Graphics Reference
In-Depth Information
Figure 3.22.
Possible line/circle intersections.
3.6.1.5. Theorem.
Let
C
be a nondegenerate conic in
P
2
. Any line
L
in
P
2
intersects
C
in 0, 1, or 2 points. Given any point
p
on
C
, there is one and only one line that inter-
sects
C
in that single point.
Proof.
This fact is easily checked if
C
is a circle. See Figure 3.22. The general case
follows from the fact that any nonempty nondegenerate conic is projectively equiva-
lent to a circle.
Definition.
If a line
L
meets a nondegenerate conic
C
in a single point
p
, then
L
is
called the
tangent line
to
C
at
p
. This definition applies to both the affine and projec-
tive conics.
3.6.1.6. Theorem.
If a nondegenerate conic is defined in homogeneous coordinates
by the equation
T
p
Q
= 0,
then, in terms of homogeneous coordinates, the equation of the tangent line
L
to the
conic at a point
P
0
is
T
p
Q
=
0
.
(3.68)
0
In particular, [
L
] = [
p
0
Q].
Proof.
The line defined by equation (3.68) clearly contains [
p
0
]. It therefore suffices
to show that if another point satisfied equation (3.68), then the conic would be degen-
erate. See [PenP86].
3.6.1.7. Corollary.
The equation of the tangent line at a point (x
0
,y
0
) of a conic
defined by equation (3.35) is
(
)
(
)
ax
++
hy
f x
+
hx
+
by
+
g y
++ +=.
fx
gy
c
0
0
0
0
0
0
0
Proof.
Obvious.
