Graphics Reference
In-Depth Information
Before proceeding with the line-cylinder intersection, let's test the above equations for a
line-ellipsoid intersection. Eric Haines evaluates a line-ellipsoid intersection in
An Introduction
to Ray Tracing
[Glassner, 1989], using a totally different technique. The initial conditions are as
follows:
t
=
4
i
+
5
j
−
3
k
v
=
0577
i
+
0577
j
−
0577
k
a
=
12 b
=
24 c
=
8
and the ellipsoid's center is
69
−
2
The values of are -10.3 and 11.1, where the latter identifies the real point of intersection,
which turns out to be 104114
94.
Using the technique described above, we begin by subjecting the ray to the inverse of the
translation applied to the ellipsoid:
−
⎡
⎤
⎡
⎤
100 6
010 9
001
100
−
6
⎣
⎦
⎣
⎦
010
9
001 2
000 1
−
T
−
1
T
=
=
2
000 1
−
The untransformed ray is
=
+
−
+
+
−
p
4
i
5
j
3
k
0577
i
0577
j
0577
k
(6.20)
and the transformed ray is
p
=−
2
i
−
4
j
−
k
+
0577
i
+
0577
j
−
0577
k
Substituting these values in Eq.(6.19), we obtain
−
2
×
0577
12
2
4
×
0577
24
2
1
×
−
0577
8
2
−
−
−
=
0577
2
8
2
0577
2
12
2
0577
2
24
2
−
+
+
0577
2
12
2
2
2
2
0577 0577
−
0577
−
0577
−
0577 0577
−
0577
2
8
2
0577
2
24
2
2
−
4
−
4
−
1
1
−
2
−
+
+
−
−
−
12
2
×
24
2
24
2
×
8
2
8
2
×
12
2
±
0577
2
8
2
0577
2
12
2
0577
2
24
2
−
+
+
±
√
0008092024
00030052
−
0000016055
−
0000225781
−
0000325125
=
0008092024
±
√
0007525063
0008092024
00030052
=
=−
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