Graphics Programs Reference
In-Depth Information
radii of the two circles. If a point
Q
happens to be located 30% of the distance between
circles
A
and
B
, then it is moved 30% of the distance between circles
a
and
b
. Its scale
factor is [(1
0
.
3)
r
A
+0
.
3
r
B
]. This simplifies the computations but
introduces inaccuracies because the interpolation between circles is linear. However,
the inaccuracies can be reduced as much as desired by precomputing the radii of more
circles.
The circle that corresponds to 89
◦
is large and the circle for 90
◦
has infinite radius.
Points whose
θ
is between 89
◦
and 90
◦
will be moved to the radius-
k
circle and placed in
the narrow region between the (89
/
2)
◦
circle and the outer edge. Such points increase
the inaccuracies of this method, but this may be acceptable because this region suffers
from maximum distortion anyway.
Table 4.6 lists large and small radii for five angles and for
k
=10. Thecodethat
performs the computations is also listed.
−
0
.
3)
r
a
+0
.
3
r
b
]
/
[(1
−
◦
n
R
n
r
n
00
0
0
1
22.5
4.142
1.989
k = 10;
angl = {22.5, 45., 67.5, 89.};
k Tan[angl Degree]
k Tan[angl/2 Degree]
2
45
10
4.142
3
67.5
24.142
6.682
4
89
572.9
9.827
Table 4.6
Code for Table 4.6
A given point (
x, y
)isatadistance
d
=
x
2
+
y
2
from the origin. This distance is
compared with all the radii in the table. If
d
equals an
R
n
, then the point is multiplied
by the scale factor
r
n
/R
n
. Otherwise, we find the smallest
R
n
such that
R
n
<d<R
n
+1
.
Therelativedistanceofthepointfrom
R
n
is (
d
R
n
). As an example
(recall that our
table is b
ased on
k
= 10), consider the point (15
,
10). Its distance from
−
R
n
)
/
(
R
n
+1
−
the origin is
√
15
2
+10
2
18. Thus, it is between
R
2
=10and
R
3
= 24. We compute
(18
−
10)
/
(24
−
10)
≈
0
.
57, which tells us that the point is located 57% of the distance
from
R
2
to
R
3
.
The scale factor of the point is given by [(1
≈
−
0
.
57)4
.
142+0
.
57
·
6
.
682]
/
[(1
−
0
.
57)10+
0
.
57
24]=5
.
59
/
18 = 0
.
31,soithastobemovedto0
.
31(15
,
10) = (4
.
66
,
3
.
11) on the
radius 10 circle, where its new distance from the origin is 5.6, or 57% of the distance
from
r
2
=4
.
142 to
r
3
=6
.
682.
·
A story of particular facts is a mirror which obscures and distorts that which should
be beautiful; poetry is a mirror which makes beautiful that which it distorts.
—Percy Bysshe Shelley,
A Defence of Poetry
.
Angular Fisheye Projection
The hemispherical fisheye projection assigns more importance to those image parts
located near the line of sight of the viewer. These parts are displayed in detail, while
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