Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE ELLIPSE

A' of each successive ellipse are farther away from 0, and
their distances from 0, namely,
00' = OA' m
1 - e2 1 — E
increase without limit. Thus, as e approaches 1, the ellipse
approaches as its limit the parabola whose directrix is D and
whose focus is F.
10. New Geometrical Construction for the Ellipse. Parametric Representation. Let it be required to draw an ellipse
when its axes, AA' and BB', are given.
Describe circles of radii a= OA and
b = OB, with the origin 0 as the
common center. Draw any ray from
0, making an angle 4 with the posi-
tive axis of x, as shown in the A' 0 A
figure. Through the points Q and R
draw the parallels indicated. Their \
point of intersection, P, will lie on
the ellipse. For, if the coordinates of FIG. 15
Pbe denoted by (x, y), it is clear that
(1) x = a cos y, y=b sin 4.
From these equations > can be eliminated by means of the
trigonometric identity
sin2 4) + cos2 4 = 1.
Hence
X2 y =1.
(2) + 1
a2 b2
Conversely, any point (x, y) on the ellipse (2) has corresponding to it an angle 4), for which equations (1) are true.
Equations (1) afford what is known as a parametric representation of the coordinates of a variable point (x, y) of the
ellipse in terms of the parameter 4b. When b = a, the ellipse
becomes a circle, and the equations (1) become

(3)

x = =acos, y = asin O.