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 75

If the generating point moves in a plane, and in such a way that
its distance from a fixed point in the plane is equal to its distance
from a fixed straight line in the plane, it will generate a curve of
single curvature called the parabola.
If the generating point moves in a plane, and in such a way that
the difference of its distances from two fixed points in the plane is
a constant quantity, it will generate a curve of single curvature
called the hyperbola.
If the generating point moves in such a way as to retain a constant distance from a fixed straight line, and to have a uniform
motion both around and in the direction of the straight line, it
will generate a curve of double curvature called the helix.
206. Representation of Curves. Curved lines like straight lines
may be represented by their projections on H and V, and when so
represented they are in general definitely determined.
These projections are the loci of the projections of the generating point in its consecutive positions, and are usually curved lines.
If the curve is of single curvature and its plane is parallel to
the plane of projection, its projection upon this plane is a curve of
the same character and magnitude as the original curve.
If the curve is of single curvature and its plane is perpendicular to
the plane of projection, its projection upon this plane is a straight line.
The projection of a circle upon a plane to which the plane of
the circle is oblique is an ellipse.*
The projection of an ellipse upon a plane to which the plane of
the ellipse is oblique is either a circle or an ellipse.*
The projection of a curve of double curvature is always a curve
whatever the relation of the curve to the plane of projection. The
projection of the helix upon a plane perpendicular to its axis is a
circle.
207. Tangents to Curves. A straight line is tangent to a curve
at a given point when it represents the rectilinear path in which
the generating point is moving at the instant it passes through the
point of tangency.
* The truth of these statements will be more apparent after the student has
obtained some knowledge of the nature of the intersections of cylindrical surfaces
by planes. See Chapter XVI.