Graphics Reference
In-Depth Information
Far plane
v
C
Near plane
Up
A
B
P
Look
u
w
f
n
Figure 13.6: The
uvw
frame for a camera, the
look
and
vup
vectors, and the points P
,
A
,
B
,
and C.
In writing the equations, we'll use
vup
and
look
to indicate the up vector
and look direction, respectively; these shorter names make the equations more
comprehensible. The point
P
is just a shorter name for the camera's
Position
.
We'll build the orthonormal basis,
u
,
v
,
w
, in reverse order. First,
w
is a unit
vector pointing opposite the look direction, so
w
=
−
look
=
S
(
look
)
.
(13.2)
look
To construct
v
, we first project
vup
onto the plane perpendicular to
w
, and
hence perpendicular to the
look
direction as well, and then adjust its length:
v
=
vup
−
(
vup
·
w
)
w
(13.3)
v
v
=
=
S
(
v
)
.
(13.4)
v
Finally, to create a right-handed coordinate system, we let
u
=
v
×
w
.
(13.5)
Inline Exercise 13.1:
Some camera software (like Direct3D, but not OpenGL)
starts by letting
w
=
S
(
look
)
, without negation.
(a) Show that this makes no difference in the computation of
v
.
(b) Show that in this case, if we want
u
,
v
to retain the same orientation on the
view plane (i.e.,
u
pointing right,
v
pointing up), then the computation of
u
becomes
u
=
w
v
.
(c) Is the resultant
uvw
-coordinate system right- or left-handed?
×
Now we'll compute the four points
P
,
A
,
B
, and
C
. The only subtlety con-
cerns determining the length of edges
AB
and
AC
. The edge
AB
subtends half the
horizontal field of view at
P
, and is at distance
f
from
P
,so
tan
θ
h
2
=
AB
f
, so
(13.6)
AB
=
f
tan
θ
h
2
,
(13.7)
θ
h
denotes the horizontal field of view angle, converted to radians,
θ
h
=
FieldOfView
180
,
where
(13.8)
