Graphics Reference
In-Depth Information
6.4 A line intersecting an ellipse in
R
2
section we develop the previous idea of the line-circle intersection with a line-ellipse intersection.
We begin with the equation of an ellipse centered at the origin
x
2
a
2
+
y
2
b
2
=
1
(6.8)
where a and b are the maximum x- and y-radii, respectively.
Y
b
P
v
p
T
t
a
X
Figure 6.5.
Figure 6.5 shows the transformed line and the untransformed ellipse, and we see that the
coordinates of P must satisfy Eq. (6.8). Therefore,
x
v
2
a
2
y
v
2
b
2
x
t
+
y
t
+
+
=
1
Expanding and simplifying gives
b
2
x
t
+
2
x
v
+
a
2
y
t
+
2
y
v
=
a
2
b
2
2x
t
x
v
+
2y
t
y
v
+
2
b
2
x
v
+
a
2
y
v
+
2b
2
x
t
x
v
+
2a
2
y
t
y
v
+
a
2
y
t
+
b
2
x
t
−
a
2
b
2
=
0
which is a quadratic in , and solved using
±
√
B
2
=
−
B
−
4AC
2A
where
b
2
x
v
+
a
2
y
v
A
=
2b
2
x
t
x
v
+
2a
2
y
t
y
v
B
=
a
2
y
t
+
b
2
x
t
−
a
2
b
2
C
=
