3D Viewing

1
3D Viewing
2
Viewing Specification Eye Coordinates

• Suppose that modeling transformation has been
  done and now all objects are in WC.
• We need viewing specification or camera
  information.
• VRP view reference point or eye position.
• AT a reference point for objects. (AT is a
  reference point toward which the eye is aimed. AT
  is typically somewhere in the middle of the
  scene.)
• VUP view up vector.
• VPN view plane normal. (orientation of
  projection point) VRP-AT
• VRP, u, v, n defines eye coordinates (EC),
  which is also called camera coordinates or view
  coordinates.
• In EC, camera is at the origin and looks in the
  n direction.

y
y
VUP
v
VRP
AT
u
n
z
z
x
x
VRP
3
Eye Coordinates Teapot Example

• Lets focus on the teapot centered at WC.

v
u
n
VRP
4
Viewing Transformation

• We have two coordinates WCO,x,y,z and
  ECVRP,u,v,n.
• Lets superimpose EC onto WC a translation
  followed by a rotation.
• Such a transformation (which is a rigid motion)
  simply reinterprets the model from WC into EC,
  and is called viewing transformation.

y
v
u
VRP
n
x
z
5
Viewing Transformation (contd)

• viewing transformation in a single 4x4 matrix
• Why from WC to EC? Its for easy projection and
  clipping. You will see pretty soon!
• Modeling and viewing transformations are often
  combined into a matrix Mmodelview.

viewing transformation
modeling transformation
in WC
in EC
in LC
modelview transformation
in LC
in EC
6
View Volume

• Thanks to the viewing transformation, all
  vertices are newly defined in ECVRP,u,v,n, and
  we can forget about the WCO,x,y,z.
• Then, lets use x,y,z and u,v,n
  interchangeably because we are familiar with
  x,y,z.
• Suppose a huge scene. In general, we cannot see
  all objects in the scene. We need to specify how
  much of the scene we can see. View volume does it.

7
Perspective View Volume

• We can imagine a window through which we can see
  the world.
• The window is parallel to xy-plane.
• Then, for perspective projection, the output
  image is made by the projectors (from COP) that
  pass through the window.
• Such a perspective view volume makes an infinite
  pyramid.
• We can bound near and far sides of the infinite
  pyramid.
• The perspective view volume makes a truncated
  pyramid, frustum.
• Only the objects inside the frustum are visible.

y
COP
window
z
x
window
near plane
far plane
COP
8
Perspective View Volume (contd)

• The frustum is defined by l, r, b, t, n, f
  which stands for left, right, bottom, top, near,
  far. We set n and f to be positive such that
  they are the perpendicular distances from COP to
  the two planes.
• Note that the pyramid is not upright, which is
  unrealizable in the camera model but is useful
  for simulating human stereo vision, etc.
• We can find that the vector connecting VRP and AT
  is not really the viewing direction. The vector
  is for establishing z-axis of EC.
• Note that window is implicitly defined through
  the near plane.

-z
z -f
y
far plane
near plane
(r, t, -n)
COP
x
(l, b, -n)
z
9
Viewing Examples Model and EC

• Given the model in the figure, suppose that
  VRP(0,0,40), AT(0,0,0), and VUP(0,10,0).
• Lets see some view volume examples with this
  model.

y
10
VUP
vy10
-n-z
AT
10
x
vy
v
20
-20
30
-10
ux10
COP
u
ux
z40
z
n
nz
10
Perspective Projection Example I
(60, 60, -40)

• ? example 1
• We got the following image, where the viewport
  corresponds to the view volumes window.

60
l 0, r 6 b 0, t 6 n 4, f 40
far plane
40
20
y10
viewport
x10
20
40
60
near plane
-20
(r, t, -n) (6, 6, -4)
COP
(l, b, -n) (0, 0, -4)
11
Perspective Projection Example II
(60, 60, -40)

• example 2
• What if the window size is reduced by 1/4?
• We got the following image.

