Graphics Reference
In-Depth Information
(
)
=¢¢+¢¢
2
2
qxy
¢¢
,
ax
by
.
2
The numbers a¢ and b¢ are just the eigenvalues of the transformation associated to B
and are the roots of the characteristic polynomial for B. Since the change corresponds
to a
linear
change of variables, in this new coordinate system equation (3.35) will
have been transformed into an equation of the form (3.37). We now have all the pieces
of the puzzle.
3.6.4. Theorem.
Define numbers D, D, and I for equation (3.35) by
ahf
hbg
fgc
ah
hb
ab
2
D=
,
D
=
=
-
h
,
and
I
=
a
+
b
.
(1) The quantities D, D, and I are invariant under a change of coordinates via a
rigid motion (translation or rotation).
(2) If Dπ0, then equation (3.35) defines a nondegenerate conic. More precisely,
(a) D > 0: We have an ellipse if ID<0 and the empty set otherwise.
(Note that since a and b have the same sign in this case, the sign
of ID is the same as the sign of bD or aD.)
(b) D < 0: We have a hyperbola.
(c) D = 0: We have a parabola.
(3) If D=0, then equation (3.35) factors into two factors of degree one (the con-
verse is also true) and defines the empty set, a point, or a pair of lines. The
pair of lines may be parallel, intersecting, or coincident. More precisely,
(a) D > 0: We get a single point.
(b) D < 0: We get two intersecting lines.
(c) D = 0:
b π 0: There are three cases depending on E = g
2
- bc.
E > 0: We get two parallel lines.
E < 0: We get the empty set.
E = 0: We get a single line.
b = 0: Then h = 0. There are three cases depending on F = f
2
- ac.
F > 0: We get two parallel lines.
F < 0: We get the empty set.
F = 0: We get a single line.
Proof.
Part (1) follows from properties of the determinant and the trace function (I
is the trace of the matrix B). The main ideas behind the proof of (2) have been sketched
above. The condition on the product ID in (2)(a) is equivalent to saying that I and D
have opposite signs. The effect that the signs of I and D have on the conic is best seen
