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.

60

DESCRIPTIVE GEOMETRY

Principle. The projection of a straight line upon a plane is the
line in which a plane drawn through the given line perpendicular
to the given plane intersects the given plane (see Section 23).
Analysis 1. If through any point of the given line we draw a
straight line perpendicular to the given plane, the plane of the
given line and this auxiliary line will be the projecting plane of
the given line, and will intersect the given plane in the required
projection.
Analysis 2. Since the projection of a straight line upon a plane
is a straight line and contains the projections of all the points of
the given line, the required
S. s ~n1, ~ projection is the straight line
\ /l X determined by the projections
~ bm, \ of any two points of the
given line.
Construction. See Fig. 70.
~/ Y I ^ ^e, BLet [I-N represent the given
CG- g!-/ - Aj -( — ~> L line and let S represent the
epCI'' given plane. Following AnF, Y \ x alysis 2, project upon the
[/1/ /1, \A 'R given plane any two points
/, P'i of the given line, as M and
i/ /- N. The projection of M
a! FIG. 70 upon S is MS, horizontally
projected at ms, and vertically projected at ms' (see Sections 5 and 151).
The projection of N upon S is NS, horizontally projected at ns,
and vertically projected at ns'. Therefore ms,-ns, is the horizontal
projection and ms'-ns' is the vertical projection of the required
projection.
Check. Assume a third point O upon the given line. Project
this point upon S, and note if this projection falls upon the
projection already found.
162. Problem 100. Project the line [M= - 2, - 3, - 8; N= 3,
-9,- 4] upon the plane S -- 8, 315~, 330~, following Analysis 1.
163. Problem 101. Project the line [M = -6, 6, 8; N= 6, - 3,
6] upon the plane S = 0~, -3, 6.