60
l 0, r 3 b 0, t 3 n 4, f 40
40
(30, 30, -40)
20
viewport
20
40
60
-20
(6, 6, -4)
COP
(3, 3, -4)
12
Perspective Projection Example III
(60, 60, -40)
60

• example 3
• What if the far plane moves towards COP?
• ? We got the following image.

l 0, r 6 b 0, t 6 n 4, f 15
40
20
viewport
20
40
60
-20
(6, 6, -4)
COP
13
Parallel Projection Oblique vs. Orthographic

• Recall that view plane and window are parallel to
  xy-plane.
• Lets consider the cross-section of EC (seen
  along y direction).
• Orthographic projection is a special parallel
  projection where the view plane is orthogonal to
  DOP.
• Dont get confused with parallel projection
  where COP is at ? and COP as EC origin. COP is
  basically for defining EC, and the parallel
  projection is inherently unrealistic.

x

• ? perspective
• ? parallel
• - orthographic
• - oblique

z
Assuming the DOP is parallel to xz-plane.
x
x
x
DOP
DOP
DOP
z
orthographic
oblique
z
z
14
Parallel View Volume

• In parallel projection, the view volume is an
  infinite parallelepiped (not necessarily a box),
  which is parallel to DOP.
• We can also bound near and far sides of the
  infinite parallelepiped to make the parallel view
  volume. (Imagine that the window slides along DOP
  to make both planes.)
• Only the objects inside the view volume are
  visible.

window
near plane
far plane
15
Parallel View Volume (contd)

• Oblique projection is much less common, and so
  lets consider only orthographic projection where
  DOP is orthogonal to the view plane.
• Note that DOP is along the z-axis.
• Then, the view volume has to be aligned with
  xyz-axes.
• Two end points in a diagonal are enough to define
  the view volume.
• As in the perspective projection, we set n and f
  to be positive.
• Note that the view volume is defined by l, r, b,
  t, n, f for both perspective projection and
  orthographic projection. Of course, the set l,
  r, b, t, n, f has different semantics in the two
  cases.

(r, t, -f)
DOP
y
parallel view volume
(l, b, -n)
x
z
16
Orthographic Projection Examples

• Lets use the same example.

• example 2

l 0, r 20 b 0, t 20 n 15, f 40

• example 3

l 5, r 20 b 5, t 20 n 15, f 40

• example 4

• example 1

l 5, r 20 b 5, t 20 n 25, f 40
l 0, r 20 b 0, t 20 n 4, f 40
17
Culling Clipping

• Where are we now?
• Objects are defined in EC through modeling and
  viewing transformations.
• Projection type (parallel or perspective) is
  specified.
• View volume is given as l, r, b, t, n, f in EC.
• The view volume specifies how much of the scene
  we can see. So, polygons completely inside the
  view volume are processed for display.
• How about the scene outside the view volume?
• Polygons completely outside the view volume are
  culled.
• Polygons intersecting the boundary of the view
  volume are clipped against the boundary, and then
  processed for display.

18
Clipping Teapot Example

• Suppose that, in orthographic projection, we have
  a view volume intersecting the teapot.

clipped polygon mesh to be rendered
19
Projection Transformation

• OK, we just specified the view volume l, r, b,
  t, n, f in EC. The next step is projection
  transformation.
• Lets first tackle the orthographic projection.
• Operations such as clipping can be easily
  implemented in canonical view volume(CVV), which
  is defined as a 2x2x2 cube with its center at the
  origin, i.e. xyz-1,1.
• Projection transformation or specifically
  orthographic transformation transforms the box
  view volume into CVV.

