Graphics Reference
In-Depth Information
where
λ
is a scalar, and
v
= 1
(12.13)
For an intersection at
P
2
=
r
2
or
2
r
2
=0
q
=
r
or
q
q
−
Using the cosine rule
2
=
2
+
2
q
λ
v
s
−
2
λ
v
s
cos(
θ
)
(12.14)
2
=
λ
2
2
+
2
q
v
s
−
2
v
s
λ
cos(
θ
)
substituting (12.13) in (12.14)
2
=
λ
2
+
2
q
s
−
2
s
λ
cos(
θ
)
(12.15)
identify cos(
θ
)
.
v
=
s
v
cos(
θ
)
s
therefore
.
cos(
θ
)=
s
v
s
(12.16)
substituting (12.16) in (12.15)
2
=
λ
2
.
2
q
−
2
s
v
λ
+
s
therefore
2
r
2
=
λ
2
.
2
r
2
= 0
q
−
−
2
s
v
λ
+
s
−
(12.17)
(12.17) is a quadratic where
(
s
.
v
±
.
v
)
2
λ
=
s
−
s
2
+
r
2
(12.18)
and
s
=
c
−
t
the discriminant of (12.18) determines whether the line intersects, touches or
misses the sphere.
The position vector for
P
is given by
p
=
t
+
λ
v
where
(
s
.
v
±
.
v
)
2
λ
=
s
−
s
2
+
r
2
and
s
=
c
−
t
