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Following a Curve

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. 


Figure 23: Hypocycloid spirolateral 190


Figure 24: Epicycloid spirolateral 190

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


160                                                245                                                  390
Figure 25: Hypocycloid spirolaterals


5901,2,4,5                                             5901,3,4,5                                              272 
Figure 26: Hypocycloid spirolaterals


245                                                         230                                                        390
Figure 27: Epicycloid spirolaterals


160                                                          260                                                      5901,2,4,5
Figure 28: Epicycloid spirolaterals

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
 


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