Graphics Reference
In-Depth Information
For a miss condition
.
v
)
2
2
+
r
2
<
0
(
s
−
s
For a touch condition
.
v
)
2
2
+
r
2
=0
(
s
−
s
For an intersect condition
.
v
)
2
2
+
r
2
>
0
(
s
−
s
To test these formulae we will create all three scenarios and show that the
equations are well behaved.
Figure 12.13 shows a sphere with three lines represented by their direction
vectors
λ
v
1
,λ
v
2
and
λ
v
3
.
The sphere has radius
r
= 1 and is centred at
C
with position vector
c
=
i
+
j
whilst the three lines
L
1
,L
2
and
L
3
miss, touch and intersect the sphere
respectively.
The lines are of the form
p
=
t
+
λ
v
therefore
p
1
=
t
1
+
λ
v
1
p
2
=
t
2
+
λ
v
2
p
3
=
t
3
+
λ
v
3
where
1
√
2
i
+
1
√
2
j
t
1
=2
iv
1
=
t
2
=2
iv
2
=
j
1
√
2
i
+
1
√
2
j
t
3
=2
iv
3
=
−
Y
P
3
¢
l
v
2
l
v
1
l
v
3
r
C
P
2
P
3
c
T
t
X
Z
L
1
L
3
L
2
Fig. 12.13.
Three lines that miss, touch and intersect the sphere.
