All of the Web Sketchpad models below (and here) are based on the distance definition of a hyperbola: the set of points P such that the difference in the distances from P to two fixed points (the foci) is constant.
Experiment with the interactive models and see if you can prove that each method presented on pages 2-7 generates hyperbolas. The constructions on pages 5 and 6 are based on illustrations found in the 17th-century Dutch mathematician Frans van Schooten’s manuscript, Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus (A Treatise on Devices for Drawing Conic Sections). Van Schooten’s beautiful pictures are below. You’ll find more of van Schooten’s constructions in my book Exploring Conic Sections with The Geometer’s Sketchpad.