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.

SINGLE CURVED SURFACES

101

244. Problem 151. Given a cone whose axis is oblique to Hi and
V and whose base is a circle on V; required to draw the two projections of the cone and to assume a point upon the surface.
245. Problem 152. Given a cone whose axis is perpendicular to
H, whose vertex is 1 unit below H, and whose right section at the
distance of 8 units below H is a circle 6 units in diameter; required
to draw the two projections of the cone and to assume a point upon
the surface.
246. Problem 153. Given a cone whose base is a circle on H and
whose axis is in a profile plane and oblique to H; required to draw the
two projections of the cone and to assume a point upon the surface.
247. The Convolute. The convolute is a single curved surface
which may be generated by a straight line moving tangentially to
a curve of double curvature. The generating line is called the
generatrix, the curved line is called the directrix, and the various
positions occupied by the generatrix are called the rectilinear
elements of the surface.
Since a rectilinear tangent to a curved line contains two consecutive points of the curve, two consecutive tangents must have a
point in common and therefore intersect. The consecutive elements of the convolute, then, will intersect, and since the directrix
is of double curvature, only those elements which are consecutive
will, in general, intersect.
If the directrix is a helix, the convolute is called a helical
convolute.
248. To represent the Helical Convolute. The helical convolute
is represented in Fig. 87, where for purposes of clearness the distances between consecutive positions of the generatrix are greatly
magnified.
The curve a,-B-C-D-E-Frepresents the helical directrix. The
straight lines a,,-B-C, a,,,-C-D, a,,,,-D-E, etc., represent positions
of the generatrix or rectilinear elements of the surface.
It will be noticed here, as stated above, that only consecutive
elements intersect; for example, a,,,-C-D intersects its preceding
consecutive element, a,,-B-C, at C; it also intersects its succeeding
consecutive element, a,,,,-D-E, at D; but it does not intersect the
element av-E-F, which is not consecutive to it.