3D Viewing Cont'

1
3D Viewing (Cont.)

• Credits Andries van Dam of Brown University,
  Angel Book slides.

2
3D Viewing the Synthetic
Camera

• Programmers reference model for specifying 3D
  view projection parameters to the computer
• General synthetic camera each package has its
  own but they are all (nearly) equivalent.
  position of camera
• orientation
• field of view (wide angle, normal)
• depth of field (near distance, far distance)
• focal distance
• tilt of view/film plane (if not normal to view
  direction, produces oblique projections)
• perspective or parallel projection? (camera near
  objects or an infinite distance away)

3
View Volumes

• A view volume contains everything visible from
  the point of view or direction
• what does the camera see?
• Conical view volumes
• approximates what eye sees
• expensive math (simultaneous quadratics) when
  clipping objects against cones surface
• Can approximate with rectangular cone instead
  (called a frustum)
• works well with a rectangular viewing window
• simultaneous linear equations for easy clipping
  of objects against sides

4
Graphics (rendering) pipeline
Viewing Geometry Processing World Coordinates
(floating point)
Rendering Pixel Processing Screen Coordinates
(integer)
Conservative VSD Selective traversal of object
database (or traverse scene graph to get CTM)
Transform Vertices To canonical view
volume
Light at Vertices Calculate light intensity at
vertices (lighting model of choice)
Conservative VSD Back-face Culling
Conservative VSD View Volume Clipping
Image- precision VSD Compare pixel
depth (Z-buffer)
Shading Interpolate color values (Gouraud) or
normals (Phong)
per polygon
per pixel of polygon
avd November 9,
2004 VSD 6/48
avd November 5,
2002 VSD 5/45
5
View Volume

• We need to know six things about our synthetic
  camera model in order to take a picture
• Position of the camera (from where its looking)
• The Look vector specifies in what direction the
  camera is pointing
• The cameras Orientation is determined by the
  Look vector and the angle through which the
  camera is rotated about that vector, i.e., the
  direction of the Up vector

Position
6
View Volume

• Aspect ratio of the electronic film ratio of
  width to height
• Height angle determines how much of the scene we
  will fit into our view volume larger height
  angles fit more of the scene into the view volume
  (width angle determined by height angle and
  aspect ratio)
• the greater the angle, the greater the amount of
  perspective distortion
• Front and back clipping planes limit extent of
  cameras view by rendering (parts of) objects
  lying between them and throwing away everything
  outside of them

7
Position

• Determining the Position is analogous to a
  photographer deciding the vantage point from
  which to shoot a photo
• Three degrees of freedom x, y, and z coordinates
  in 3-space
• This x, y, z coordinate system is right-handed
  if you open your right hand, align your palm and
  fingers with the x axis, and curl your fingers
  towards the y axis, your thumb will point along
  the z axis

8
Orientation
camera Position

• Orientation is specified by a point in 3D space
  to look at (or a direction to look in) and an
  angle of rotation about this direction
• Default (canonical) orientation is looking down
  the negative z-axis and up direction pointing
  straight up the y-axis
• In general the camera is located at the origin
  and is looking at an arbitrary point with an
  arbitrary up direction
• This is a little abstracteasier formulation?

9
Look and Up Vectors

• Look Vector
• the direction the camera is pointing
• three degrees of freedom can be any vector in
  3-space
• Up Vector
• determines how the camera is rotated around the
  Look vector
• for example, whether youre holding the camera
  horizontally or vertically (or in between)
• projection of Up vector must be in the plane
  perpendicular to the look vector (this allows Up
  vector to be specified at an arbitrary angle to
  its Look vector)

Up vector
Projection of up vector
10
View Angle

• Determines amount of perspective distortion in
  picture, from none (parallel projection) to a lot
  (wide-angle lens)
• In a frustum, two viewing angles width and
  height angles
• Choosing View angle analogous to photographer
  choosing a specific type of lens (e.g., a
  wide-angle or telephoto lens)

11
View Angle

• Lenses made for distance shots often have a
  nearly parallel viewing angle and cause little
  perspective distortion, though they foreshorten
  depth
• Wide-angle lenses cause a lot of perspective
  distortion

Resulting picture
12
Front and Back Clipping Planes

• Volume of space between Front and Back clipping
  planes defines what camera can see
• Position of planes defined by distance along Look
  vector
• Objects appearing outside of view volume dont
  get drawn
• Objects intersecting view volume get clipped

13
Front and Back Clipping Planes

• Reasons for Front (near) clipping plane
• Dont want to draw things too close to the camera
• would block view of rest of scene
• objects would be prone to distortion
• Dont want to draw things behind camera
• wouldnt expect to see things behind the camera
• in the case of the perspective camera, if we were
  to draw things behind the camera, they would
  appear upside-down and inside-out because of
  perspective transformation
• Reasons for Back (far) clipping plane
• Dont want to draw objects too far away from
  camera
• distant objects may appear too small to be
  visually significant, but still take long time to
  render
• by discarding them we lose a small amount of
  detail but reclaim a lot of rendering time
• alternately, the scene may be filled with many
  significant objects for visual clarity, we may
  wish to declutter the scene by rendering those
  nearest the camera and discarding the rest

