Graphics Reference
In-Depth Information
Although this may look a little artificial right now, it prevents a factor of 2 from enter-
ing equations at a later stage and so we shall use this form of the equation from now
on. In any case, to understand the solution set of our quadratic equation, the idea will
be to transform it into a simpler one. We shall show that in the nondegenerate cases
a simple change in coordinate systems will change the general quadratic equation
(3.35) into one of the equations (3.32), (3.33), or (3.34). The main problem is finding
a change in coordinates that will eliminate the xy term.
We start off by describing two straightforward simple-minded but ad hoc
approaches to solve our problem. The second is actually good enough to give us the
formulas that one uses to convert (3.35) to one of the standard forms. However, there
are also some well-defined invariants associated to (3.35), which can tell us right away
what sort of curve one has without actually transforming the equation. To be able to
prove that these invariants work as specified is what motivates us to present a third,
more elegant approach to analyzing (3.35) using homogeneous coordinates and the
theory of quadratic forms.
The first simple-minded approach uses a form of “completing the square.” Prob-
ably the most straightforward way to do this is to rewrite the equation as
2
2
h
a
ab
-
h
È
Í
˘
˙
2
ax
+
y
+
yfx yc
++ +=
22
0
1
a
1
where a
1
satisfier a = a
2
. The substitution
xax
h
a
¢=
-
y
1
1
y
¢=
y
will then produce an equation in x¢ and y¢ that has no x¢y¢ term. Although this sub-
stitution is satisfactory in some applications and produces simple formulas, it has the
disadvantage that the linear transformation of coordinates to which it corresponds is
not a rigid motion and may deform shapes in undesirable ways. Thus, since the xy
term arises from a rotation of the axes of the conic, a second and “better” way to
eliminate this term is via a rotation about the origin. Consider the substitution
xx
=¢
cos
q
-¢
y
sin
q
yy
=¢
sin
q
+¢
y
cos
q
which corresponds to rotating the conic about the origin through an angle -q. This
substitution will transform equation (3.35) into an equation in x¢ and y¢ for which the
coefficient of the x¢y¢ term is
(
)
(
)
2
2
2
ba
-
sin
qq
cos
+
2
h
cos
q
-
sin
q
.
Setting this expression to zero and using some simple trigonometric identities gives
the equation
