The Catholic University of America School of Engineering Department of Electrical Engineering and Computer Science CSC  513:  Fundamentals  of  Computer  Graphics   Assignment  1   1)  Write  the  parametric  form  of  a  ray  with  source  p  and  direction  d.   2)  Assuming  d   in  the  above  ray  is  a  unit  vector,  write  an  algorithm  that,  for  an  arbitrary  non-­‐negative   integer  n,  generates  n  evenly  spaced  points  along  the  ray  at  1  unit  intervals.   3)  Write  the  parametric  form  of  the  unit  circle.   4)  Given  an  arbitrary  positive  integer  n,  write  an  algorithm  that  generates  n  evenly  spaced  points  that  are   sampled  along  the  unit  circle.   5)  Suppose  you  have  access  to  the  functions  scale(sx,  sy),  translate(tx,  ty),  and  rotate(α)  which  generate  the   corresponding  2D  homogeneous  transformation  matrices.  Using  these  functions,  write  the  expression  of  a   transformation  T  such  that  if  p  lies  on  the  unit  circle,  Tp  will  lie  on  an  ellipse  centered  at  c,  with  a  major   radius  r1,  minor  radius  r2  and  rotated  at  an  angle  of  α.  You  do  not  have  to  compute  the  full  matrix.  You  may   leave  it  expressed  using  the  above  functions.  Hint:  remember  that  order  is  important  and  that  operations   associate  from  the  "inside  out".   6)  Write  the  implicit  form  of  the  unit  sphere.     7)  Given  an  arbitrary  point  p,  write  a  test  to  determine  if  p  is  inside  the  unit  sphere.   8)  Suppose  you  are  given  an  affine  transformation  T  that  maps  the  unit  sphere  to  some  arbitrarily  located   and  oriented  ellipsoid.  Give  an  expression  which,  given  an  arbitrary  point  p,  determines  if  p  is  inside  said   ellipsoid.  Hint:  use  the  result  from  the  previous  exercise.   9)  You  are  given  the  vertices  of  a  convex  polygon  in  the  2D  plane  in  counter-­‐clockwise  order  as  (p1,  ...,  pn).   The  coordinates  of  vertex  pi  are  (xi,  yi).     9a)   (3   marks)   Give   an   expression   for   the   coordinates   of   the   outward-­‐facing   normal   ni   of   the   edge   connecting  pi  and  pi+1.   9b)  (3  marks)  Let  q  =  (xq,  yq)  be  an  arbitrary  point  on  the  plane  containing  the  gi ...

1. The Catholic University of America
School of Engineering
Department of Electrical Engineering and Computer Science
CSC
513:
Fundamentals
of
Computer
Graphics
Assignment
1
1)
Write
the
parametric
form
of
a
ray
with
source
p
and
direction

2. d.
2)
Assuming
d
in
the
above
ray
is
a
unit
vector,
write
an
algorithm
that,
for
an
arbitrary
non-­‐negative
integer
n,
generates
n
evenly
spaced
points
along
the
ray
at
1
unit

3. intervals.
3)
Write
the
parametric
form
of
the
unit
circle.
4)
Given
an
arbitrary
positive
integer
n,
write
an
algorithm
that
generates
n
evenly
spaced
points
that
are
sampled
along
the

4. unit
circle.
5)
Suppose
you
have
access
to
the
functions
scale(sx,
sy),
translate(tx,
ty),
and
rotate(α)
which
generate
the
corresponding
2D
homogeneous
transformation
matrices.
Using
these
functions,
write
the
expression
of
a

5. transformation
T
such
that
if
p
lies
on
the
unit
circle,
Tp
will
lie
on
an
ellipse
centered
at
c,
with
a
major
radius
r1,
minor
radius
r2
and
rotated
at
an
angle
of
α.

6. You
do
not
have
to
compute
the
full
matrix.
You
may
leave
it
expressed
using
the
above
functions.
Hint:
remember
that
order
is
important
and
that
operations
associate
from
the
"inside
out".

