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.

110

DESCRIPTIVE GEOMETRY

Lay off upon a'-b' downward from c' the distance c'-cv equal to
the total fall of the generatrix per revolution, and through cV draw
cV-dv parallel to c'-d'. The line cV-dV is the vertical projection of
the generatrix at the end of one circuit.
The point D (d,, d'), which is the point in which the generatrix
in its original position pierces H, will generate a helix whose
A/1,~~ ~horizontal projection is the circle
- — \ d - d, —d,,,-d,,,,-., and whose ver/ j A \ tical projection d'-d"-d""-d"-d
a j. (,d, may be found by Section 214.:\; \ i/ '/ When the horizontal projection
\xt,! \ ' /! ~of the point D takes the position.'/? /'"<-^- d,, or when the horizontal projec-G / -- 'd' - -L tion of the generatrix takes the,\\,> '^ /.position c,-d,, the vertical projecr\il t / tion, d", of D in this position will
\>cVg / ~ fall upon the vertical projection of
\\ /C" - the helix generated by D and on
\CrY / the straight line through d, perNc""/ pendicular to G-L.
This shows that in moving from
c"/ the first position to this position,b all points of the generatrix have
moved downward a distance equal
FIG. 114 to the distance of d" below G-L.
Therefore, to obtain the vertical projection of C for this position of
the directrix, lay off upon.a'-b' downward from c' a distance c'-c"
equal to the distance of d" below G-L. The point c" is the vertical projection sought, and the straight line c"-d" is the vertical
projection of the generatrix in this position. In the same way we
may find the horizontal and vertical projections of the generatrix
in other positions, as may be seen from the diagram.
The generatrix in its first position pierces H at d,, in the second
position it pierces H at e,, in the third position it pierces H at
f, etc. Since these various positions of the generatrix represent
elements of the surface, the curve d,-e,-f,-g,-~. will represent
the base of the surface.