20
Projection Transformation (contd)

• Obviously we need scaling. However, scaling is
  about origin.
• As the 1st step, lets translate the view
  volumes center to the origin.
• We know that the view volumes center is
  ((lr)/2,(bt)/2,-(nf )/2)).
• So, the translation is Tpar T(-(lr)/2,-(bt)/2,
  (nf )/2))

y
x-1,1 y-1,1 z-1,1
xl,r yb,t z-n,-f
x
z
y
x
Tpar makes ?
z
21
Projection Transformation (contd)

• Note that the CVV has the size of 2x2x2. So,
  lets scale!
• Width along x r - l should be scaled into 2.
• So, scaling factor along x direction Sx 2/(r-l)
• Similarly, Sy 2/(t-b) and Sz 2/(f-n)
• MorthoSpar Tpar is the orthographic
  (projection) transformation from the box view
  volume into CVV, or from EC to clip coordinates
  (CC).
• After orthographic transformation, clipping is
  done. Then, the clipped model is within CVV whose
  dimensions are all 1 along ?x/?y/?z axes. Its
  called that the model is in normalized device
  coordinates(NDC).
• Some people say we have a 3D film (generated by
  projection).

y
x
z
r - l
22
Projection Transformation (contd)

• Orthographic transformation with teapot example.

v
u
n
translation
scaling
CVV in NDC
23
Projection Transformation (contd)

• Given l, r, b, t, n, f, the orthographic
  transformation Mortho is computed as Spar Tpar.
• Mortho can be simply rewritten as follows.

x? (2/(r-l))x - (rl)/(r-l) y? (2/(t-b))y -
(tb)/(t-b) z? (2/(f-n))z (fn)/(f-n)
24
Projection Transformation (contd)

• Some systems such as OpenGL use -z direction (0 0
  -1) as DOP. (Compare this with positive-z
  projection direction in EC.) For this purpose, 3D
  reflection is required.

reflection wrt xy-plane
25
Window Coordinates and Viewport

• A window at your screen (not the window
  associated with the view volume) is associated
  with window coordinates (WdC).
• A viewport is defined at WdC.
• The viewport is not necessarily the entire
  window, and is defined by its lower left corner
  (xmin ymin), width, and height.
• Viewport transformation transforms the x- and
  y-coordinates of CVV to coincide with the
  viewport.

viewport
window
height
(xmin,ymin)
width
26
Viewport Transformation

• The viewport transformation is a scaling followed
  by a translation.
• For scaling, Sxwidth/2 and Syheight/2.
• Translation moves the center at the origin to the
  viewport center (ox,oy)(xminwidth/2,
  yminheight/2).
• So, Mviewport T(ox,oy,0) S(Sx, Sy,1). Simply,
  x?x(width/2)ox and y?y(height/2)oy. Dont
  do matrix multiplication.

translation
scaling
27
Viewport Transformation (contd)

• WdC is 3D. WdC is also called 3D screen
  coordinates.
• In some systems such as OpenGL, Mviewport encodes
  z-coordinates. However, we simply preserve the
  z-coordinates. Either way, the z-coordinates are
  necessary for hidden line/surface removal, etc.
  You will see soon. Be patient. ?
• In sumary, Mviewport Mortho Mmodelview
  transforms an LC point into WdC. All done!!!!

viewport
y

oy
Mviewport
ymin
x
ox
z
xmin
28
Viewport Transformation (contd)

• Viewport transformation with teapot example
• Final images

CVV
29
Orthographic Projection Summary

• The modeling transformation moves from LC to WC.
• Viewing specifies the location/orientation of the
  viewer or camera, and viewing transformation
  moves from WC to EC.
• The two transformations are combined into
  Mmodelview.
• The view volume is defined by l, r, b, t, n, f.
  
• The orthographic transformation Mortho changes
  the view volume into CVV. Its said the model is
  then in clip coordinates(CC).
• Clipping is done.
• The viewport transformation Mviewport transforms
  vertices in CVV into 3D points in WdC.

orthographic transformation
modeling viewing transformation
in EC
in LC
viewport transformation
in CC
in WdC
clipping
in NDC
30
Projection Transformation for Perspective
Projection

• Now, lets tackle the perspective projection
• Note that the modelview transformation does not
  depend on the projection type. What distinguishes
  the perspective projection from the parallel
  projection is the projection transformation.
• Perspective projection transformation from the
  frustum to CVV.
• It is often simply called a perspective
  transformation.

