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Arbitrary 3-D View-Plane

Arbitrary 3-D View-Plane. 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)

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Arbitrary 3-D View-Plane

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  1. Arbitrary 3-D View-Plane

  2. 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) Specifying An Arbitrary 3-D View

  3. Specifying A View

  4. 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, x=z, x = -z, z = zmin, z = zmax Normalizing Transformation

  5. 1 0 0 -VRPx ) 0 1 0 -VRPy ) 0 0 1 -VRPz ) 0 0 0 1 ) 1. Translate VRP to origin

  6. 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 2. Rotate VRC

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

  8. 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 )

  9. Center line of window lies along the vector [CW - PRP], this is the direction of projection, DoP. 4. Shear such that the center line of the view volume becomes the z-axis PRP

  10. ( (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]. Shear (cont.)

  11. ( 1 0 SHx 0 ) We want SH*DoP = 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 Shear (cont.)

  12. y = v - v 5. Scale max min 2 z=-PRPn + F z= -PRPn + B z=-PRPn Y axis +1 y = -v + v max min Before Scale y= -z 2 Front Back Clipping Plane Clipping z=-1 Plane -Z -1 View plane y= z After Scale -1

  13. 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) 5. Scale (cont.)

  14. (Sx 0 0 0 ) (0 Sy 0 0 ) (0 0 Sz 0 ) (0 0 0 1 ) 5. Scale (cont.) Sx = 2PRPn / [(umax - umin) (-PRPn + B)] Sy = 2PRPn / [(vmax - vmin) (-PRPn + B)] Sz = -1 / [(vmax - vmin) (-PRPn + B)]

  15. Nper = [Sper ][SHper ][T(-PRP) ][R ][T(-VRP)] Total Composite Transformation

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