I took some time to actually understand the perspective transformations properly using the excellent ImageMagick examples (mainly created by Anthony Thyssen) as a reference. The basic idea of perspective distortion seems simple: to distort the control points to new locations.

His example uses a checkerboard as the image to manipulate and, based on an array of settings (source, destination) sets control points for the image to twist and turn from. The final output is echoed from a call to distortImage.