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 105

The point x', the point in which a straight line through x, perpendicular to G-L intersects y'-w'-r'-q', is the required vertical
projection.
The greater the number of rectilinear elements assumed, the
more accurately will the point x' be located.
256. The Warped Surface with two Curvilinear Directrices and a
Plane Directer. The warped surface with two curvilinear directrices
and a plane directer is generated by a moving straight line which
always touches two curvilinear directrices and remains parallel to
a plane directer.
257. To assume a Rectilinear Element of a Warped Surface with
Two Curvilinear Directrices and a Plane Directer.
Analysis and Construction. See Fig. 111. Let M-N and O-P
represent the two given curvilinear directrices, and let S represent
the given plane directer.
In the plane S assume a
number of straight lines, as
E-B, E-l, E-2, E —3, etc.
Through any point A on
one of the directrices jM-NL X /
draw the straight lines A-L, i ',
A-K, A-G, A-F, etc., par- -
allel respectively to E-B, /: '
B-1, B-2, B-E3, etc. These
lines through the point A4 I \
lie in a plane parallel to S..> // I'
A-L intersects the hori- I
zontal projecting surface of //
O-P at L; A-K intersects/ /
the same surface at K;,
FIr. 111
A-G intersects the same
surface at G; and so on. L-K-G-F is the intersection of the plane
through the point A with the horizontal projecting surface of O-P.
The point X, in which O-P crosses L-K-G-F, is the point in which
0-P intersects the plane through the point A. The straight line
A-X is the required element, since it is in a plane parallel to S and
touches the two directrices M-N and 0-P.