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					Computer Graphics

    Projections
                 3D Viewing
• Inherently more complex than 2D case.
  – Most display devices are only 2D
• Need to use a projection to transform 3D
  object or scene to 2D display device.




                                             2
         Generalised Projections.
• Transforms points in a coordinate system of
  dimension n into points in one of less than n (ie
  3D to 2D)
• Projectors emanate from a centre of projection,
  pass through every point in the object and
  intersect a projection surface to form the 2D
  projection.

                                                  3
                        Projections.
• Henceforth refer to planar geometric projections as just:
  projections.
• Two classes of projections :
     – Perspective.
     – Parallel.                            Parallel
                            A                          A   Parallel
                                             A
                  A
 Centre of
Projection.                                                B
                       B       B                          Centre of
                                                  B       Projection
    Perspective                                            at infinity
A Taxonomy of Projections
                                 Planar geometric projections.




                      Parallel                            Perspective


   Orthographic                         Oblique             1 point

        Axonometric       Cabinet       Cavalier            2 point

         Isometric                                          3 point

Elevations
           Perspective Projections.
• Defined by projection plane and centre of projection.

• Visual effect is termed perspective foreshortening.
   – The size of the projection of an object varies inversely with
     distance from the centre of projection.
   – Similar to a camera - Looks realistic !

• Not useful for metric information
   – Parallel lines do not in general project as parallel.
   – Angles only preserved on faces parallel to the projection
     plane.
   – Distances not preserved

                               Graphics                          6
           Perspective Projections
• A set of lines not parallel to
  the projection plane
  converge at a vanishing
  point.
   – Can be thought of in 3D as the
     projection of a point at
     infinity.
   – Homogeneous coordinate is 0
     (x,y,0)




                                      7
             1-Point Projection

Projection plane cuts 1
axis only.
1-Point Perspective


                      A painting (The
                      Piazza of St. Mark,
                      Venice) done by
                      Canaletto in 1735-
                      45 in one-point
                      perspective
2-Point Perspective

                            y




                      z          x




                  Projection plane
2-Point Perspective
                  Painting in two point
                  perspective by
                  Edward Hopper
                  The Mansard Roof
                  1923 (240 Kb);
                  Watercolor on paper,
                  13 3/4 x 19 inches;
                  The Brooklyn
                  Museum, New York
3-Point Perspective
Generally held to add little beyond 2-point perspective.

                A painting (City
                Night, 1926) by
                Georgia O'Keefe, that
                is approximately in
                three-point
                perspective.
                                                  y
                                             z             x




                                        Projection plane
            Parallel Projections
• Specified by a direction to the centre of projection,
  rather than a point.
   – Centre of projection at infinity.
• Orthographic
   – The normal to the projection plane is the same as the
     direction to the centre of projection.
• Oblique
   – Directions are different.
Orthographic Projections
                  Most common orthographic
                  Projection :

                  Front-elevation,
                  Side-elevation,
                  Plan-elevation.

                  Angle of projection parallel to
                  principal axis; projection plane
                  is perpendicular to axis.

                  Commonly used in technical
                  drawings




                                           15
           Isometric Projection
• Most common axonometric projection
• Projection plane normal makes equal
  angles with each axis.
• i.e normal is (dx,dy,dz), |dx| = |dy|=|dz|
• Only 8 directions that satisfy this
  condition.
                      Isometric Projection

                                                            y

                          y



Normal                                             120º          120º


                                                          120º
                                  x            z                        x
         Projection             All 3 axes equally foreshortened
           Plane
                          z     - measurements can be made
                                - Hence the name iso-metric
           Oblique projections.
• Projection plane normal differs from the direction
  of projection.
• Usually the projection plane is normal to a
  principal axis.
   – Projection of a face parallel to this plane allows
     measurement of angles and distance.
   – Other faces can measure distance, but not angles.
   – Frequently used in textbooks : easy to draw !
 Oblique projection

Normal
Parallel to x axis
                         y




                             x
            Projection
              Plane
                         z
 Geometry of Oblique Projections
• Point P=(0,0,1) maps to:
          P’=(l.cosa, l.sina, 0) on xy plane,
    and P(x,y,z) onto P’(xp,yp,0)
                                      1   0 l cosa 0
x p  x  z (l cosa )                 0   1 l sin a 0
                         and   M ob                 
y p  y  z (l sin a )                0   0    0    0
                                                     
                                      0   0    0    1
Perspective Projection – Simplest
              Case
Centre of projection at the origin,
Projection plane at z=d.
                                                    Projection
                                                    Plane.
           y
                                                           P(x,y,z)



                      x
                                      Pp(xp,yp,d)




                       d                                              z
     Perspective Projection – Simplest
                   Case
From similar triangles :
xp x yp y                                   x
   ;  
d  z d   z
                                                        xp     P(x,y,z)

       dx    x         dy    y
xp             ; yp                                                 z
        z    z/d         z    z/d
                                                    d

       y                                        d
                             P(x,y,z)
                                                                  z
           x
                    Pp(xp,yp,d)
                                                        yp   P(x,y,z)
           d                      z
                                        y
                  Perspective Projection

x                                                              
                                              T
                  d 1  d .x d.y            x y z z T
                       T
           yp                           d 1
     p
                         z
                                 z        
                                                     d

         The transform                       d
                      ation can be represente as a 4x4 matrix :
                  1   0   0 0
                  0   1 0 0
         M per              
                  0   0 1 0
                             
                  0   0 1/d 0
           Perspective Projection
Represent the general projected point Pp  X       Z W
                                                        T
                                                Y
                 1   0 0     0  x 
                 0   1 0     0  y 
Pp  M per  P                
                 0   0 1     0  z 
                                
                 0   0 1/d   0  1 


 X       Z W   x             z / d
                  T                      T
       Y                  y   z


                                                       24
   Orthographic Projection

Orthographic Projection onto a plane at z = 0.
           xp = x , yp = y , z = 0.

                                         1   0 0 0
                                         0   1 0 0
                               M orth            
                                         0   0 0 0
                                                  
                                         0   0 0 1

				
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posted:3/14/2012
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Description: graphics lecture