Graphics Reference
In-Depth Information
Note in the two examples how the frame that defines the camera coordinate
system also defines the transformation from world coordinates to camera coordinates
and conversely. The frame is the whole key to camera coordinates and look how simple
it was to define this frame!
The View Plane Coordinate System.
The origin of this coordinate system is the
point in the view plane a distance d directly in front of the camera and the x- and
y-axis are the same as those of the camera coordinate system. More precisely, if
(
u
1
,
u
2
,
u
3
,
p
) is the camera coordinate system, then (
u
1
,
u
2
,
p
+d
u
3
) is the view plane
coordinate system.
4.3
Vanishing Points
If there is no clipping, then after one has the camera coordinates of a point, the next
problem is to project to the view plane z = d. The central projection p of
R
3
from the
origin to this plane is easy to compute. Using similarity of triangles, we get
(
)
=¢
(
)
=
(
)
p xyz
,,
x y d
, ,
¢
dxzdy z d
,
, .
(4.5)
Let us see what happens when lines are projected to the view plane. Consider
a line through a point
p
0
=
(x
0
,y
0
,z
0
), with direction vector
v
=
(a,b,c), and
parameterization
()
=
(
()
() ()
)
=+
pt
xt yt zt
,
,
pv
t
.
(4.6)
0
This line is projected by p to a curve p¢(t) = (x¢(t),y¢(t),d) in the view plane, where
d
x t
z t
+
+
d
y t
z t
+
+
0
0
0
0
¢
()
=
¢
()
=
xt
and
y t
;
(4.7)
It is easy to check that the slope of the line segment from p¢(t
1
) to p¢(t
2
) is
¢
()
-
()
¢
()
-
()
=
yt
yt
yc bz
xc az
-
-
2
1
0
0
,
xt
xt
2
1
0
0
which is independent of t
1
and t
2
. This shows that the curve p¢(t) has constant slope
and reconfirms the fact that central projections project lines into lines (but not
necessarily onto).
Next, let us see what happens to p¢(t) as t goes to infinity. Assume that c π 0. Then,
using equation (4.7), we get that
(
¢
()
¢
()
)
=
(
)
lim
xt yt
,
dacdbc
,
(4.8)
t
Æ•
This limit point depends only on the direction vector
v
of the original line. What this
means is that all lines with the same direction vector, that is, all lines parallel to the