-z
z-f
y
far plane
(r, t, -n)
near plane
COP
x
(l, b, -n)
z
31
3D Shearing

• Lets see the frustum along -y direction
• So, shearing along x is needed. OK, but is that
  it? No

-z
z-f
y
far plane
(r, t, -n)
near plane
COP
x
(l, b, -n)
z
32
3D Shearing (contd)
y

• Similarly, let us see along -x.
• H?xy(p,q) makes the vector CC, which connects COP
  and the near planes center ((lr)/2, (bt)/2,
  -n), be (0, 0, -n).

-z
-z
z-f
y
far plane
near plane
(r, t, -n)
COP
x
(l, b, -n)
z
xpz
x
0
p
0
1
H?xy(p,q)
in summary z unaffected x? x pz y? y qz
yqz
y
0
q
1
0

z
z
0
1
0
0
1
1
1
0
0
0
33
3D Shearing (contd)

• Lets compute p and q.
• If the frustum happens to be upright, CCxCCy0,
  and so pq0, which means that H?xy(p,q) I and
  shearing has no effect.

CC near-plane-center - COP
should be
34
Understanding 3D Shearing

• Shear a coordinate value with respect to another
  coordinate.
• Six basic shearing matrices Hxy(s), Hxz(s),
  Hyx(s), Hyz(s), Hzx(s), and Hzy(s).
• Commonly used is shearing two coordinates values
  wrt the remaining.
• Determinant of any shearing matrix H1, and so
  shearing is volume-preserving. (No scaling!)

35
Shearing followed by Scaling

• 3D shearing has made the skewed frustum upright.
• Now, lets make the frustums side faces be of
  45 slope.
• Such a task can be performed by scaling.

y
y
y
To be done!
45
Done!
-z
-z
-z
original frustum
upright frustum
45-slope frustum
36
Scaling

• Note that shearing does not change the size of
  the near plane.
• Both (r-l)/2 and (t-b)/2 should be the same as n
  for making a 45-slope frustum. We then need
  scaling!
• Sx n/(r-l)/2 2n/(r-l)
• Sy n/(t-b)/2 2n/(t-b)
• Sz 1.

(r-l)/2
-z
y
(t-b)/2
near plane
n
x
COP
z
37
Shearing Scaling Teapot Example
scaling
shearing
38
Distortion

• From the 45 frustum into CVV Lets represent
  (x2 y2 z2) in terms of (x1 y1 z1).
• 
  

y
p
-z1
y
z-f
1
n
(x1,y1,z1)
(x2,y2,z2)
z
z
1
-1
z1
-n
z-n
-1
The point p (whose y value is -z1) should go to
y1. ? -z11 y1 y2 ? y2 -y1/z1 Similarly,
x2 -x1/z1
39
Distortion (contd)

• We found that x2 -x1/z1 and y2 -y1/z1 .
• How about z2?
• Note that a triangle in the frustum should be
  transformed also into a triangle in CVV.
• The triangle in CVV is on a plane axbyczd0.
• and so
• OK, we found that z2AB/z1.
• Lets compute A and B.

should be for the formula to be planar
plane
40
Distortion (contd)

• Lets compute A and B for z2AB/z1.
• Note that z1-f, -n and z2-1,1.
• So, if z1 -f, then z2 -1. Similarly, if z1 -n,
  then z21.
• Solving the two equations, we get A(nf)/(n-f)
  and B2nf/(n-f).
• (x2 y2 z2 1) (-x1/z1 -y1/z1 AB/z1 1)
  (x1 y1 -Az1-B -z1)
• Multiplying every component by -z1, we can
  represent the distortion transformation by a
  matrix multiplication.
• Its possible thanks to homogeneous coordinate
  system.

called Mdistort
not (0 0 0 1) !!
41
Why called Distortion?