14
View Volume Specification

• From Position, Look vector, Up vector, Aspect
  ratio, Height angle, Clipping planes, and
  (optionally) Focal length together specify a
  truncated view volume
• Truncated view volume is a specification of
  bounded space that camera can see
• 2D view of 3D scene can be computed from
  truncated view volume and projected onto film
  plane
• Truncated view volumes come in two flavors
  parallel and perspective

Truncated view volume means we only need to
render what the camera can see
15
Truncated View Volume for
Orthographic Parallel Projection

• Limiting view volume useful for eliminating
  extraneous objects
• Orthographic parallel projection has width and
  height view angles of zero

16
Truncated View Volume
(Frustum) for Perspective
Projection
17
Wheres My Film?

• Real cameras have a roll of film that captures
  pictures
• Synthetic camera film is a rectangle on an
  infinite film plane that contains image of scene
• Why havent we talked about the film in our
  synthetic camera, other than mentioning its
  aspect ratio?
• How is the film plane positioned relative to the
  other parts of the camera? Does it lie between
  the near and far clipping planes? Behind them?
• Turns out that fine positioning of Film plane
  doesnt matter. Heres why
• for a parallel view volume, as long as the film
  plane lies in front of the scene, parallel
  projection onto film plane will look the same no
  matter how far away film plane is from scene
• same is true for perspective view volumes,
  because the last step of computing the
  perspective projection is a transformation that
  stretches the perspective volume into a parallel
  volume
• In general, it is convenient to think of the film
  plane as lying at the far clip plane

18
Sources

• Carlbom, Ingrid and Paciorek, Joseph, Planar
  Geometric Projections and Viewing
  Transformations, Computing Surveys, Vol. 10, No.
  4 December 1978
• Kemp, Martin, The Science of Art, Yale University
  Press, 1992
• Mitchell, William J., The Reconfigured Eye, MIT
  Press, 1992
• Foley, van Dam, et. al., Computer Graphics
  Principles and Practice, Addison-Wesley, 1995
• Wernecke, Josie, The Inventor Mentor,
  Addison-Wesley, 1994

19
The OpenGL Camera

• In OpenGL, initially the object and camera frames
  are the same
• Default model-view matrix is an identity
• The camera is located at origin and points in the
  negative z direction
• OpenGL also specifies a default view volume that
  is a cube with sides of length 2 centered at the
  origin
• Default projection matrix is an identity

20
Default Projection

• Default projection is orthogonal

clipped out
2
z0
21
Moving the Camera Frame

• If we want to visualize object with both positive
  and negative z values we can either
• Move the camera in the positive z direction
• Translate the camera frame
• Move the objects in the negative z direction
• Translate the world frame
• Both of these views are equivalent and are
  determined by the model-view matrix
• Want a translation (glTranslatef(0.0,0.0,-d))
• d gt 0

22
Moving Camera back from Origin
frames after translation by d
d gt 0

• default frames

23
Moving the Camera

• We can move the camera to any desired position by
  a sequence of rotations and translations
• Example side view
• Rotate the camera
• Move it away from origin
• Model-view matrix C TR

24
OpenGL code

• Remember that last transformation specified is
  first to be applied

glMatrixMode(GL_MODELVIEW) glLoadIdentity() glTra
nslatef(0.0, 0.0, -d) glRotatef(90.0, 0.0, 1.0,
0.0)
25
The LookAt Function

• The GLU library contains the function gluLookAt
  to form the required modelview matrix through a
  simple interface
• Note the need for setting an up direction
• Still need to initialize
• Can concatenate with modeling transformations

glMatrixMode(GL_MODELVIEW) glLoadIdentity() gluL
ookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0., 1.0. 0.0)
26
gluLookAt

• glLookAt(eyex, eyey, eyez, atx, aty, atz, upx,
  upy, upz)

27
Other Viewing APIs

• The LookAt function is only one possible API for
  positioning the camera
• Others include
• View reference point, view plane normal, view up
  (PHIGS, GKS-3D)
• Yaw, pitch, roll
• Elevation, azimuth, twist
• Direction angles

28
Projections and Normalization

• The default projection in the eye (camera) frame
  is orthogonal
• For points within the default view volume
• Most graphics systems use view normalization
• All other views are converted to the default view
  by transformations that determine the projection
  matrix
• Allows use of the same pipeline for all views

xp x yp y zp 0
29
Homogeneous Coordinate Representation
default orthographic projection

• xp x
• yp y
• zp 0
• wp 1

pp Mp
M
In practice, we can let M I and set the z term
to zero later
30
Simple Perspective

