Graphics Reference
In-Depth Information
Let the reflecting line be defined as
p
=
t
+
v
(5.1)
where is a scalar, and the line's normal vector is
v
⊥
n
=
The incoming ray passes through R with direction
v
in
, and the reflected outgoing ray originates
at P with direction
v
out
.
From Fig. 5.1 we observe that
=
v
in
+
w
n
(5.2)
and
w
=
v
out
−
n
(5.3)
where is a scalar.
Equating (5.2) and (5.3) gives
v
in
+
n
=
v
out
−
n
and
v
out
=
v
in
+
2
n
(5.4)
We now need to find .
We multiply Eq. (5.4) throughout by
n
and obtain
n
·
v
out
=
n
·
v
in
+
2
n
·
n
but
n
·
v
out
=
n
·
−
v
in
=−
n
·
v
in
Therefore,
−
n
·
v
in
=
n
·
v
in
+
2
n
·
n
and
n
·
v
in
=−
2
(5.5)
n
Substituting Eq. (5.5) in Eq. (5.4) gives
2
n
n
·
v
in
v
out
=
v
in
−
(5.6)
2
n
