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.

GENERATION AND CLASSIFICATION OF LINES 81

In the same way let D, E, F, etc., represent consecutive points
of the curve and let d,,, e,,, f,, etc., represent respectively their
horizontal projections.
Connect a, and B by a straight line; also connect a, and 6, by a
straight line. These two lines together with the horizontal projecting line of B form a right-angled triangle in which a,-b, is the
horizontal projection of a,-B.
Practically speaking, the two lines a,-B and a,-b, are identical
with the arcs of which they are chords. For this reason the angle
B-a,-b, measures the slope of the helix, that is, the constant inclination of the curve to H, or the angle which a rectilinear tangent to
the helix at any point makes with H.
Connect B and C by a straight line; also connect b, and c,, by a
straight line. These two lines together with the two projecting
lines B-b, and C-e,, form a quadrilateral in which B-C will make
the same angle with H as that made by a,-B, since the uniform
motions of the generating point give to each elementary portion of
the helix the same inclination to H.
If the plane of the triangle a,-B-b, be revolved about B-b, as an
axis until it coincides with the plane of the quadrilateral B-c,!, the
line a,-b, will take the position a,,-b,, which is a continuation of
c,,-b,; and the line a,-B will take the position a,,-B, which is a
continuation of C-B.
If the plane of the triangle a,,-C-c,, be revolved about C-c,, as
an axis until it coincides with the plane of the quadrilateral C-d,,,
the line a,-b,-c,, will take the position a,,,-bt-c,, which is a continuation of d,,-c,,; and the line a,,-B-C will take the position
a,,,-C, which is a continuation of D-C.
The line a,-B is tangent to the helix at the point B, since it
contains B and its consecutive point C. The line a,,-b, is the horizontal projection of this tangent and is itself tangent to the horizontal projection of the helix at the point b,. The tangent a,,-B
pierces H at a,,, at a distance from b, equal to the rectification of
the arc a,-b.
Again, a,,,-C is tangent to the helix at the point C, since it contains C and its consecutive point D. The line a,,-b6-c~ is the
horizontal projection of this tangent and is itself tangent to the