Planar scenes would appear to be ideally suited for self-calibration because, by eliminating the problems of occlusion and parallax, high accuracy two-view relationships can be calculated without restricting motion to pure rotation. Unfortunately, the only monocular solutions so far devised involve costly non-linear minimisations which must be initialised with educated guesses for the calibration parameters. So far this problem has been circumvented by using stereo, or a known calibration object. In this work we show that when there is some control over the motion of the camera, a fast linear solution is available without these restrictions. For a camera undergoing a motion about a plane-normal rotation axis (typified for instance by a motion in the plane of the scene), the complex eigenvectors of a plane-induced homography are coincident with the circular points of the motion. Three such homographies provide sufficient information to solve for the image of the absolute conic (IAC), and therefore the calibration parameters. The required situation arises most commonly when the camera is viewing the ground plane, and either moving along it, or rotating about some vertical axis. We demonstrate a number of useful applications, and show the algorithm to be simple, fast, and accurate.