On Helical Projection and Its Application in Screw Modeling

As helical surfaces, in their many and varied forms, are finding more and more applications in engineering, new approaches to their efficient design and manufacture are desired. To that end, the helical projection method that uses curvilinear projection lines to map a space object to a plane is examined in this paper, focusing on its mathematical model and characteristics in terms of graphical representation of helical objects. A number of interesting projective properties are identified in regard to straight lines, curves, and planes, and then the method is further investigated with respect to screws. The result shows that the helical projection of a cylindrical screw turns out to be a Jordan curve, which is determined by the screw's axial profile and number of flights. Based on the projection theory, a practical approach to the modeling of screws and helical surfaces is proposed and illustrated with examples, and its possible application in screw manufacturing is discussed.