Curves by Inversion

Another approach to curving spirolaterals is to apply a transformation, a projection, or mapping from a linear image to a curved one.  In researching these possibilities, the first one encountered that was interesting was the concept of inversion, as described by Dixon  and Lawrence.   This mathematical method turns lines into circles.  The transformation based on the image midpoint is as follows:

   X = (x * r2) / (x2 + y2) 
   Y = (y * r2) / (x2 + y2)

Figure 15 demonstrates this transformation on a simple square spirolateral.

Inversion by definition reconstructs the spirolateral as curves.  After transforming a variety of spirolaterals, it was found that ones that are open in the center generate the most interesting results.  Ones that have lines that cross the image midpoint, generate very strange results, which are visually not interesting.  Symmetry also added to the quality of the results, as did the selection of the line thickness.  Figure 16 displays a sample of spirolateral inversions.

Figure 15: Inversion of spirolateral 190

7901,3,4,6,7                                                            245                                                                        230
Figure 16: Spirolateral inversions



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