Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

102

DESCRIPTIVE GEOMETRY

If we conceive a large number of these tangents or elements of
the surface to be drawn at small distances apart, we may form an
idea of the character of the surface.
The curved line a,-a,,-a,,,-a,,,-, etc., traced through the various
points in which the elements pierce H, is the intersection of the
surface with H and may be taken as the base of the surface.
From the method by which these points a,,, a,,, a,,,,, etc., are
found (see Section 216), it will be seen that the base of the helical
convolute, in case the axis of the helical directrix is taken perpendicular to H, is the involute of that circle which represents the
horizontal projection of the directrix.
In Fig. 87 only that portion of the surface extending between
the helical directrix and the base on H is considered.
Since the generatrix extends without limit in both directions
from the point of tangency on the helix, there will be generated
simultaneously on opposite sides of the helical directrix two distinct portions of the surface. These two portions of the surface
are called nappes of the surface, and their line of separation, which
is the helical directrix, is called the edge of regression.
From the nature of the helical directrix and the relation of the
generatrix to the directrix it is evident that if the generation be
extended beyond one circuit of the axis, the surface will consist
of a series of overlapping surfaces whose base or intersection with
H will have the form of a spiral.
249. To assume a Rectilinear Element of the Helical Convolute.
Draw a rectilinear tangent to the helical directrix (see Section 216).
250. To assume a Point upon the Surface of the Helical Convolute.
First assume an element of the surface and then assume a point
upon the element.