Graphics Reference
In-Depth Information
Exercise 13.1:
(a) Suppose that a perspective camera has horizontal and vertical
field-of-view angles
θ
h
and
θ
v
. What is the aspect ratio (width/height) of the film?
(b) Show that if
θ
v
are both small, then the film aspect ratio and the ratio
θ
h
/θ
v
are approximately equal.
Exercise 13.2:
Equations 13.2-13.5 show how to determine the
uvw
frame
from the look and up directions. Show that the following approach yields the same
results:
θ
h
and
w
=
−
look
(13.22)
look
t
=
w
×
vup
(13.23)
t
u
=
(13.24)
t
v
=
u
×
w
,
(13.25)
Also explain why it's not necessary to normalize
v
.
Exercise 13.3:
We noted that as the viewpoint in a perspective view moved
farther and farther from the film plane, the view approached a parallel view. Con-
sider the case where the eye is at position
(
0, 0,
n
)
, the near plane is at
z
=
0, and
the far plane is at
z
=
θ
h
= arctan(
f
)
so that the
−
1 so that
f
=
n
+
1. Let
θ
v
=
viewing area on the far plane is
1. Write down the product
M
pp
M
per
,
as a function of
n
, and see what happens in the limit as
n
−
1
≤
x
,
y
≤
. Explain the result.
Exercise 13.4:
Just as a projective transformation of the plane is determined
by its value on four points, a projective transformation of the line is determined
by its value on three points. Such a projective transformation always has the form
t
→∞
at
+
b
→
ct
+
d
, where
a
,
b
,
c
, and
d
are real numbers with
ad
−
bc
≥
0.
(a) Suppose you want to send the points
t
=
0, 1,
to 3, 7, and 2, respectively.
Find values of
a
,
b
,
c
, and
d
that make this happen. The value at
t
=
∞
∞
is defined
→∞
/
as the limit of values as
t
, and turns out to be
a
c
.
(b) Generalize: If we want
t
=
0, 1,
∞
to be sent to
A
,
B
, and
C
, find the appropri-
ate values of
a
,
b
,
c
, and
d
.
Exercise 13.5:
Create examples to show that a connected
n
-sided polygon in
the plane, when clipped against a square, can produce up to
disconnected
pieces within the square (ignore the parts that are “clipped away”). What is the
largest number of pieces that can be produced if the polygon is convex? Explain.
Exercise 13.6:
Construct a pinhole camera from a shoebox and a sheet of
tissue paper by cutting off one end of the shoebox and replacing it with tissue
paper, punching a tiny hole in the other end, and taping the top of the box in place.
Stand inside a darkened room that looks out on a bright outdoor scene; look at the
tissue paper, pointing the pinhole end of the box toward the window. You should
see a faint inverted view of the outdoor scene appear on the tissue paper. Now
enlarge the hole somewhat, and again view the scene; notice how much blurrier
and brighter the image is. What happens if you make the pinhole a square rather
than a circle?
Exercise 13.7:
Find a photograph of a person, and estimate the distance from
the camera to the subject—let's say it's 3 meters. Have a friend stand at that dis-
tance, and determine at what distance you would have to place the photograph
n
/
2
