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In-Depth Information
We now substitute the positive value of in Eq. (6.20):
p
=
4
i
+
5
j
−
3
k
+
1110577
i
+
0577
j
−
0577
k
which makes the intersection point, 104 114
−
94.
6.10 A line intersecting a cylinder
Another primitive used in ray tracing is the cylinder. So let's consider a cylinder with radius r
and infinite length with its rotational axis aligned with the z-axis, as shown in Fig. 6.13.
Y
T
t
λ
v
P
p
O
r
Q
v
X
Z
Figure 6.13.
The line equation is given by
p
=
t
+
v
(6.21)
where is a scalar and
1.
The point P x
P
y
P
is the point of intersection, and the line QP is constructed such that it
is perpendicular to the z-axis, which makes its length equal to r.
We can now state that
v
=
2
x
p
+
y
p
=
r
2
r
=
(6.22)
But from Eq. (6.21), we see that
x
p
=
x
t
+
x
v
y
p
=
y
t
+
y
v
and substituting these in Eq. (6.22), we obtain
x
v
2
y
v
2
x
t
+
+
y
t
+
=
r
2
(6.23)
