Graphics Reference
In-Depth Information
2
2
x
a
y
b
The ellipse:
+=
1
,
ab
≥>
0
(3.33)
2
2
Focus:
(c, 0)
2
a
c
(
)
Directrix:
x
=
if
c
>
0
c
a
, where c ≥ 0 and c
2
= a
2
- b
2
Eccentricity:
e
=
The segments [(-a,0),(a,0)] and [(0,-b),(0,b)] are also sometimes called the
major
and
minor axis
of the ellipse, respectively, or simply the
principal axes
.
2
2
x
a
y
b
The hyperbola:
-=
1
,
ab
,
>
0
(3.34)
2
2
Focus:
(c, 0)
2
a
c
Directrix:
x
=
c
a
, where c
2
= a
2
+ b
2
Eccentricity:
e
=
The segments [(-a,0),(a,0)] and [(0,-b),(0,b)] are also sometimes called the
trans-
verse
and
conjugate axis
of the hyperbola, respectively. The
asymptotes
of the
hyperbola are the lines y =±bx/a.
Because of the symmetry present in the case of a hyperbola and ellipse, they actu-
ally have two foci and directrices. The second of the pair is obtained by reflecting the
ones given above about the y-axis.
The discussion above shows that conic sections are solutions to quadratic equa-
tions. To connect the two we start from the other direction.
An (
affine
)
conic
is any subset of
R
2
Definition.
defined by an equation of the
form
2
2
ax
+++++=,
bxy
cy
dx
ey
f
0
where (a,b,c) π (0,0,0).
Our object is to show that the terms “conic” and “conic section” refer essentially
to the same geometric spaces. To that end it is convenient to rewrite the equation
defining a conic in the form
2
2
ax
+
2
hxy
+
by
+
2
fx
+
2
gy
+
c
=
0
.
(3.35)
