Graphics Reference
In-Depth Information
string around the near branch of the hyperbola with vertex
F
1
(at point
A
in the figure),
then around the outside of the ellipse (at point
B
in the figure), and finally attach it
to
F
2
. If we pull the string tight at some point
P
, then the locus of those points
P
will
trace out one quarter of an ellipsoid. By attaching the string at corresponding differ-
ent locations we can get the rest of the ellipsoid. Because of the similarity of this con-
struction with that for the ellipse, one makes the following definitions.
Definition.
The ellipse and hyperbola used to construct the ellipsoid are called the
focal curves
(the
focal ellipse
and
focal hyperbola
) of the ellipsoid. In general, given any
quadric, we say that two conics in orthogonal principal planes for this quadric are
focal curves
for the quadric if they are confocal with the intersection of the principal
planes with the quadric. Two quadric surfaces with the same focal curves are called
confocal
.
Only ellipsoids and hyperboloids of one or two sheets have focal curves. The family
of all confocal quadrics of one of those three types that have a fixed pair of focal curves
fill up all of space. The tangent planes of the three just-mentioned confocal families
are mutually orthogonal at a point of intersection. (For a precise definition of a
tangent plane see Chapter 8.)
Next, we state the analog of Theorem 3.6.4 for the surface case. We rewrite
equation (3.69) as
2
2
2
ax
+++ + + ++ ++=.
by
cz
2
hxy
2
fyz
2
gzx
2
lx
2
my
2
nz
d
0
(3.71)
Define matrices A and B by
ahf l
hbgm
fgcn
lmnd
Ê
ˆ
ahf
hbg
fgc
Ê
ˆ
Á
Á
Á
˜
˜
˜
Á
Á
˜
A
=
and
B
=
˜
.
(3.72)
Ë
¯
Ë
¯
3.7.2. Theorem.
Define D, D, I, and J for equation (3.71) by
ahf l
hbgm
fgcn
lmn f
ahf
hbg
fgc
()
=
()
=
D=
det
A
,
DB
=
det
,
()
=++
2
2
2
I
=
tr B
a
b
c
,
and
J
= + + - - -
bc
ca
ab
f
g
h
.
(1) The quantities D, D, I, and J are invariant under a change of coordinates via
rigid motions (translations and/or rotations).
(2) If D π 0, then let r
1
, r
2
, and r
3
be the nonzero eigenvalues of the matrix B in
(3.72).
