Projections

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Projections
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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

• Orthographic Direction of projection is
  orthogonal to the projection plane
• Elevations Projection plane is perpendicular to
  a principle axis
• Front
• Top (Plan)
• Side
• Axonometric Projection plane is not orthogonal
  to a principle axis
• Isometric Direction of projection makes equal
  angles with each principle axis.
• Oblique Direction of projection is not
  orthogonal to the projection plane projection
  plane is normal to a principle 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 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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One- Point Projections
(x, y, z)

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

Z
(xproj, yproj, 0)
(0, 0, -d)
-Z
(1 0 0 0) (x) (x) (0 1 0 0) (y)
(y) (0 0 0 0) (z) (0) (0 0 1/d 1) (1) (z/d
1)
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Projections

• 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)

(1 0 0 0) (x) (x) (0 1 0 0) (y)
(y) (0 0 1 0) (z) (z) (0 0 1/d 0) (1) (z/d)
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 and back clipping planes (hither and yon)

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Specifying A View
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Normalizing Transformation for Perspective
Projection

• 1. Translate VRP to origin
• 2. 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
• 3. Translate so that the CoP given by the PRP is
  at the origin
• 4. Shear such that the center line of the view
  volume becomes the z-axis
• 5. Scale so that the view volume becomes the
  canonical view volume y z, y -z, xz, x
  -z, z zmin, z zmax

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1. Translate VRP to origin

• 1 0 0 -VRPx )
• 0 1 0 -VRPy )
• 0 0 1 -VRPz )
• 0 0 0 1 )

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2. Rotate VRC

• We want to take 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

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2. Rotate VRC (cont.)

• ux uy uz 0 )
• vx vy vz 0 )
• nx ny nz 0 )
• 0 0 0 1 )

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3. Translate so that the CoP given by the
PRP is at the origin

• 1 0 0 -PRPu )
• 0 1 0 -PRPv )
• 0 0 1 -PRPn )
• 0 0 0 1 )

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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 the direction of projection, DoP.

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Shear (cont.)

• ( (umax umin)/2 ) ( PRPu )
• CW ( (vmax vmin)/2 ) PRP ( PRPv )
• ( 0 ) ( PRPn )
• ( 1 ) ( 1 )
• ( (umax umin)/2 - PRPu )
• DoP CW-PRP ( (vmax vmin)/2 -
  PRPv )
• ( 0 - PRPn )
• ( 1 )
• The shear matrix must take this direction of
  projection and shear it to the z-axis , DoP'
  0, 0, DoPz.

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Shear (cont.)

• ( 1 0 SHx 0 ) We want SHDoP DoP'
• SH ( 0 1 SHy 0 )
• ( 0 0 1 0 )
• ( 0 0 0 1 )
• ( 1 0 SHx 0 ) ( (umax umin)/2 - PRPu )
  (0)
• ( 0 1 SHy 0 ) ( (vmax vmin)/2 - PRPv
  ) (0)
• ( 0 0 1 0 ) ( 0 - PRPn ) (DoPz)
• ( 0 0 0 1 ) ( 1 ) (1)
• SHx -DoPx/DoPz, SHy -DoPy/DoPz

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5. Scale
y v - v
max
min
2
z -PRPn B
z-PRPn F
z-PRPn
y -v v
max
min
2
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5. Scale (cont.)

• Scale is done in two steps
• 1. First scale in x and y
• xscale 2PRPn/(umax - umin)
• yscale 2PRPn/(vmax - vmin)
• 2. Scale everything uniformly such that the back
  clipping plane becomes z -1
• xscale -1 / (-PRPn B)
• yscale -1 / (-PRPn B)
• zscale -1 / (-PRPn B)

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Total Composite Transformation

• Nper Sper SHper T(-PRP) R T(-VRP)