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.

38

DESCRIPTIVE GEOMETRY

radius of the arc in which the point moves, will be equal to the
distance of the vertical projection of the point from G-L.
88. Problem 26. Given the straight line [M= -6, 6, 0; N= 6, - 2,
0] and the point 0 = 0, 6, 5; required to revolve 0 about 3V-N into H.
89. Problem 27. Given the straight line [M= - 6, - 6, 0; N= 6, 2,
0] and the point 0 = 0,- 6, 4; required to revolve 0 about M-Ninto H.
90. Problem 28. Given the straight line [X =- 6, 2, 0;.V= - 6,
- 4, 0] and the point = 0, 4, - 6; required to revolve 0 about
MT-N into H.
91. Problem 29. Given a straight line in V and a point in space;
required to revolve the point about the line as an axis into V.
Analysis. From what has been said in connection with Problem
25, it will be evident that the vertical trace of the plane of revolution must pass through the vertical projection of the given point
and must be perpendicular to the given line; also that the radius
of the arc in which the point moves will be equal to the hypotenuse

0o,
m,
m,

d I
I I,
e c /
Iv
FIG.

of a right-angled triangle whose base
is equal to the distance of the vertical projection of the point from the
L line, and whose altitude is equal to the
distance of the horizontal projection
s,, of the point from G-L.,! -~ Construction. In Fig. 46 let M-N
represent the line in V and let 0 represent the point in space. Through
o' draw o'-a' perpendicular to mn'-n'.

This must be the vertical trace of the plane of revolution. Make
a-oV equal to the hypotenuse of a right-angled triangle c-d-e, whose
base c-d is equal to o'-a' and whose altitude c-e is equal to b-o,.
It will be noticed, as in Problem 25, that if the point 0 is so
situated that its vertical projection falls on mn'-n', the base of the
triangle will vanish and the hypotenuse will be equal to the altitude. In such a case then the radius of the arc of revolution
will be equal to the distance of the horizontal projection of the
point from G-L.
92. Problem 30. Given the straight line [M= - 6, 0, - 5; N= 6, 0,
4] and the point 0 = 0, 4, 6; required to revolve 0 about M1-N into V.