Lecture 5 Viewing and projections

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Lecture 5Viewing and projections

• Viewing
• In OpenGL
• The camera frame
• Other view APIs
• Projections
• Perspective projections
• Parallel projections
• View volumes
• Parallelpiped
• Frustum

2
Viewing

• Modeling coords
• What we model object in
• Independent of viewing
• OpenGL uses the modeling part of the model-view
  matrix to convert from the modeling frame to the
  world frame
• OpenGL camera
• Placed at origin of world frame pointing in
  negative z direction

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Viewing

• If we model scene around origin
  the camera will not see all objects so we
  must
• 1) move camera back, or
• 2) move scene to in front of camera
• Sequence of operations
• 1.Begin with model_view matrix identity matrix
• 2.Define objects using glVertex
• 3.Apply model_view matrix to move world frame to
  camera frame (I.e. move objects to eye space)
• If we now position an object, is w.r.t. the
  repositioned world frame

These are equivalent operations
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Camera positioning

• OpenGL Example
• This moves points Or we can think
  of it as world space
  positioning the camera

glMatrixMode (GL_MODEL_VIEW) glLoadIdentity
() glTranslate (0, 0, -d) glRotate (0, -90, 1,
0)
z-d
-90deg
zd
90deg
M TR T(0,0,-d) Ry(-90)
M-1 R-1T-1 Ry(90) T(0,0,d)
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ExampleFind isometric view (p205)
y
45deg
(1,1,1)
(-1,1,1)
z
y
?
?
(0,1,sqrt(2))
d sqrt(11sqrt(2)sqrt(2))
sqrt(12) sqrt(3)
? cos? sqrt(2)/sqrt(3) sqrt(6)/3 sin?
sqrt(1)/sqrt(3) sqrt(3)/3
? ? acos(sqrt(6)/3) 35.26
z
y
z
R
RxRy
sqrt(2)/2 0 sqrt(2)/2 0 0 1
0 0 -sqrt(2)/2 0 sqrt(2)/2 0 0
0 0 1
1 0 0 0 0 sqrt(6)/3
sqrt(3)/3 0 0 -sqrt(3)/3 sqrt(6)/3 0 0 0
0 1
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Defining a camera frame

• 1. Position camera (VRP)
• set_view_reference_point (x, y, z)
• 2. Specify view plane by its normal (VPN)
• set_view_plane_normal (nx, ny, nz)
• 3. Specify camera up vector (VUP)
• set_view_up (vupx, vupy, vupz)
• (VUP must be orthogonal to VPN, so project to VP)
• 4. This gives us two orthogonal vectors, v and n
• find the third, u v x n

vup
n
v
VP
vrp
u
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Creating a camera frame

• Given p, n (unit), and up
• Find view matrix, V, which takes x,y,z to u,v,n
  
• x,y,z are world coords, u,v,n are eye coords
• V RT

ux uy uz 0 vx vy vz 0 nx ny nz 0 0 0
0 1
1 0 0 -vrpx 0 1 0 -vrpy 0 0 1 -vrpz 0 0
0 1
pu pv pn 1
px py pz 1
V
Need to find u and v
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Creating a camera frame

• Find v
• Find u

1. Project up onto n ((upn)/(nn)n) 2.
Subtract the result (the comp. of up in the
direction of n) from up to get v, which is
orthogonal to n and thus lies in the plane
VP v up - (upn)/(nn)n v (I -
nnT/nTn) up 3. Normalize
up
v
u
VP
n
This is the projection onto VP. What does this
remind you of?
1. Take the cross product u v x n 2. Normalize
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OpenGL camera specification

• Instead of arbitrarily specifying n,
  we can specify an eye position and a
  lookat position in world coords and
  compute n
• n eye - at (in OpenGL, vrp eye)
• In OpenGL, well use this convenient function
• gluLookAt (eyex, eyey, eyez, atx, aty, atz, upx,
  upy, upz)

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Other view APIs

• Flight simulation
• Orientation of an airplane is called its attitude
• Defined by 3 angles independent angles around x,
  y, and z axes (Euler angles)
• pitch (x), yaw (y), and roll (z)

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Other view APIs

• Celestial navigation
• Positions given in polar coordinates
• elevation - angle between the optical
  axis and the horizontal plane of view
• - sweeps optical axis up and down
• azimuth - angle between the optical
  axis and some arbitrary reference
  direction in the horizontal plane
• - sweeps optical axis left to right.
• twist angle - rotates camera about optical axis

ELEVATION ANGLE
AZIMUTH ANGLE
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Projections

• A projection is
• A mapping from Rn ? Rn
• we often assume z0, but we are still in 3D
• Idempotent gt PPP P
• subsequent projections have no effect
• Two main types we will use in CG
• Parallel
• Perspective
• Can be combined for a generalized projection

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Projections

• Taxonomy of projections

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Projections
Perspective
(eye, origin of camera frame)
COP
view plane (VP)
view volume, or frustum
Parallel
As COP moves to infinity, rays become parallel
we say DOP (direction of proj.)
Both perspective and parallel projections are
planar geometric projections because the surface
is a plane and the projectors are lines
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Perspective projections

• Characteristics
• Parallel lines of the object that are not
  parallel to view plane converge to a vanishing
  point
• Closer objects look larger than farther objects
• Natural view, used in rendering and animation
• Does not preserve lengths or angles
• Types
• One, two, three point perspectives
• defined by number of principal axes cut by proj.

