Graphics Reference
In-Depth Information
Hyperbola
Parabola
Ellipse
Circle
(a)
(b)
(c)
(d)
Figure 3.18.
Conic sections.
directrix
|pf|
dist (p,L)
= e = eccentricity
p
f
focus
Figure 3.19.
Some notation for conics.
L
distance of each of its points from a fixed point
p
to the distance of the point from a
fixed line
L
not passing through
p
is constant.
Proof.
See [Eise39].
Definition.
Using the notation in Theorem 3.6.1 with respect to some noncircular,
nondegenerate conic section
C
, the fixed point
p
is called the
focus
, the fixed line
L
is called the
directrix
, and the constant e is called the
eccentricity
of
C
. The conic
section is called an
ellipse
,
parabola
, or
hyperbola
depending on whether e < 1, e = 1,
or e > 1, respectively. Ellipses and hyperbolas are often called
central conic sections
.
Define the
focus
of a circle to be its center and its
eccentricity
to be 0. (We can make
the definition of a circle match that of the other conic sections completely by think-
ing of the
directrix
of a circle as a line at infinity and the limiting case for an ellipse
where we let its directrix move further and further away from the focus.)
See Figure 3.19. Conic sections can also be constructed via string constructions.
See [HilC99]. For example, we can trace out an ellipse by tying the two ends of a string