Last Time

1
Last Time

• Canonical view pipeline

Local Coordinate Space
World Coordinate Space
View Space
3D Screen Space
Display Space
Projection
2
Today

• General Orthographic
• Perspective viewing
• Simple case
• Completely general case

3
Simple Orthographic Projection

• Specify the region of space that we wish to
  render as a view volume
• Assume that world coordinate axes line up with
  the image plane coordinate axes
• Viewer is looking in the z direction, with x to
  the right and y up
• The view volume has
• a near plane at zn
• a far plane at zf , (f lt n)
• a left plane at xl
• a right plane at xr, (rgtl)
• a top plane at yt
• and a bottom plane at yb, (bltt)

y
(l,t,f)
z
x
(r,b,n)
4
Rendering the Volume

• To project, map the view volume onto the
  canonical view volume
• After that, we know how to map the view volume to
  the window
• The mapping looks similar to the one for
  canonical?window

5
General Orthographic Projection

• We could look at the world from any direction,
  not just along z
• The image could rotated in any way about the
  viewing direction x need not be right, and y
  need not be up
• How can we specify the view under these
  circumstances?

6
Specifying a View

• The location of the eye in space
• A point in space for the center of projection,
  (ex,ey,ez)
• The direction in which we are looking gaze
  direction
• Specified as a vector (gx,gy,gz)
• This vector will be normal to the image plane
• A direction that we want to appear up in the
  image
• (upx,upy,upz), This vector does not have to be
  perpendicular to n
• We also need the size of the view volume
  l,r,t,b,n,f
• Specified with respect to the eye and image
  plane, not the world

7
Getting there

• We wish to end up in the simple situation, so
  we need a coordinate system with
• A vector toward the viewer
• One pointing right in the image plane
• One pointing up in the image plane
• The origin at the eye
• We must
• Define such a coordinate system, view space
• Transform points from the world space into view
  space
• Apply our simple projection from before

8
View Space

• Given our camera definition
• Which point is the origin of view space?
• Which direction is the normal to the view plane,
  w?
• How do we find the right vector, u?
• How do we find the up vector, v?
• Given these points, how do we do the
  transformation?

9
View Space

• The origin is at the eye (ex,ey,ez)
• The normal vector is the normalized viewing
  direction
• It is actually more general to think of it as the
  vector perpendicular to the image plane, as you
  must do in the homework
• We know which way up should be, and we know we
  have a right handed system, so uupw,
  normalized
• We have two vectors in a right handed system, so
  to get the third vwu

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World to View

• We must translate so the origin is at (ex,ey,ez)
• To complete the transformation we need to do a
  rotation
• After this rotation
• The direction u in world space should be the
  direction (1,0,0) in view space
• The vector v should be (0,1,0)
• The vector w should be (0,0,1)
• The matrix that does that is

11
All Together

• We apply a translation and then a rotation, so
  the result is
• And to go all the way from world to screen

12
OpenGL and Transformations

• OpenGL internally stores several matrices that
  control viewing of the scene
• The MODELVIEW matrix is intended to capture all
  the transformations up to the view space
• The PROJECTION matrix captures the view to screen
  conversion
• You also specify the mapping from the canonical
  view volume into window space
• Directly through function calls to set up the
  window
• Matrix calls multiply some matrix M onto the
  current matrix C, resulting in CM
• Set view transformation first, then set
  transformations from local to world space last
  one set is first one applied

13
OpenGL Camera

• The default OpenGL image plane has u aligned with
  the x axis, v aligned with y, and n aligned with
  z
• Means the default camera looks along the negative
  z axis
• Makes it easy to do 2D drawing (no need for any
  view transformation)
• glOrtho() sets the view-gtcanonical matrix
• Modifies the PROJECTION matrix
• gluLookAt() sets the world-gtview matrix
• Takes an image center point, a point along the
  viewing direction and an up vector
• Multiplies a world-gtview matrix onto the current
  MODELVIEW matrix
• You could do this yourself, using glMultMatrix()
  with the matrix from the previous slides

14
Left vs Right Handed View Space

• You can define u as right, v as up, and n as
  toward the viewer a right handed system u?vw
• Advantage Standard mathematical way of doing
  things
• You can also define u as right, v as up and n as
  into the scene a left handed system v?uw
• Advantage Bigger n values mean points are
  further away
• OpenGL is right handed
• Many older systems, notably the Renderman
  standard developed by Pixar, are left handed

