The Polar Coordinates filter is another one of those gems that a lot of folks shy away from because it doesn't make much sense at first glance. When you choose Filter ^ Distort ^ Polar Coordinates, Photoshop presents two radio buttons, as shown in Figure 11-31. You can either map an image from rectangular to polar coordinates or from polar to rectangular coordinates.
All right, time for some global theory. The first image in Figure 11-32 shows a stretched detail of the world map from the Digital Stock library. This map falls under the heading of a Mercator projection, meaning that Greenland is all stretched out of proportion, looking as big as the United States and Mexico combined.
The reason for this has to do with the way different mapping systems handle longitude and latitude lines. On a spherical globe, lines of latitude converge at the poles. On a Mercator map, they run absolutely parallel. Because the Mercator map exaggerates the distance between longitude lines as you progress away from the equator, it likewise exaggerates the distance between lines of latitude. The result is a map that becomes infinitely enormous at each of the poles.
When you convert the map to polar coordinates (by selecting the Rectangular to Polar radio button in the Polar Coordinates dialog box), you look down on it from the extreme North or South Pole. This means that the entire length of the top edge of the Mercator map becomes a single dot in the exact center of the polar projection. The length of the bottom edge of the map wraps around the entire perimeter of the circle. The second example in Figure 11-32 shows the result. As you can see, the Rectangular to Polar option is just the tool for wrapping text around a circle.
If you select the Polar to Rectangular option, the Polar Coordinates filter produces the opposite effect. Imagine for a moment that the conical gradation shown in the upper-left corner of Figure 11-33 is a fan spread out into a full circle. Now imagine closing the fan, breaking the hinge at the top, and spreading out the rectangular fab ric of the fan. The center of the fan unfolds to form the top edge of the fabric, and what was once the perimeter of the circle is now the bottom edge of the fabric. Figure 11-33 shows two examples of what happens when you convert circular images from polar to rectangular coordinates.
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