• Center of projection at the origin
• Projection plane z d, d lt 0

31
Perspective Equations

• Consider top and side views

xp
yp
zp d
32
Homogeneous Coordinate Form
M

• consider q Mp where

? p
q
33
Perspective Division

• However w ? 1, so we must divide by w to return
  from homogeneous coordinates
• This perspective division yields
• the desired perspective equations
• We will consider the corresponding clipping
  volume with the OpenGL functions

xp
yp
zp d
34

• glMatrixMode(GL_PROJECTION)
• glLoadIdentity()
• Above commands affect the projection matrix.
• Purpose is to define a viewing volume (includes
  type of projection)
• glFrustum()
• gluPerspective()
• glOrtho()

35
OpenGL Orthogonal Viewing

• glOrtho(left,right,bottom,top,near,far)

near and far measured from camera
36
OpenGL Perspective

• glFrustum(left,right,bottom,top,near,far)

37
Using Field of View

• With glFrustum it is often difficult to get the
  desired view
• gluPerpective(fovy, aspect, near, far) often
  provides a better interface

front plane
aspect w/h
38
Normalization

• Rather than derive a different projection matrix
  for each type of projection, we can convert all
  projections to orthogonal projections with the
  default view volume
• This strategy allows us to use standard
  transformations in the pipeline and makes for
  efficient clipping

39
Pipeline View
modelview transformation
projection transformation
perspective division
4D ? 3D
nonsingular
clipping
projection
3D ? 2D
against default cube
40
Notes

• We stay in four-dimensional homogeneous
  coordinates through both the modelview and
  projection transformations
• Both these transformations are nonsingular
• Default to identity matrices (orthogonal view)
• Normalization lets us clip against simple cube
  regardless of type of projection
• Delay final projection until end
• Important for hidden-surface removal to retain
  depth information as long as possible

41
Orthogonal Normalization

• glOrtho(left,right,bottom,top,near,far)

normalization ? find transformation to
convert specified clipping volume to default
42
Orthogonal Matrix

• Two steps
• Move center to origin
• T(-(leftright)/2, -(bottomtop)/2,(nearfar)/2))
• Scale to have sides of length 2
• S(2/(left-right),2/(top-bottom),2/(near-far))

P ST
43
Final Projection

• Set z 0
• Equivalent to the homogeneous coordinate
  transformation
• Hence, general orthogonal projection in 4D is

Morth
P MorthST
44
Simple Perspective

• Consider a simple perspective with the COP at the
  origin, the near clipping plane at z -1, and a
  90 degree field of view determined by the planes
• x ?z, y ?z

45
Perspective Matrices

• Simple projection matrix in homogeneous
  coordinates
• Note that this matrix is independent of the far
  clipping plane

M
46
Generalization

• N

after perspective division, the point (x, y, z,
1) goes to
x x/z y y/z Z -(ab/z)
which projects orthogonally to the desired point
regardless of a and b
47
Picking a and b
If we pick
a
b
the near plane is mapped to z -1 the far plane
is mapped to z 1 and the sides are mapped to x
? 1, y ? 1
Hence the new clipping volume is the default
clipping volume
48
Normalization Transformation
distorted object projects correctly

• original clipping
• volume

original object
new clipping volume
49
Normalization and Hidden-Surface Removal

• Although our selection of the form of the
  perspective matrices may appear somewhat
  arbitrary, it was chosen so that if z1 gt z2 in
  the original clipping volume then the for the
  transformed points z1 gt z2
• Thus hidden surface removal works if we first
  apply the normalization transformation
• However, the formula z -(ab/z) implies that
  the distances are distorted by the normalization
  which can cause numerical problems especially if
  the near distance is small

50
OpenGL Perspective

• glFrustum allows for an unsymmetric viewing
  frustum (although gluPerspective does not)

51
OpenGL Perspective Matrix

• The normalization in glFrustum requires an
  initial shear to form a right viewing pyramid,
  followed by a scaling to get the normalized
  perspective volume. Finally, the perspective
  matrix results in needing only a final orthogonal
  transformation

P NSH
our previously defined perspective matrix
shear and scale
52
Why do we do it this way?

• Normalization allows for a single pipeline for
  both perspective and orthogonal viewing
• We stay in four dimensional homogeneous
  coordinates as long as possible to retain
  three-dimensional information needed for
  hidden-surface removal and shading
• We simplify clipping

53
Graphics (rendering) pipeline
Viewing Geometry Processing World Coordinates
(floating point)
Rendering Pixel Processing Screen Coordinates
(integer)
Conservative VSD Selective traversal of object
database (or traverse scene graph to get CTM)
Transform Vertices To canonical view
volume
Light at Vertices Calculate light intensity at
vertices (lighting model of choice)
Conservative VSD Back-face Culling
Conservative VSD View Volume Clipping
Image- precision VSD Compare pixel
depth (Z-buffer)
Shading Interpolate color values (Gouraud) or
normals (Phong)
per polygon
per pixel of polygon
avd November 9,
2004 VSD 6/48
avd November 5,
2002 VSD 5/45