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.

82

DESCRIPTIVE GEOMETRY

horizontal projection of the helix at the point c,,. The tangent
a,,-C pierces H at a,,,, at a distance from c,, equal to the rectification
of the arc a,-b,-c,.
That portion of the tangent a,,-B between the points a,, and B
is the rectified length of the curve a,-B, and that portion of the
tangent a,,,-C between the points a,,, and C is the rectified length
of the curve a,-B-C.
We have now considered three consecutive points, a,, B, and C,
of the helix, and it will be evident that the same process of reasoning may be applied indefinitely to the consecutive points of the
curve.
We may conclude, then, that the horizontal projection of a rectilinear tangent to a helix at any point on the curve, provided the
axis of the helix is perpendicular to H, will be tangent to the horizontal projection of the helix at the horizontal projection of the
point of tangency; also that the tangent itself will pierce H upon
the horizontal projection of the tangent and at a distance from the
horizontal projection of the point of tangency equal to the rectification of that portion of the horizontal
projection of the helix between the point
l ab, d, in which the helix pierces H and the
horizontal projection of the point of. i,!tangency.
" 'a' i ' 8 d' a L By use of these principles we can draw..:',,,. the two projections of a rectilinear tangent to the helix when the axis of the
helix is assumed perpendicular to H.:^^", Construction. Let the helix be represented as in Fig. 88. Assume any point,
-b....-. -j as E, upon the curve (see Section 215).
At e, draw e,-f, tangent to the horizontal
FIG. 88 projection of the helix. Upon this tangent lay off from e, a distance el-f, equal to the rectification of
the arc e,-c,, where d, is the point in which the helix pierces H,
and where e, is the horizontal projection of the point of tangency.
The line e,-f, is the horizontal projection of the required tangent,
and f, is the point in which this tangent pierces H. The vertical