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Circular Transformations

Continuing the search for other methods to curve spirolaterals, Dixon in a section on transformations outlines a method he calls anitMercator.  Horizontal lines become circles concentric with the coordinate origin, vertical lines become radial, and slanting lines become logarithmic spirals.  The transformation in polar coordinates is as follows:

   A = k * x                      where: k = 2p/(xmax-xmin)
   R = exp(k * y)

Figure 17 demonstrates this transformation on a simple square spirolateral.

This transformation is effected by the line thickness and also by the offset from the origin.  The origin is positioned in the lower-left corner of the image.  This location results in the image being bent clockwise starting with the left corner.  Figure 18 displays the effect of increasing the offset. 

Figure 19 displays a sample of spirolateral transformed by antiMercator. 


Figure 17: antiMercator spirolateral 190


Figure 18: antiMercator offsets on spirolateral 190


                 160                                                           230                                                390 
Figure 19: antiMercator spirolaterals

While investigating the antiMercator transformation, one alternate method was found by removing the exponential function.  The circular form remains without the logarithmic spiral effect.  Since no formal name has been found for this transformation, it will be referred to as simply Circular. 

Figure 20 demonstrates this circular transformation on a simple square spirolateral.

This transformation differs from the antiMercator in that the horizontal line spacing is of a more constant distance for the center, so that original distances are better represented.  This transformation also changes as the offset increases.  Figure 21 displays the effect of increasing the offset. 

Figure 22 displays a sample of spirolateral transformed by the Circular transformation.


Figure 20: Circular spirolateral 190


Figure 21: Circular offsets on spirolateral 190


        160                                                       245                                                             230 
Figure 22: Circular spirolaterals


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BitArt, Robert J. Krawczyk
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