Graphics Reference
In-Depth Information
which reveals the following values for :
x
v
a
2
+
2
x
v
y
v
x
t
y
t
x
t
x
v
a
2
b
2
y
v
b
2
−
y
t
y
v
−
+
±
a
2
b
2
=
(6.9)
x
v
a
2
+
y
v
b
2
The value of the discriminant of Eq. (6.9) determines whether the line intersects, is tangential,
or misses the ellipse:
Miss condition
2
x
v
y
v
x
t
y
t
x
v
a
2
+
y
v
b
2
−
< 0
a
2
b
2
Tangential condition
2
x
v
y
v
x
t
y
t
x
v
a
2
+
y
v
b
2
−
=
0
a
2
b
2
Intersect condition
2
x
v
y
v
x
t
y
t
x
v
a
2
+
y
v
b
2
−
> 0
a
2
b
2
We can check the validity of Eq. (6.9) by setting a
=
b
=
r, which reveals
x
v
r
2
+
2
x
v
y
v
x
t
y
t
x
t
x
v
r
2
y
v
r
2
−
y
t
y
v
r
2
−
+
±
r
4
=
x
v
r
2
+
y
v
r
2
2
x
v
y
v
x
t
y
t
=−
x
t
x
v
+
y
t
y
v
±
r
2
−
which is the same as Eq. (6.7).
