Graphics Reference
In-Depth Information
Therefore,
1
−
0
+
0
−
0
1
2
=
=
1
+
1
and
1
2
1
2
x
P
=
y
P
=
which is correct.
2
6.3 A line intersecting a circle in
R
Now let's consider the case of a line intersecting a circle in
2
, for which there are three possible
scenarios: the line intersects the circle at two points; the line is tangential to the circle at a single
point; or the line does not intersect the circle at all.
The equation of a circle whose center is located at the origin is given by
R
x
2
y
2
r
2
+
=
(6.6)
where r is the radius. But in reality, the circle could be located away from the origin at x
C
y
C
,
in which case the circle's equation becomes
x
C
2
y
C
2
−
+
−
=
r
2
x
y
However, as mentioned above, we can preserve the simplicity of the original circle equation
by translating the line equation in the opposite direction. So, for the moment, let's investigate
the intersection of a line with Eq. (6.6), assuming that the line equation will be subject to the
inverse of the transform applied to the circle.
We begin with the usual line equation
p
=
t
+
v
where is a scalar and
v
=
1.
Y
P
λ
v
p
T
t
X
Figure 6.3.