Projection
Projection
Overview
o Orthographic projection
o Perspective projection
o Projection in OpenGL
o We know how to transform objects to View space
o We now want to ‘take a photo’ of the objects in View
space - mapping from 3D to 2D is called projection:
o Parallel projection maps objects directly onto the
screen without affecting their relative size, often
used in architectural and CAD applications
o Perspective projection gives realistic views, but
does not preserve proportions - projections of
distant objects are smaller than those near objects
of the same size. This is called foreshortening.
Projection View Volume/Frustum
Different Types of Projections
o In Orthographic parallel
projection, view plane is
perpendicular to the
direction of projection (think
about a torch casting
parallel rays on the view
plane)
o In Perspective projection,
objects are projected onto
the view plane along lines
which converge at the
observer, or Centre of
Projection (CoP)
Q
P
Q
Viewing Frustum
zv
P1
P1
P2
yv
P2
view plane
Perspective view volume is
defined by field of view or planes
(near, far, left, right, top and bottom)
xv
Image Plane
yv
Eye Position
P
P1’
P2’
CoP/
Eye/
Camera
view plane
Orthographic view volume
is a rectangular block defined
by near, far, left, right,
top and bottom planes
zv
xv
Eye Position Field of view
Orthographic Projection
Image Plane
Viewing Frustum
Perspective Projection
o Looking along X axis, similar triangles gives:
Looking along z-axis we can see point P on the image
plane, which is the 2D projection of a 3D point Q in camera
space, and xP = xQ, yP = yQ (which means P and Q
have the same x, y values)
P(xp,yp)
y
P(x,y,z)
y d
dy
  y 
y z
z
p
p
yV
yv
d
Q
P(xp, yp)
P
Viewing Frustum
zv
xv
Image Plane
xV
-z
Similarly, looking along Y axis
x
x
d
dx
  x 
x z
z
p
p
Eye Position
1
Projection in OpenGL
Projection in OpenGL
o An OpenGL projection matrix takes points in view space
and converts them to points in normalised device
space or canonical space
o The canonical space is a 2x2x2 cube, centered at the
coordinate origin of the view space, with X, Y, Z each
ranging from -1 to 1
o Thus only parallel orthographic projection is needed to
obtain 2D images of a 3D scene
Object
Coordinate
Space
View
Space
Canonical
Space
OpenGL transforms all objects to a canonical space,
e.g., the perspective viewing frustum is transformed to
the canonical space – note all the objects undergo the
same transformations
Screen
Space
Canonical to Screen Transform
Canonical to Screen Transform
y
o The last step is to position the 2D image on the display
screen by calling glViewport() in OpenGL, to
transform canonical coordinates to display coordinates
o Parameters of glViewport() describe position of
screen space within the window and the width and
height of the screen, measured in pixels on the screen
y
x
screen
(1,1)
x
y
(xmax,ymax)
(xmin,ymin)
x
(-1,-1)
 x
 pixel
 y pixel

 z pixel

 1
 
   xmax  xmin  2
 
0

 
0
 
0
 
0
0 0 

y

y
2
0 0 
 max min 

0
1 0 
0
0 1 
1 0 0
0 1 0
0 0 1
0 0 0
 xmax  xmin 
 ymax  ymin 
0
1
2  xcanonical

2  ycanonical

 zcanonical

1






Projections in OpenGL
o glMatrixMode(GL_PROJECTION);
o glLoadIdentity();
o
o //glOrtho(‐windowWidth/2, windowWidth/2, ‐windowHeight/2, windowHeight/2, 1.0, 1000.0);
o
o
o
o
o
gluPerspective(60.0, windowWidth/windowHeight, 1, 1000.0);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
}
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