

Peritrochoid (or rather with cusps Pericycloid)

Angle of the connecting line between the centers of the revolving and of the stationary wheel as well as the horizontal axis
φ (Phi): 1 Period = 1 Rotation
Radius of the fixed wheel
rR 4
Radius of the revolving wheel
rG 1
Distance of the point which generates the trochoid to the center of the revolving wheel
a: 4
 
Explanation of the table: the first 3 rows shows curtate trochoids
Quantity of
self-intersecting points 		Quantity of
inflection points 		Special cases
presented graphically
any (0 ... 16) any
(0 oder 8)
0 0  
0 0 4 approximated linear guideways
0 8  
0 0 4 cusps
40 
404 self-tangential points
120 
8 (+4=12)02 self-tangential points as well as
4-fold self-intersecting points
16 0  
Starting angle of the revolving wheel
γ0 (Gamma0):

This animation draws a cycloid respectively a trochoid generated by a rolling circle, which rolls on, around or inside another circle. Depending on the choice of the user it is an epicycloid, pericycloid or hypocycloid or rather in the general form with loops or waves instead of cusps called an epitrochoid, peritrochoid or hypocycloid. The user can vary the transmission ratio and the distance of the point generating the trochoid to the center of the circle so that all shapes of cycloids can be represented.

Show zoom and save dialog
 

Double generation of trochoids

All trochoids can be generated by another alternative pair of wheels:
    This Epitrochoide with the transmission ratio i= 4 / 1 and the distance a= 4 is identical to the
Peritrochoide with the transmission ratio i= 4 / 5 and the distance a= 1.25