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Projections

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DoP = [CW-PRP] = ( (vmax vmin)/2 - PRPv ) ( 0 - PRPn ) ( 1 ) ... this direction of projection and shear it to the z-axis , DoP' = [0, 0, DoPz] ... – PowerPoint PPT presentation

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Title: Projections


1
Projections
2
Planar Projections
Perspective Distance to CoP is finite
Parallel Distance to CoP is infinite
3
Projections
4
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

5
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

6
Perspective Projections
7
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)
8
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
9
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)

10
Specifying A View
11
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

12
1. Translate VRP to origin
  • 1 0 0 -VRPx )
  • 0 1 0 -VRPy )
  • 0 0 1 -VRPz )
  • 0 0 0 1 )

13
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

14
2. Rotate VRC (cont.)
  • ux uy uz 0 )
  • vx vy vz 0 )
  • nx ny nz 0 )
  • 0 0 0 1 )

15
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 )

16
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.

17
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.

18
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

19
5. Scale
y v - v
max
min
2
z -PRPn B
z-PRPn F
z-PRPn
y -v v
max
min
2
20
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)

21
Total Composite Transformation
  • Nper Sper SHper T(-PRP) R T(-VRP)
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