Title: Projections
1Projections
2Planar Projections
Perspective Distance to CoP is finite
Parallel Distance to CoP is infinite
3Projections
4Parallel 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
5Perspective 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
6Perspective Projections
7One- 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)
8Projections
- 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
9Specifying 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)
10Specifying A View
11Normalizing 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
121. Translate VRP to origin
- 1 0 0 -VRPx )
- 0 1 0 -VRPy )
- 0 0 1 -VRPz )
- 0 0 0 1 )
132. 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
142. Rotate VRC (cont.)
- ux uy uz 0 )
- vx vy vz 0 )
- nx ny nz 0 )
- 0 0 0 1 )
153. 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 )
164. 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.
17Shear (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.
18Shear (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
-
195. Scale
y v - v
max
min
2
z -PRPn B
z-PRPn F
z-PRPn
y -v v
max
min
2
205. 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)
21Total Composite Transformation
- Nper Sper SHper T(-PRP) R T(-VRP)