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.

76 DESCRIPTIVE GEOMETRY
In Fig. 80 let A-B-D represent a curve generated by a point
moving under some law. Suppose that when the generating point
reaches the position B, the law under which it is moving ceases to
act and the point is allowed to move freely in the direction B-,,
in which it is moving at this instant. The straight line B-E is
tangent to the curve at the point B.
Or, a straight line is tangent to a curve at a given
point when the line contains the given point and its conD secutive point. In Fig. 81 let B represent any point on
1B the curve A-B-D. Through B draw any secant, as B-F,
cutting the curve at B and F. Now if the point B remain
fixed and the point F be made to move along the curve
toward B, the secant will gradually approach the position
of a straight line tangent to the curve at the point B.
FIG. 80 Finally, when the point F has taken the position consecutive to B, or practically the position B itself, the line B-F will
take the position B-E and be tangent to the curve at the point B.
Two curves are tangent to each other when they have two consecutive points in common, or when they are both tangent to the
same line at a common point.
A straight line which is tangent to a curve of
single curvature lies in the plane of the curve. G/
If two lines, straight or curved, are tangent to
each other, their projections will also be tangent
to each other, since the projections of the two B
consecutive points common to the two lines will
also be consecutive and be common to the two
projections of these lines. \
208. Problem 141. To draw a rectilinear tangent
FIG. 81
to an ellipse at a point assumed on the curve.*
Construction.f In Fig. 82 let A-B and D-E represent the axes
of the ellipse, let F and F' represent the foci of the ellipse, and
let P represent the point assumed on the curve.
* The method of representing the ellipse, the parabola, and the hyperbola is
fully explained in text-books on elementary mechanical drawing.
t The proof of the methods here given for drawing rectilinear tangents to the
ellipse, the parabola, and the hyperbola may be found in text-books on analytic
geometry.