1
Assignment 4 (NonProgramming & Programming)
Worth:
10 points
Due Date and Time:
Before class begins at 12:20 pm on 06/28/2007. Copy zipped
programming submission to CS250 web page. Place your stapled nonprogramming
submission on instructor’s desk in Michelangelo.
Topics
•
Introduction to Graphics Pipe
•
Transforms: Scaling, Rotation, and Translation
•
View Transform
•
Perspective Transform
•
Viewport Transform
•
First Person Camera
•
Hidden Surface Removal using BackFace Removal & Depth Buffer
•
Triangle Rasterization with Linear (Depth and Color) Interpolation
NonProgramming Statement
1.
Let
r
q
p
C
F
ˆ
,
ˆ
,
ˆ
,
=
be the notation used to describe an orthonormal righthanded
reference frame where
r
q
p
C
ˆ
,
ˆ
,
ˆ
,
are the origin,
−
X
axis (oriented right),
−
Y
axis
(oriented into paper), and
−
Z
axis (oriented up), respectively. The implicit world
reference frame is defined as
k
j
i
O
F
W
ˆ
,
ˆ
,
ˆ
,
=
where
( )
0
,
0
,
0
=
O
,
( )
0
,
0
,
1
ˆ
=
i
,
( )
0
,
1
,
0
ˆ
=
j
, and
( )
1
,
0
,
0
ˆ
=
k
. In world frame, a camera is located at position
W
C
with view and up vectors
v
r
and
up
, respectively. Let
w
v
u
C
F
W
V
ˆ
,
ˆ
,
ˆ
,
=
be the
view frame within static world frame
W
F
with origin at
W
C
,
−
X
axis
u
ˆ
oriented
right,
−
Y
axis
v
ˆ
oriented into paper, and
−
Z
axis
w
ˆ
oriented up.
1.1. To completely define
V
F
, compute normalized basis vectors
u
ˆ
and
w
ˆ
. Draw a
neat picture to illustrate your answer. (2.5 points)
1.2. Next, compute the affine matrix required to transform a point from world frame
W
F
into view frame
V
F
.
Draw a neat picture to illustrate your answer. (2.5
points)
1.3. In view frame
V
F
, horizontal fieldofview is
o
θ
, aspect ratio is
a
, near and far
clip planes are
n
and
∞
units from viewpoint along
−
Y
axis
v
ˆ
, respectively.
Assume near clip plane is also the projection plane. Righthanded NDC frame is
defined by the following inequalities:
1
0
≤
≤
x
(width),
1
0
≤
≤
z
(height),
1
0
≤
≤
y
(depth). Starting from first principles, derive the perspective matrix
required to transform a view frame point into clip frame. Draw a neat picture to
illustrate your answer. (5 points)
1.4. Now, suppose in view frame
V
F
, the bottomleft and topright coordinates of
view window are
( )
b
n
l
,
,
and
( )
t
n
r
,
,
, respectively. Let far clip plane be
f
units from viewpoint along
−
Y
axis
v
ˆ
. Righthanded NDC frame is defined by
the following inequalities:
1
1
≤
≤
−
x
(width),
1
1
≤
≤
−
z
(height),
f
y
n
≤
≤
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(depth). Starting from first principles, derive the homogeneous perspective
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 Spring '07
 Ghali
 Linear Algebra, Computer Graphics, 3D computer graphics, view frame, neat picture

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