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.

REPRESENTATION OF WARPED SURFACES

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The points y, and y' should lie in the same straight line perpendicular to G-L, and thereby check the work.
The greater the number of auxiliary lines drawn through the point
A, the more accurately may the required element be determined.
272. Problem 161. Assume a rectilinear element upon the hyperboloid of one nappe whose three directrices are [A - 6, 6, 2;
B =-1, -1, 6], [C=-2, -2, 1; =2, 5, 6], and [E = 0, 6,
1; F=6, 3, 6].
273. Problem 162. Assume a rectilinear element upon the hyperboloid of one nappe whose three directrices are [A =- 6, - 2,
6; B=-1, 5, 2], [C = 0,2, 2; = 0, 2, 6], and [E=2, - 2, 1;
F= 6, 4, 6].
274. Problem 163. Given a warped surface with three curvilinear
directrices, one in H, another in V, and the third in the third quadrant;
required to assume a rectilinear element of the surface.
275. The Helicoid. The helicoid is a warped surface generated
by a straight line moving uniformly around and along a rectilinear
directrix which it intersects and with which it makes a constant
angle.
276. To represent the Helicoid. The helicoid is represented by
locating the directrix or axis (which is usually taken perpendicular
to H), a number of the more important rectilinear elements of the
surface, and the base or intersection of the surface with H.
To do this we must know the angle which the generatrix makes
with the directrix, and the vertical distance through which the
generatrix moves for each circuit of the directrix.
In Fig. 114 let A-B represent the directrix, or axis, assumed
perpendicular to H. Through any point C (et, c') on the axis
draw C-D parallel to V and making the given angle with the axis.
Since C-D is taken parallel to V its horizontal projection c,-d, will
be parallel to G-L, and its vertical projection c'-d' will make the
same angle with a'-b' that the generatrix makes with the directrix.
Since the generatrix moves uniformly around and along the
directrix, each point of the generatrix will generate a helix whose
pitch will be equal to the total rise or fall of the generatrix per
revolution, and whose radius will be equal to the distance of the
point from the directrix.