The monograph explores the surface formed by two families of circles, the Dupin cycloid, named after the great French scientist Pierre Charles Franois Dupin, a student of Gaspard Monge ., who discovered it. The properties of the Dupin cycloid, its application to various geometric constructions, as well as the possibility of practical application in engineering and architecture are considered. The monograph may be of interest to professionals in the field of engineering geometry (specialty 05. 01. 01 "Engineering geometry and computer graphics"), as well as graduate students, teachers and those interested in geometric constructions .