Abstract
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.
Highlights
Projection, or three-dimensional (3D) projection, is a general method of mapping space points to a 2D plane
Further study is still needed to put it into practice, it is evident that the helical projection method does have advantages in describing the feature of complex helical surfaces
In the light of the need to overcome the weakness of conventional projection methods in solving engineering problems associated with helical surfaces, this paper investigated the helical projection method focusing on its projective properties and potential use in component modeling and representation
Summary
Projection, or three-dimensional (3D) projection, is a general method of mapping space points to a 2D plane. Since its invention, this method has been widely used for graphic representation and solution of space problems in engineering. The projection, either orthographic or perspective, is produced by rays of imaginary straight lines While it proves prestigious in most cases, the weakness of this method was detected with respect to curved surfaces. Tevlin [1] proposed the use of cylindrical helixes in projection, which brought about a new type of projection, namely, helical projection In his work, he demonstrated the new method with respect to the solution of a number of geometrical problems pertinent to helixes and helical surfaces
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