The Hypocycloid

                                                                                             by

                                                                                  Edmund Koontz
 
 

History

     In 1594 Galileo and Mersenne discovered the cycloid.  Roemer in 1674 while studying different styles of gear teeth discovered the hypocycloids.  Johann Bernoulli applied the hypocycloid to practical use in 1691.  Euler while working on an optical problem in 1745 studied the deltoid curve, a specific kind of hypocycloid.
 
 
 
 

.....................................................................

                        Galileo Galilei                Marin Mersenne             Johann Bernoulli                Leonhard Euler
 
 
 

Definition

     A hypocycloid is a special plane curve formed by the locus of a point on a small circle that rolls within a larger circle.  The number of cusps is determined by the ratio of the larger circles radius to that of the smaller circles.  If the ratio is three to one the small circle will rotate three times.  The ratios of the large circles radius to that of the small circles radius is identical to the ratio of the large circles circumference to that of the small circles circumference.  The formation of the three cusped hypocycloid is illustrated below.
 
 

Parametrization

     The radius of the large circle is "a" and that of the small circle is "b".  Throughout the development of the parametric equations describing hypocycloids the variables from the diagram below will be used.  "t" is the angle that is formed between the fixed "a" and the line connecting the center of the large circle to that of the small circle.  Theta is the angle, in radians, that the small circle has rotated.
 
 

     The parametric equations that describe the motion of the small circle as it rotates about the center of the large circle are:

                                                                                x = (a-b)*cos(t)
                                                                                  y = (a-b)*sin(t)

     The circle that is formed has radius (a-b).  The parametric equations for the point, that is traced, on the small circle with the center of the circle at the origin are given below.

                                                                               x = b*cos(theta)
                                                                                 y = - b*sin(theta)

     To find the parametric equations for the hypocycloid the two sets of parametric equations are added.

                                                                    x = (a-b)*cos(t) + b*cos(theta)
                                                                      y = (a-b)*sin(t) - b*sin(theta)

     The difference between the periods of "t" and "theta" can be described by the following relationship.

                                                                           theta = (a/b - 1)*t

     Substituting for "theta" gives the parametric equations for the hypocycloid.

                                                                    x = (a-b)*cos(t) + b*cos((a/b - 1)*t)
                                                                    y = (a-b)*sin(t) - b*sin((a/b - 1)*t)

Special Cases
a = b             cardiod
a = 2b           nephroid
a = 3b           tricuspoid
a = 3b           deltoid
a = (3/2)b     deltoid
a = 4b           astroid

References

Weisstein, E. "Hypocycloid"  http://www.astro.virginia.edu/~eww6n/math/Hypocycloid.html

Johnson, M. "The Deltiod Curve"
http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Sp98/MikeJ/deltiod.htm

Whitman, N. "Hypocycloid"
http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Sp99/Nick/Hypocycloid.htm

MacTutor History of Mathematics Archive. "Hypocycloid."
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hypocycloid.html