The antiMercator and Circular transformation consist of computing polar coordinates based on the original image and then converting them to Cartesian coordinates.  An alternate method is to not convert the polar to rectangular Cartesian coordinates but to follow a curve.  Using the angle and radius computation from the Circular transformation, the Cartesian coordinates are computing with the following curves, as in Lawerence:

Hypocycloid: X = R * (cos(A)*(n-1) + cos(A*(n-1)))        where: n = number of cusps, 3 and 4
                       Y = R * (sin(A)*(n-1) - sin(A*(n-1))) 

Epicycloid:    X = R * (cos(A)*(n+1) - cos(A*(n+1)))        where: n = number of cusps, 2, 3, and 4 
                      Y = R * (sin(A) (n+1) - sin(A*(n+1)))

The major difference between these two curves is that the Hypocycloid produces concave curves and the Epicycloid produces convex ones. 

Figures 23 and 24 demonstrate these transformations on a simple square spirolateral. 

Figures 25 to 28 display sample of spirolaterals using the Hypocycloid and Epicycloid curves.

These transformations produce much more delicate curves due to the effect of the line thickness.  The Hypocycloid is particularly interesting because of the very sharp point where its curves meet.  It is also the transformation that produces concave curves.  The Epicycloid is somewhat similar to the antiMercator and Circular transformation except the unequal treatment of vertical and horizontal lines, as they generate a changing radius.  Both generate quite a series of unexpected results