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

45

to H and makes an angle of 45 degrees with V, are m,-n,, and
m-nfn, where n,-n, is 4 units long and makes an angle of
45 degrees with G-L, and where mf-n" is parallel to G-L.
When the line is revolved to its final position, n" will move in the
arc of a circle to n', and n,, will move in a straight line parallel to
G-L, to n,. The required projections are then m,-n, and m'-n'.
The line just located is in the third quadrant and runs from the
given point upward to the left and away from V.
Consider other positions which the line may occupy and yet
satisfy the requirements of the problem.
115. Problem 53. Construct Problem 52 when 1= — 3, 6, - 2.
116. Problem 54. Construct Problem 52 when l = 4, 6, 4.
117. Problem 55. Construct Problem 52 whean 71=- 4, - 6, 2.
118. Problem 56. Draw the two projections of a straight line 6
units long, passing through a point M1= O, -3, -1, making an
angle of 60 degrees with H, and lying in a profile plane.
Analysis. First draw the projections of the line when it passes
through the given point, is parallel to V, and makes the required
angle with H. Then revolve the line about an axis through the
given point perpendicular to 1I until the horizontal projection of
the line makes the proper angle with G-L (see Section 104).
119. Problem 57. Given a rectangular card whose dimensions
are 6 units by 4 units, whose surface is parallel to V, whose long
edges are perpendicular to H, and whose upper right-hand vertex is
a point B = 2 -, -1, -1; required (1) to draw the two projections
of the card in the given position, and (2) to revolve the card about its
right-hand vertical edge as an axis, through angles of 30 degrees,
45 degrees, 60 degrees, and 90 degrees, and to draw the corresponding projections.
Construction. See Fig. 56. The projections of the card in its
first position are a,-b, and a'-b'-d'-e', where b, and b' are the projections of B, where a'-b' is 4 units long, and where a'-e' is 6
units long.
When the card is revolved about B-D as an axis through an
angle of 30 degrees, the two projections of the card are a,,-b, and
a"-b'-d'-e", where the angle a,-b,-a,, is 30 degrees, and where a"t
and e" are in a straight line through a,, perpendicular to G-L.