Circular Transformations Continuing the search for other methods to curve spirolaterals, Dixon in a section on transformations outlines a method he calls antiMercator. 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/(xmaxxmin)
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 lowerleft 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.
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.
