Projection

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Projection

• Pradondet Nilagupta

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Definitions

• Projection a transformation that maps from a
  higher dimensional space to a lower dimensional
  space (e.g. 3D-gt2D)
• Center of projection (CoP) the position of the
  eye or camera with respect to which the
  projection is performed (also eyepoint, camera
  point, proj. reference point)
• Projection plane in a 3D-gt2D projection, the
  plane to which the projection is performed (also
  viewplane)

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Planar Projections
Perspective Distance to CoP is finite
Parallel Distance to CoP is infinite
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Projections
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Parallel Projections (1/2)

• Orthographic Direction of projection is
  orthogonal to the projection plane
• Elevations Projection plane is perpendicular to
  a principal axis
• Front
• Top (Plan)
• Side
• Axonometric Projection plane is not orthogonal
  to a principal axis
• Isometric Direction of projection makes equal
  angles with each principal axis.

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Parallel Projections (2/2)

• Oblique Direction of projection is not
  orthogonal to the projection plane projection
  plane is normal to a principal axis
• Cavalier Direction of projection makes a 45
  angle with the projection plane
• Cabinet Direction of projection makes a 63.4
  angle with the projection plane

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Perspective Projections

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

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Perspective Projections
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One-Point Projection
(x, y, z)

• Center of Projection on the negative z-axis with
  viewplane in the x-y plane.
• xprojected xd/(dz) x/(1(z/d))
• yprojected yd/(dz) y/(1(z/d))

Z
(xproj, yproj, 0)
(0, 0, -d)
-Z
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Another one-Point Projection

• Center of Projection at the origin with viewplane
  parallel to the x-y plane a distance d from the
  origin.
• xprojected dx/z x/(z/d)
• yprojected dy/z y/(z/d)

Z
-Z
Mper Points plotted are x/w, y/w where w z/d
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Specifying An Arbitrary 3-D View

• Two coordinate systems
• World reference coordinate system (WRC)
• Viewing reference coordinate system (VRC)
• First specify a viewplane and coordinate system
  (WRC)
• View Reference Point (VRP)
• View Plane Normal (VPN)
• View Up Vector (VUP)
• Specify a window on the view plane (VRC)
• Max and min u,v values (Center of the window
  (CW))
• Projection Reference Point (PRP)
• Front (F) and back (B) clipping planes (hither
  and yon)

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Specifying A View
v
VUP
(umax, vmax)
VRP
CW
(umin, vmin)
u
VPN
n
PRP
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Normalizing (Viewing) Transformation

• Translate VRP to origin
• Rotate the VRC system such that the VPN (n-axis)
  becomes the z-axis, the u-axis becomes the x-axis
  and the v-axis becomes the y-axis
• Translate so that the CoP given by the PRP is at
  the origin
• Shear such that the center line of the view
  volume becomes the z-axis
• Scale so that the view volume becomes the
  canonical view volume, bounded by six planes y
  z, y -z, xz, x -z, z zmin, z zmax

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1. Translate VRP to origin
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2. Rotate VRC (1/2)

• We want to transform
• u into (1, 0, 0)
• v into (0, 1, 0)
• n into (0, 0, 1)
• First derive n, u, and v from user input
• n VPN / VPN
• u (Vup x n) / Vup x n
• v n x u
• u, v, and n are unit (normalized) vectors

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2. Rotate VRC (2/2)

• Special orthogonal matrix

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3.Translate so that the PRP is at the origin
Note The PRP is given in View Reference
Coordinates. The originof the VRC (the VRP) is
currently at the origin of world coordinates,so
this translation will move the PRP to the world
origin.
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4. Shear such that the center line of the view
volume becomes the z-axis
Center line of window lies along the vector CW,
-PRP, this is thedirection of projection, DoP
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4. Shear (cont.)

• The shear matrix must take this direction of
  projection and shear it to the z-axis , DoP'
  0, 0, DoPz.

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4. Shear (cont.)

