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.

104

DESCRIPTIVE GEOMETRY

connect A and B by a straight line. A-B is an element of the surface, since it touches the two directrices and lies in the plane T
which is parallel to the plane directer.
255. To assume a Point upon the Surface of the Hyperbolic
Paraboloid.
Analysis 1. First assume an element of the surface and then
assume a point upon this element.
Analysis 2. Assume at random the horizontal projection of the
point. Through the horizontal projecting line of the point draw
a plane and find the intersection of this plane with
Q n, 0 /. < athe surface. This inter~-f' ~'- 4'^/- section must contain the, -a ' -^ -'-~b_- point upon the surface,
7rn-,K ---- ^ \_ and the vertical projection
so<i.. of the point must lie both
a m mit T- -—. —t-a - L 1upon the vertical projec' ' -.......p- tion of this intersection
" -— k..L-~_: _/y~::~ i!! i~b, and upon the vertical pro-;t - w e2'e j- jection of the horizontal:,\ o~~ projecting line of the
^^. q \ point.
0':~4\ Construction. See Fig.
110. Let M-N and O-P
FIG. 110
represent the two rectilinear directrices, and let N-O and M-P represent two rectilinear
elements of the surface.
Any plane parallel to the two lines N-0 and M-P is the plane
directer of the surface.
Assume at random x, as the horizontal projection of the required
point. Through the horizontal projecting line of X pass any plane,
as S.
By Section 254 assume a number of rectilinear elements of the
surface in the vicinity of X. These elements are A-B, D-E, F-G,
K-L, etc. These elements, beginning with A-B, intersect the
plane S in the points Y, W, R, Q, etc. The line Y-W-R-Q is the
intersection of S and the warped surface, and must contain X.