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(a) D<0:
I D > 0, J > 0: We have an ellipsoid.
Not both I D > 0 and J > 0: We have an hyperboloid of two sheets.
(b) D>0:
I D > 0, J > 0: The empty set.
Not both I D > 0 and J > 0: We have an hyperboloid of one sheet.
(c) D=0:
I D > 0, J > 0: A single point.
Not both I J > 0 and J > 0: We have a cone.
In case (a) and (b), equation (3.71) can be reduced to
D
2
2
2
rx
+
r y
+
rz
+
=
0
.
1
2
3
D
and in case (c) to
2
2
2
rx
+
r y
+
rz
=
0
.
1
2
3
(3) If D = 0, then equation (3.71) defines a paraboloid or cylindrical surface gen-
erated by a conic in a plane (types (8) and (12) in Theorem 3.7.1) unless the
surface is degenerate. More precisely, let r i be the nonzero eigenvalues of the
matrix B in (3.72).
(a) Dπ0:
Equation (3.71) can be reduced to
-
D
2
2
rx
+
r y
+
2
rr z
=
0
.
1
2
12
D<0, J > 0: We have an elliptic paraboloid.
D>0, J < 0: We have an hyperbolic paraboloid.
(b) D=0:
If
ah l
hbm
lmd
af l
fcn
lnd
bgm
gcn
mn d
=
=
= 0,
then the surface is degenerate and reduces to a plane or a pair of planes.
Otherwise, it is a cylindrical surface of one of the following types:
J > 0: a real elliptic cylinder or the empty set.
J < 0: a hyperbolic cylinder.
J = 0: a parabolic cylinder.
Alternatively, let k be the number of nonzero eigenvalues of the matrix B
in (3.72). Then equation (3.71) can be reduced to one of the following:
 
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