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Perspective projections

• One-point
• One principle axis cut by projection plane
• One axis vanishing point
• Two-point
• Two principle axes cut by projection plane
• Two axis vanishing points
• Three-point
• Three principle axes cut by projection plane
• Three axis vanishing points

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Perspective projections
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Perspective projections
Assume that the view plane (VP) is
orthogonal to the z axis at z
d. Find the projection p of a point p By
similar triangles, xp/d x/z gt xp (dx)/z
x/(z/d) yp/d y/z gt yp (dy)/z y/(z/d)
p(x,y,z)
d is a scale factor applied to x and
y, division by z causes distant objects to
appear smaller than closer objects perspective
proj is irreversible (do you see why?)
pp(xp,yp,zp)
z
z0 COP
zd VP
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Perspective projections

• The fact that many points map to one
  point is a problem
• We need depth info for hidden surface removal
• Homogeneous coordinates allow a fix
• Use p (x, y, z, w) instead of p (x, y, z, 1)

x y z z/d
x/(z/d) y/(z/d) d 1
xp yp zp 1
1 0 0 0 0 1 0 0 0 0 1 0 0 0
1/d 0
x y z 1



Note w z/d cannot be zero (gt pts on plane z0
do not project) (This division is not
technically part of the projection.)
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Parallel projections

• Characteristics
• Maps 3D points to view plane (VP) along lines
  parallel to VP
• Preserves relative proportions of objects
• used in drafting, esp. for plan views (top, side,
  etc.) because it gives accurate measurements of
  lengths and angles
• does not give a realistic appearance
• Defined by a projection vector
• Two types, orthographic and oblique

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Parallel projections

• Orthographic parallel projection
• DOP is orthographic to projection plane
• Elevations
• proj. plane is perpendicular to a principle axis.
  
• front, top (plan), side
• Axonometric
• proj. plane is not orthogonal to a principle axis
  
• more than one face of an object will be in image
• does not preserve distances or angles (in
  general, isometric is an exception)

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Parallel projections

• Axonometric projections
• Isometric
• DOP makes equal angles with each principle axis
• Most common axonometric projection
• Maintains relative proportions
• Dimetric
• Proj. plane placed symmetrically to 2 faces
• Trimetric
• General case

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Parallel projections

• Oblique parallel projection
• DOP is not orthogonal to the projection plane
  projection plane is normal to a principle axis.
• less natural, eye and camera lens are ? parallel
  to image plan, circles are projected to ellipses
• Cavalier
• DOP makes 45deg. angle with the proj. plane
• Cabinet
• DOP makes a 63.4deg. angle with the proj. plane

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Projections

• Isometric projection - architectural
• http//www.mda1.demon.co.uk/html/htmpd/smary400.h
  tm
• Projection examples - varied http//cleo.murdoch.e
  du.au/asu/pubs/tlf/tlf99/ns/ostrogonac.html
• Projection tutorial http//mane.mech.Virginia.EDU/
  engr160/Graphics/Projection.html

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Parallel projections

• Assume the VP is at z 0
• Then the projection is easy,

pp(xp,yp,zp)
p(x,y,z)
DOP
z
z0 VP
xp yp zp 1
1 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1
x y z 1
x y 0 1


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Generalized projections
1 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1
Assumes DOP parallel to z axis
Mparallel
1 0 0 0 0 1 0 0 0 0 1 0 0 0
1/d 0
Assumes COP at origin
Mperspective
A more robust formulation not only removes these
restrictions but also integrates perspective and
parallel projections into a single matrix
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View volumes

• Two types
• View volume for orthographic projection is a
  right parallelpiped
• View frustum for perspective is a clipped pyramid
• Defined by six clip planes, left/right, top/
  bottom, near/far (also hither/yon front/back)

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View volumes in OpenGL

• Frustum
• glFrustum(xmin, xmax, ymin, ymax, near, far)
• near and far must be positive distances measured
  from COP, they describe planes parallel to z0
• frustum does not have to be a right frustum,
  i.e., xmin/xmax and ymin/ymax
• Alters CTM
• glMatrix(GL_PROJECTION)
• glLoadIdentity()
• glFrustum(xmin, xmax, ymin, ymax,near, far)

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View volumes in OpenGL

• Alternatively, we can specify volume
  using the angle, or field, of view
• gluPerspective (fovy, aspect, near, far)
• fovy is angle between top and bottom clip planes
• aspect is the aspect ratio (width/height) of VP
• near and far are the same as before
• Alters CTM

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Perspective view frustum

• Define frustum by field of view and
  near and far clip planes

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View volumes in OpenGL

• Volume
• glOrtho (xmin, xmax, ymin, ymax, near, far)
• near must be less than far, but the distances do
  not need to be positive w.r.t. COP
• we dont have to worry about division by zero
• arguments are otherwise the same as in glFrustum