15
Perspective Projection

• Abstract camera model - box with a small hole in
  it

• Pinhole cameras work in practice - camera
  obscura, etc

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Distant Objects Are Smaller
17
Parallel lines meet
common to draw film plane in front of the focal
point
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Vanishing points

• Each set of parallel lines (direction) meets at
  a different point The vanishing point for this
  direction
• Classic artistic perspective is 3-point
  persepctive
• Sets of parallel lines on the same plane lead to
  collinear vanishing points the horizon for that
  plane
• Good way to spot faked images

19
Basic Perspective Projection

• Assume you have transformed to view space, with x
  to the right, y up, and z back toward the viewer
• Assume the origin of view space is at the center
  of projection (the eye)
• Define a focal distance, d, and put the image
  plane there (note d is negative)

20
Basic Perspective Projection

• If you know P(xv,yv,zv) and d, what is P(xs,ys)?
• Where does a point in view space end up on the
  screen?

P(xv,yv,zv)
P(xs,ys)
yv
d
-zv
xv
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Basic Case

• Similar triangles gives

yv
P(xv,yv,zv)
P(xs,ys)
d
-zv
View Plane
22
Simple Perspective Transformation

• Using homogeneous coordinates we can write

23
Perspective View Volume

• Recall the orthographic view volume, defined by a
  near, far, left, right, top and bottom plane
• The perspective view volume is also defined by
  near, far, left, right, top and bottom planes
  the clip planes
• Near and far planes are parallel to the image
  plane zvn, zvf
• Other planes all pass through the center of
  projection (the origin of view space)
• The left and right planes intersect the image
  plane in vertical lines
• The top and bottom planes intersect in horizontal
  lines

24
Clipping Planes
Left Clip Plane
Near Clip Plane
xv
Far Clip Plane
View Volume
l
n
-zv
f
r
Right Clip Plane
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Where is the Image Plane?

• Notice that it doesnt really matter where the
  image plane is located, once you define the view
  volume
• You can move it forward and backward along the z
  axis and still get the same image, only scaled
• But we need to know where it is to define the
  clipping planes
• Assume the left/right/top/bottom planes are
  defined according to where they cut the near clip
  plane
• Or, define the left/right and top/bottom clip
  planes by the field of view

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Clipping Planes
Left Clip Plane
Near Clip Plane
xv
Far Clip Plane
View Volume
FOV
-zv
f
Right Clip Plane
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OpenGL

• gluPerspective()
• Field of view in the y direction (vertical
  field-of-view)
• Aspect ratio (should match window aspect ratio)
• Near and far clipping planes
• Defines a symmetric view volume
• glFrustum()
• Give the near and far clip plane, and places
  where the other clip planes cross the near plane
• Defines the general case
• Used for stereo viewing, mostly

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Perspective Projection Matrices

• We want a matrix that will take points in our
  perspective view volume and transform them into
  the orthographic view volume
• This matrix will go in our pipeline just before
  the orthographic projection matrix

(r,t,n)
(l,b,n)
(r,t,n)
(l,b,n)
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Mapping Lines

• We want to map all the lines through the center
  of projection to parallel lines
• Points on lines through the center of projection
  map to the same point on the image
• Points on parallel lines map orthographically to
  the same point on the image
• If we convert the perspective case to the
  orthographic case, we can use all our existing
  methods
• The intersection points of lines with the near
  clip plane should not change
• The matrix that does this, not surprisingly,
  looks like the matrix for our simple perspective
  case

30
General Perspective

• This matrix leaves points with zn unchanged
• It is just like the simple projection matrix, but
  it does some extra things to z to map the depth
  properly
• We can multiply a homogenous matrix by any number
  without changing the final point, so the two
  matrices above have the same effect

31
Complete Perspective Projection

• After applying the perspective matrix, we still
  have to map the orthographic view volume to the
  canonical view volume

32
OpenGL Perspective Projection

• For OpenGL you give the distance to the near and
  far clipping planes
• The total perspective projection matrix resulting
  from a glFrustum call is

33
Near/Far and Depth Resolution

• It may seem sensible to specify a very near
  clipping plane and a very far clipping plane
• Sure to contain entire scene
• But, a bad idea
• OpenGL only has a finite number of bits to store
  screen depth
• Too large a range reduces resolution in depth -
  wrong thing may be considered in front
• See Shirley for a more complete explanation
• Always place the near plane as far from the
  viewer as possible, and the far plane as close as
  possible