7. 6)
Write
the
implicit
form
of
the
unit
sphere.
7)
Given
an
arbitrary
point
p,
write
a
test
to
determine
if
p
is
inside
the
unit
sphere.
8)
Suppose
you
are

8. given
an
affine
transformation
T
that
maps
the
unit
sphere
to
some
arbitrarily
located
and
oriented
ellipsoid.
Give
an
expression
which,
given
an
arbitrary
point
p,
determines
if
p
is
inside
said
ellipsoid.
Hint:

9. use
the
result
from
the
previous
exercise.
9)
You
are
given
the
vertices
of
a
convex
polygon
in
the
2D
plane
in
counter-­‐clockwise
order
as
(p1,
...,
pn).
The
coordinates
of
vertex
pi

10. are
(xi,
yi).
9a)
(3
marks)
Give
an
expression
for
the
coordinates
of
the
outward-­‐facing
normal
ni
of
the
edge
connecting
pi
and
pi+1.
9b)
(3
marks)

11. Let
q
=
(xq,
yq)
be
an
arbitrary
point
on
the
plane
containing
the
given
polygon.
Let
l
be
the
line
containing
pi
and
pi+1,
and
let
ni
be
the
outward-­‐facing
normal
vector
as
given

12. in
part
a).
What
is
a
test
that
determines
whether
or
not
q
lies
on
the
outward
side
of
l
(the
side
toward
which
ni
points)?
9c)
(5
marks)
Provide
an
algorithm
that

13. determines
whether
a
2D
point
q
is
inside,
outside,
or
on
the
boundary
of
the
given
polygon.
Hint:
Each
edge
is
contained
in
an
infinite
line.
Each
infinite
line
divides
the
2D
plane
into

14. two
half-­‐planes:
the
"left"
half-­‐plane
and
the
"right"
half-­‐plane
(left
and
right
are
defined
with
respect
to
a
counter-­‐clockwise
direction
of
traversal
of
the
vertices).
The
key
insight
you
should
use
is
that
the

15. interior
of
a
convex
polygon
is
the
intersection
of
the
left
half-­‐planes
of
each
edge
of
the
polygon.
10)
In
stereo
rendering,
two
cameras
are
needed,
with
slightly
different
vantage
points
and
view

16. directions;
each
is
used
to
render
an
image
for
the
corresponding
eye.
This
can
be
specified
with
the
following
parameters:
• c:
The
center
of
interest,
a
point
in
world
space
that
lies

17. along
the
optical
axis
of
both
cameras.
• em:
The
midpoint
between
the
eye
positions
of
each
camera.
• t:
An
“up”
vector
that
allows
us
to
specify
a
tilt
rotation
about
the
axis
passing

18. through
c
and
em.
The
z
axis
is
a
special
case.
• s:
The
distance
between
the
two
eyes.
All
answers
should
be
given
in
terms
of
the
above

19. stereo
parameters,
as
well
as
any
intermediate
quantities
you
specify.
10a)
Given
an
expression
for
the
unit
vector
d
that
is
perpendicular
to
both
t
and
c
-­‐
em
such
that

20. (c
-­‐
em,
t,
d)
is
a
right-­‐handed
coordinate
frame.
10b)
Give
expressions
for
eL
and
eR,
the
eye
positions
of
each
camera.
10c)
Give
expressions
for
the
basis
vectors

21. that
make
up
the
two
cameras'
coordinate
frames:
(uL,
vL,
wL)
and
(uR,
vR,
wR)
10d)
Give
a
test
that
determines
whether
or
not
a
polygon
face
with
normal
n
is
a
back

22. face
that
can
be
culled
when
rendering
from
both
cameras.
11)
This
illustration
shows
a
top-­‐down
view
of
a
3D
scene
where
each
blue
edge
corresponds
to
a
planar
square
perpendicular
to

23. the
image
plane
(i.e.
coming
out
of
the
page)
and
the
eye
e
of
a
possible
viewpoint
lies
in
the
image
plane.
The
short
vectors
are
normal
vectors.

24. 11a)
Assuming
the
particular
scene
and
camera
placement
shown
above,
is
it
possible
to
exclude
any
polygons
from
rendering?
Explain
your
answer.
11b)
Draw
the
BSP
tree
for
the
above
scene
that

25. would
result
from
adding
the
polygons
to
the
tree
in
the
order
(S1,
S2,
S3,
S4,
S5,
S6).
11c)
Describe
how
your
tree
will
be
traversed
when
rendering
the
scene
from
the
eye

26. location
specified.