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.

RELATING TO POINT, LINE, AND PLANE

41

With m, as a center and with m,-o, as a radius draw the circle
o-o,-o,,. This circle represents the path of the horizontal projection of the point during revolution. Through o' draw the straight
line o'-o"' parallel to G-L. This line represents the path of the
vertical projection of the point during revolution.
Now when O occupies such a position that its horizontal projection falls at o,,, its vertical projection will take the position of in
a straight line through o,, perpendicular to G-L.
In the same way the two projections of 0 in any position of the
circuit may be found.
99. Problem 37. Given the straight line [M = 0, - 6, - 6; N= 0
- 6, - 1] and the point 0 = - 3, 2, 3; required to revolve 0 about
M-N as an axis through arcs of 30 degrees and 160 degrees.
100. Problem 38. Given the straight line [M = O, - 2, 6; N= 0,
- 2, 1] and the point 0 = 3, 4, - 2; required to revolve the point 0
about M-N as an axis through arcs of
45 degrees and 135 degrees. m,
101. Problem 39. Given a straight line r,,,, ----— i t
perpendicular to V, also given a point in
space; required to revolve the point about i
the line as an axis. G ' i n,
Analysis and Construction. In Fig. 51 /- '. - 1
let iM-N represent the given line, and /.
let 0 (o,, o') represent the given point. m n
Since the point 0 must move in a plane /,
perpendicular to MlI-N and therefore par- ', /
allel to V, and since the point 0 must
FIG. 51
retain a constant distance from the axis
Ml-N, the vertical projection of the point, namely o', must move in
the arc of a circle with center at the point where the axis pierces
V; and its horizontal projection o, must move in a straight line o,-o,,
drawn through o, parallel to G-L. Now if the point 0 occupy the
several positions indicated in the diagram, the projections will fall
at o,, o'; or,, o;,,,, o"'; etc.
102. Problem 40. Given the straight line [1 = O, - 1, - 6; N= 0,
- 6,- 6] and the point 0 = - 3,- 4, 2; required to revolve the point 0
about M2-N as an axis through arcs of 30 degrees and 90 degrees.