A general analysis for the first-order design of a three-lens afocal zoom system with one lens fixed is presented. The reasonable solution areas in the focal length diagrams with positive or negative magnification are derived and shown graphically. The relation between the two separations of the three lenses in zooming is found to be a hyperbola. According to the different locations of hyperbola centers, four cases are analyzed. From the four hyperbolic graphs, we get five different types of zoom systems. For each zoom type, we find the maximum range of magnification and the position where the maximum or minimum system length occurs during zooming. The zoom loci for the first or second lens fixed are also discussed. © 1997 Society of Photo-Optical Instrumentation Engineers.