Avector approach to tracing the path of a laser beam through an optical system containing movable plane mirrors is described, which permits a unified treatment of a number of basic mirror-scanning devices. We show that the scan field produced by the mirror-scanning system is a curved surface with a straight line as its generating element. The cross section of the scan field can be a circle, an ellipse, or a curve in the shape of an egg. Based on this understanding, some advanced topics are addressed, e.g., the relationship between the scan field and the scan pattern, the dependence of the scan pattern on the location and orientation of the observation surface, optical distortions in a scan pattern, spot-size enlargement caused by non-normal incidence of the scan beam on the observation plane, and so on. Design equations and curves are derived for the mirror-scanning devices that most frequently exist in linear and circular scan technology. Part II contains an analysis of the galvanometer-based optical scanner paddle scanner and the regular polygon. In Part III, X-Y scanning systems are studied.