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.

SURFACES OF SINGLE CURVATURE

139

a straight line and to draw a plane tangent to the cone and parallel
to the line.
328. Problem 204. Given a cone whose axis is oblique to H and V
and whose base is a circle in a plane parallel to V and in front of it;
required to assume a straight line and to draw a plane tangent to
the cone and parallel to the line.
329. Problem 205. To draw a plane tangent to a helical convolute
at a point on the surface.
Analysis. Since the surface of the helical convolute is of single
curvature, the rectilinear element of the surface through the point
of tangency will be the element of tangency, and therefore be a
line of the required plane (see Section 226).
Since the required plane will be tangent to the surface all along the
element of tangency, a straight line tangent to the surface at any
point of this element of tangency will be a line of the required plane.
Construction. Let the helical convolute be represented as in
Fig. 134. A-B represents the axis, D)-E-F-G-K represents the
helical directrix, d-l,-
m,-n,.. represents the /
base of the surface, and a' s
0, assumed as in Section 250, represents the
given point on the /
surface.
The element R-Q-U G 'q'd L
through O is a line of wI
the required plane and
pierces at q, and
pierces V at u', points in
the required traces. I\k'
Through q, draw S- lb
q,-s, tangent to the F
FIG. 134
base of the convolute
at q,. This is a line of the plane, sinc6 it is tangent to a curve of
the surface at a point on O-Q (see Section 226).
Since this tangent is also in H, it is the required horizontal trace.
The straight line through S and u' is the required vertical trace.