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.

142

DESCRIPTIVE GEOMETRY

planes, the one tangent to the cone and the other tangent to the
convolute, along such corresponding parallel elements, will be
parallel planes.
Through the given line pass a plane tangent to this auxiliary
cone. Such a plane will be parallel to the required plane, since
it contains the given line and an element on the cone which is
parallel to the element of tangency on the convolute.
Construction. Let the helical convolute be represented as in
Fig. 136. A-B represents the axis and M-N represents the given
line.
Through n', the point in which '1-NT pierces V, draw n^-ff parallel to e'-d', where e'-d' is the vertical projection of an element
a,,
/ e, d, /
/ /A
_ /
SfI
FIG. 136
of the convolute parallel to V. The elements of the right circular
cone generated by the revolution of the line nt-f about the vertical line through n' will make the same angle with H as the elements
of the convolute. The circle whose center is n, is the base of the
cone and n' is the vertex of the cone.
Through M-N and tangent to the cone draw the plane U (see
Section 321). The horizontal trace U-u, will be tangent to the
circular base. The vertical trace U-u' will pass through n'.