• Mdistort distorts the model, and is not affine in
  the sense that parallelism is not preserved.
• Distortion transformation moves COP to the ?
  position of z-axis.
• The model is distorted, but leads to a correct
  image as we are now in (orthographic) CVV.

into CVV
model distortion
42
Distortion Teapot Example
43
Perspective Division

• In homogeneous coordinates, each vertex should be
  divided by its w-component to get a corresponding
  Cartesian coordinates.
• Its called perspective division.
• An example
• n1
• f2

A(nf)/(n-f) B2nf/(n-f).
y
Q(0 3/2 3/2)
Q?(0 1 -1/3)
P?(0 1 1)
3/2 1 1/2
P(0 1 -1)
-z
-2
R(0 1/2 3/2)
R?(0 1/3 -1/3)
44
Perspective Division (contd)

• The distortion transformation converts (x y z 1)
  into (x y -Az-B -z), and the perspective division
  is done by w-z, which actually represents the
  distance from the eye.
• As the distance from the eye increases, 1/w
  approaches 0, and so x/w and y/w also approach 0.
  This is how foreshortening is achieved.

a
b
model distortion
b
a
45
Perspective Division (contd)

• Observe that Q and R are at the center of the
  frustum along z-axis but Q? and R? are not at
  the center of CVV.
• As the transformed z-coordinates move farther
  away from the near plane, the locations become
  increasingly less precise. Such a non-linearity
  caused by perspective division leads to some
  problem for hidden-surface removal. Lets revisit
  this later.

Q?(0 1 -1/3)
y
Q(0 3/2 3/2)
P?(0 1 1)
3/2 1 1/2
P(0 1 -1)
-z
-2
R(0 1/2 3/2)
R?(0 1/3 -1/3)
46
Perspective Transformation

• Below are the steps we have followed.
• The perspective transformation MpersMdistortSper
  s H?xy(p,q) transforms the perspective view
  volume into CVV.
• We could do clipping after perspective
  transformation perspective division are done.
• Mmodelview ? Mpers ? perspective division ?
  Mviewporttransforms an LC point into WdC. We
  already derived Mviewport and Mmodelview. All
  done!!!!

y
y
y
y
45
H?xy(p,q)
Mdistort
Spers
-z
-z
-z
-z
45-slope frustum
upright frustum
original frustum
CVV
47
Perspective Transformation (contd)

• Perspective transformation Mpers
  MdistortSpers H?xy(p,q)
• Given l, r, b, t, n, f, Mpers is computed as
  follows

x? (2n/(r-l))x ((rl)/(r-l))z y?
(2n/(t-b))y ((tb)/(t-b))z z?
-((nf)/(n-f))z - 2nf/(n-f) w? -z
48
Perspective Transformation (contd)

• As we discussed in orthographic transformation,
  some systems such as OpenGL includes 3D
  reflection for perspective transformation.

49
Perspective Transformation (contd)

• A 3D affine transformation is represented by a
  3x4 matrix, where its 3x3 sub-matrix is for
  linear transformation and the 4th column is for
  translation. The 4th row is just (0 0 0 1).
• Perspective transformations 4th row is not any
  more (0 0 0 1), and is not an affine
  transformation.
• However, orthographic projection is an affine
  transformation.

50
Viewport Transformation Teapot Example

• Viewport transformation for perspective
  projection is same as that for orthographic
  projection.
• Final images

CVV
51
Perspective Projection Summary

• The modeling transformation moves from LC to WC.
• Viewing specifies the location/orientation of the
  viewer or camera, and viewing transformation
  moves from WC to EC.
• The two transformations are combined into
  Mmodelview.
• The frustum is defined by l, r, b, t, n, f.
• The perspective transformation Mpers transforms
  the perspective view volume into (parallel) CVV.
• Do clipping and then perspective division. (We
  will see.)
• Finally, the viewport transformation Mviewport
  transforms vertices in CVV into 3D points in WdC.

projection transformation
modeling viewing transformation
in EC
in CC
in LC
perspective division
viewport transformation
in CC
clipping
in WdC
in NDC
52
Orthographic Projection vs. Perspective Projection
orthographic projection
modeling viewing transformation
viewport transformation
orthographic transformation
WdC
clipping
LC
EC
NDC
CC
perspective division
perspective transformation
clipping
CC
CC
perspective projection