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Z-Buffering

• We only want to render surfaces that
  are visible
• Algorithms that do this are called
  hidden-surface removal algorithms
• z-buffer algorithm replace frame buffer color if
  current depth is less than stored depth

pp?
pr
b
pg
35
pb
distance from VP
pp?
70
45
35
z
VP
frame buffer depth buffer
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Z-Buffering

• z-buffer algorithm
• Part of projection process (image space)
• (object space algorithms work on objects)
• OpenGL
• glutInitDisplayMode(GLU_RGB GLUT_DEPTH) -
  initialize buffers
• glEnable(GL_DEPTH_TEST) - enable vsp
• glClear(GL_DEPTH_TEST) - clear buffer

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Parallel projection matrices

• Projection normalization
• Convert all projections to orthogonal proj
• 1. Normalize view volume (will distort object)
• 2. Project orthogonally
• The orthogonal projection of the distorted object
  will equal the original projection of the
  original object

(x,y,z)
(xp,yp,zp)
orthogonal proj of distorted obj
oblique proj of object
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View volume normalization

• 1. Shear view volume to make ortho to VP
• Find matrix that shears volume in x and y
• Oblique projection characterized by angle
  projectors make with VP

cos? (x-xp)/len sin? z/len gt tan?
z/(x-xp) gt xp x-zcot?
perspective view
top view (from y)
co ? (y-yp)/len sin? z/len gt tan?
z/(y-yp) gt yp y-zcot?
And, zp 0
side view (from x)
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General parallel projection

• 1. Shear view volume to make ortho to VP
• Find matrix that shears volume in x and y
• Oblique projection characterized by angle
  projectors make with VP
• Find xp

len sqrt(x2z2) sub xp from x (moves xp to
origin) cos? (x-xp)/len sin? z/len gt
tan? z/(x-xp) gt cot? (x-xp)/z gt
x-xp zcot? gt xp x-zcot?
top view (from y)
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General parallel projection

• Find yp
• Find zp

len sqrt(y2z2) sub yp from y (moves yp to
origin) cos? (y-yp)/len sin? z/len gt
tan ? z/(y-yp) gt cot ? (y-yp)/z gt
y-yp zcot ? gt yp y-zcot ?
side view (from x)
And, zp 0
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View volume normalization

• OpenGL
• Canonical view volume in OpenGL is x
  y z -1 (maps to 2D screen coords)
• We specify a convenient clip volume with
  glOrtho(xmin,xmax,ymin,ymax,near,far)
• Let GL map our volume to the canonical one

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View volume normalization

• 2. Scale and translate view volume
  to make canonical M ST
• Translate to origin
• T( -(xmaxxmin)/2, -(ymaxymin)/2, -(zmaxzmin)/2
  )
• Scale to 2x2x2
• S( 2/(xmax-xmin), 2/(ymax-ymin), 2/(zmax-zmin) )

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General parallel projection
Mortho
H
TS
PMorthoSTH
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General parallel projection
1. Align proj. vector with VP normal (shear
in x,y along z)
1 0 kxz 0 0 1 kyz 0 0 0 1 0 0 0 0 1
1 0 -cot? 0 0 1 -cot? 0 0 0 1 0 0 0
0 1
Hxy

where so that kxz
-px/pz x? x - z(px/pz) kyz
-py/pz y? y - z(py/pz)
z? z
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General 3x3 shear matrix
Read entry kxy as a shear in x along y
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General parallel projection
2. Scale to normalized device coords
(NDC) M ST
1 0 0 Tx 0 1 0 Ty 0 0 1 Tz 0 0
0 1
1 0 0 -(xmaxxmin)/2 0 1 0 -(ymaxymin)/2 0
0 1 -(zmaxzmin)/2 0 0 0 1
T

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General parallel projection
Mortho
H
TS
PMorthoSTH
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General perspective projection
cop
cop
(1,1,1)
(-1,-1,-1)
PMorthNPSH
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General perspective proj.
1. Shear view volume so centerline of proj is
perp to VP (shear in x,y along z)
shear c ((lr)/2, (tb)/2, n) to c? (0,0,n)
cop
cop
H
where so that kxz
(lr)/2n x? x z(lr)/2n kyz
(tb)/2n y? y z(tb)/2n
z? z
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General perspective proj.
2. Scale sides of frustum,i.e. normalize in
x,y (want l -1, r 1, n -1)
cop
r 1
l-1
n-1
eye
eye
3. Persp proj creates rect. parallelpiped
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General perspective proj.
4. Normalize in z (scale and translate far
plane)
fz1
1 0 0 0 0 1 0
0 0 0 (fn)/(f-n) 2fn/(f-n) 0 0
1 1
Nz
Final projection matrix is
P Hxy Sxy Ppersp Nz
in text, N includes Ppersp
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General perspective projection
cop
cop
(1,1,1)
(-1,-1,-1)
PMorthNPSTH