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

111

If the angle between the generatrix and the directrix is a right
angle, the surface becomes a right conoid. The generatrix is a
straight line and moves in such a way as to touch two directrices,
one rectilinear and the other curvilinear, and remains parallel to a
plane directer H, to which the rectilinear directrix is perpendicular.
The helicoid must not be confused with the helical convolute in
which the generatrix moves tangentially to the helix and therefore
does not intersect the axis.
The helicoid is of practical interest since it has a direct application in the structure of the common screw thread, as may be seen
in Sections 282 and 283.
277. To assume a Rectilinear Element on the Surface of the
Helicoid. Represent the generatrix in one of its positions as
explained above.
278. To assume a Point upon the Surface of the Helicoid. Follow
the method explained in Section 255.
279. Problem 164. Represent the helicoid whose generatrix makes
an angle of 30 degrees with the axis and whose pitch is 6 units.
280. Problem 165. Represent the helicoid whose generatrix makes
an angle of 60 degrees with the axis and whose pitch is 4 units.
281. Problem 166. Represent the helicoid whose generatrix makes
an angle of 90 degrees with the axis and whose pitch is 6 units.
282. Problem 167. To represent a triangular-threaded screw.
In Fig. 115 let A-B represent the axis of a helicoid, and
let D-A and D-E represent two generating elements of the surface, equally inclined to the axis, parallel to V, and intersecting
at D.
Lay off upon D-A and D-E from D the equal distances D-F and
D-G, and through F and G draw the indefinite vertical line F-K.
While the generating lines D-A and D-E generate their respective helicoidal surfaces, the three points F, D, and G will generate
their respective helices, and the straight line F-K will generate a
cylindrical surface whose diameter is equal to 1,-f,.
If in this particular case the pitch of the helicoids is made equal
to the distance F-G, or some multiple of it, the surface generated
by the two sides D-F and D-G of the isosceles triangle D-F-G
will be the surface of a triangular-threaded screw.