• We want to find a shear matrix SHper such that
• SHDoP DoP0, 0, DoPz
• SHx -DoPx/DoPz, SHy -DoPy/DoPz

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5. Scale (1/3)
y v - v
max
min
current view volume
2
z -PRPn B
z-PRPn F
z-PRPn
y -v v
max
min
2
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5. Scale (2/3)
desired view volume
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5. Scale (3/3)

• Scale is done in two steps
• 1. First scale in x and y so sides of view
  volume form 45 angles
• sx 2PRPn/(umax - umin)
• sy 2PRPn/(vmax - vmin)
• 2. Scale everything uniformly such that the back
  clipping plane becomes z -1
• sx -1 / (-PRPn B)
• sy -1 / (-PRPn B)
• sz -1 / (-PRPn B)

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Total Composite Transformation

• Nper Sper SHper T(-PRP) R T(-VRP)

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3D Rendering Sequence of events

• 1. Define object in world coordinates
• 2. Define viewing parameters
• 3. Apply normalizing transformation to object
• 4. Clip object in the canonical view volume
• 5. Project object to the viewplane
• 6. Window-viewport transformation
• 7. Rasterize object

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3D Viewing example

• Object triangle w/ vertices (0,0,0), (1,0,0),
  (0,0,1)
• VRP (2,1,2)
• VPN (1,0,1)
• VUP (0,1,0)
• PRP (view coords.) (0,0,2)
• window from (-2,-1) to (2,1) (u,v coords.)
• Back clipping plane at n-4
• viewport 200x100 pixels

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3D Viewing example
Y
VRP
X
PRP
Z
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Step 1 Translate VRP to origin

• T (-2,-1,-2)
• (0,0,0) becomes (-2,-1,-2)
• (1,0,0) becomes (-1,-1,-2)
• (0,0,1) becomes (-2,-1,-1)

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Step 2 Rotate VRC into WRC

• Rotate by -45 deg. about y-axis
• (-2,-1,-2) becomes (0,-1, -2v2)
• (-1,-1,-2) becomes (v2/2,-1,-3v2/2)
• (-2,-1,-1) becomes (-v2/2,-1,-3v2/2)

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Step 3 Translate PRP to origin

• T(0,0,-2)
• (0,-1,-2v2) becomes (0,-1,-2v2-2)
• (v2/2,-1,-3v2/2) becomes
• (v2/2,-1,-3v2/2-2)
• (-v2/2,-1,-3v2/2) becomes
• (-v2/2,-1,-3v2/2-2)

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Step 4 Shear

• Not necessary because center line of view volume
  already lies on the z-axis

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Step 5a Scale to canonical view volume

• Scale by 4/4 in x, 4/2 in y
• (0,-1,-2v2-2) becomes (0,-2,-2v2-2)
• (v2/2,-1,-3v2/2-2) becomes
• (v2/2,-2,-3v2/2-2)
• (-v2/2,-1,-3v2/2-2) becomes
• (-v2/2,-2,-3v2/2-2)

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Step 5b Scale to canonical view volume

• Scale by -1/(-2-4)1/6 in all dimensions
• (0,-2,-2v2-2) becomes (0,-1/3,-v2/3-1/3)
• (v2/2,-2,-3v2/2-2) becomes
• (v2/12,-1/3,-v2/4-1/3)
• (-v2/2,-2,-3v2/2-2) becomes
• (-v2/12,-1/3,-v2/4-1/3)

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Projecting the points

• (x,y,z) projects to (x/(z/d), y/(z/d)) where d
  -1/3
• (0,-1/3,-v2/3-1/3) to (0,-1/3(v21))
• (v2/12,-1/3,-v2/4-1/3) to (v2/(9v212),
  -4/(9v212))
• (-v2/12,-1/3,-v2/4-1/3) to
• (-v2/(9v212), -4/(9v212))

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Window-viewport transform

• translate by (1/3,1/3), scale by (300,150)
• (0,-1/3(v21)) becomes (100, -50/(v21)50)
• (v2/(9v212), -4/(9v212)) becomes
• (300v2/(9v212)100, -600/(9v212)50)
• (-v2/(9v212), -4/(9v212)) becomes
• (-300v2/(9v212)100, -600/(9v212)50)

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Final rendered image
100
50
(100,29.29)
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(82.84,25.74)
(117.16,25.74)
0
200
100