Graphics Reference
In-Depth Information
n
Q
v
P
λ
v
ˆ
T
α
n
p
t
O
Figure 3.13.
First we let
−
TQ
=
v
−
TP
=
v
and
−
OQ
=
n
ˆ
From Fig. 3.13 we can state
−
OQ
=
−
OT
+
−
TQ
(3.18)
Substituting vector names in Eq. (3.18), we obtain
ˆ
=
+
n
t
v
(3.19)
As
n
is perpendicular to
v
, we have
ˆ
n
ˆ
·
v
=
0
Multiplying Eq. (3.19) throughout by
n
, we obtain
ˆ
n
ˆ
·ˆ
n
=ˆ
n
·
t
+ˆ
n
·
v
Therefore,
n
ˆ
·ˆ
n
=ˆ
n
·
t
and
=ˆ
n
·
t
(3.20)
We can regard
p
as being a linear interpolation of
t
and
n
(see Section 2.11):
ˆ
p
=
1
−
t
+
n
ˆ
(3.21)
