Graphics Reference
In-Depth Information
6.11 A line intersecting a cone
Another ray-tracing primitive is the cone, and once again this is normally aligned, like the
cylinder, with the z-axis, as shown in Fig. 6.16. The simplest form of the cone equation is
z 2
x 2
y 2
=
+
which makes the internal angle of the apex 90 .
Y
T
λ v
t
P
p
X
Z
Figure 6.16.
If the line equation is given by
=
+
p
t
v
where is a scalar and
1.
Then for an intersection at P, we have
v
=
z P =
x P +
y P
which means that
z v 2
x v 2
y v 2
z t +
=
x t +
+
y t +
Expanding and simplifying gives
z t +
2 z v =
x t +
2 x v +
y t +
2 y v
2z t z v +
2x t x v +
2y t y v +
2 x v +
z v +
y v
x t +
y t
z t
0
=
2x t x v +
2y t y v
2z t z v
+
which is a quadratic in and solved using
± B 2
=
B
4AC
2A
 
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