Projection

1
Projection

• Projection - the transformation of points from a
  coordinate system in n dimensions to a coordinate
  system in m dimensions where mltn.
• We will be talking about projections from 3D to
  2D, where the 3D space represents a world
  coordinate system and the 2D space is a window
  which is mapped to a screen viewport.

2
Specifying a Projection

• Two things must be specified
• Projection plane and a center of projection.
• Projection plane
• A 2D coordinate system onto which the 3D image is
  to be projected. Well call this the VRP for
  view reference plane.
• Center of projection
• A point in space which serves as an end point for
  projectors. Well refer to this point as the
  COP. It is also called a PRP for a projection
  reference point.

3
Projectors

• Projectors - a ray originating at the center of
  projection and passing through a point to be
  projected. Here is an example of a projection

4
Parallel Projection

• A simple case of a projection is if the
  projectors are all in parallel.
• What does this imply about the COP?

5
Direction of Projection

• We cant specify the COP for parallel projection
• Well use Direction of Project (DOP) instead

6
Some Trivia

• Planar geometric projection
• A projection onto a planar surface (planar) using
  straight lines (geometric).
• Foreshortening
• Varying lengths of lines due to angle of
  presentation and/or distance from center of
  projection. Applies to both parallel and
  perspective projections.

7
Orthographic Projections

• Orthographic projection
• parallel projection with the direction of
  projection and the projection plane normal
  parallel.
• Elevation
• an orthographic projection in which the view
  plane normal is parallel to an axis.
• The three elevations
• front-elevation
• top-elevation (plan-elevation)
• side-elevation.

8
Axonometric orthographic projections

• Axonometric orthographic projections
• Use projection planes which are not normal to an
  axis. They show more than one face of an object
  at a time. They induce uniform foreshortening
  unrelated to depth.
• AOP preserves parallelism of lines. It does not
  preserve angles.

9
Isometric projection

• Isometric projection
• Axonometric orthographic projection where the
  projection plane normal (and the direction of
  projection) makes identical angles with each
  principle axis. How many of these are there?

10
Oblique Projection

• Oblique projection
• the projection plane normal and the direction of
  projection are at angles to each other.

DOP
VPN
11
Cavalier Projection
Why?

• An Oblique projection
• DOP is at 45 degree angle to VPN
• Lines parallel to any axis are foreshortened
  equally. Lines parallel to the z axis appear at
  an angle a, which is dependent upon the direction
  of projection.
• Two common projections have a as 45 and 30.

45o
30o
12
Cavalier Projection Angles
DOP
VPN
13
Cabinet projection

• Oblique projection
• projection plane normal is at an arctan(2)
  63.4 degree angle to the projection plane.
  (typically projecting onto the x,y plane)
• Lines parallel to the axis defining the
  projection plane are foreshortened equally.
  Lines parallel to the projection plane normal are
  halved!

14
Cabinet Projection
DOP
63.4o
VPN
15
Parallel Projection

• After transforming the object to the eye space,
  parallel projection is relative easy we could
  just drop the Z
• Xp x
• Yp y
• Zp -d
• We actually want to keep Z
• why?

(Xp, Yp)
(x,y,z)
x
16
Parallel Projection (2)

• OpenGL maps (projects) everything in the visible
  volume into a canonical view volume

(xmax, ymax, -far)
(1, 1, -1)
(xmin, ymin, -near)
(-1, -1, 1)
Canonical View Volume
glOrtho(xmin, xmax, ymin, ymax,
near, far)
17
Parallel Projection (3)

• Transformation sequence
• 1. Translation (M1) (-near zmax, -far
  zmin)
• -(xmaxxmin)/2, -(ymaxymin)/2,
  -(zmaxzmin)/2
• 2. Scaling (M2)
• 2/(xmax-xmin), 2/(ymax-ymin),
  2/(zmax-zmin)

2/(xmax-xmin) 0
0 - (xmaxxmin)/(xmax-xmin) M2 x M1
0 2/(ymax-ymin) 0
- (ymaxymin)/(ymax-ymin)
0 0 2/(zmax-zmin)
-(zmaxzmin)/(zmax-zmin)
0 0 0
1
18
Perspective Projection

• Perspective projections have projectors at angles
  to each other radiating from a center of
  projection.
• Parallel lines not parallel to the projection
  plane will not appear parallel in the projection.
  

19
Vanishing Points

• If not parallel?
• If the lines are not parallel anymore, they must
  meet somewhere. In 3D space that point will be
  at infinity and is referred to as a vanishing
  point. There are an infinite number of vanishing
  points.
• Axis vanishing points
• Lines parallel to one of the major axis come to a
  vanishing point, these are called axis vanishing
  points. Only three axis vanishing points in 3D
  space.

20
Center of Projection in OpenGL

• OpenGL always puts the center of projection at
  0,0,0
• The projection plane is at z -d
• This is sometimes called the focal length or f

21
Frustums
glFrustum(left,right,bottom,top,znear,zfar)

• The region we can see is called the frustum

(right,top,-znear)
(0,0,0)
-zfar
(left,bottom,-znear)
znear and zfar are positive
22
gluPerspective

• How do we get from
• gluPerspective(fovy, aspect, znear, zfar)
• To
• glFrustum(left,right,bottom,top,znear,zfar)

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fov to near frustum
(x,y,-znear)
-z
24
Projection Structure
Pinhole Camera Model of Projection
y
P(x,y,z)
x
Proportional!
P'(x',y',z')
-d
-z
25
Matrix for Perspective Projection?

• We need division to do projection!
• But, matrix multiplication only does
  multiplication and addition
• What about

26
Homogenous Coordinates (again)

• A 3D homogeneous coordinate
• (x, y, z, w)
• We had been saying that w is 1
• But
• (x, y, z, w) corresponds to (x/w, y/w, z/w)
• Dividing by w is called homogenizing
• If w1, x,y,z are unchanged.
• But, if w-z/d?
• (x/(-z/d), y/(-z/d), z/(-z/d)) (-dx/z, -dy/z,
  -d)

27
The Entire Viewing Process

• Rotate world so that the COP is at 0,0,0 and DOP
  is parallel to the Z axis
• Apply perspective projection
• Homogenize
• Viewport transformation

28
Viewport Transformation(Window to Viewport)

• Window
• Area of the projection plane
• Typically some normalized area with 0,0 in the
  center
• Viewport
• Area of the computer display window
• Example
• (0, 0) to (640, 480)

29
Window to Viewport Example

• Assume Window (-1,-1) to (1,1)
• OpenGL calls these normalized device coordinates
• Viewport (0, 0) to (640, 480)
• OpenGL calls these window coordinates

30
Perspective Projection (6)

• Final Projection Matrix

x 2N/(xmax-xmin) 0
(xmaxxmin)/(xmax-xmin) 0 x y
0 2N/(ymax-ymin) (ymaxymin)/(ymax-ym
in) 0 y z 0 0
-(F N)/(F-N)
-2FN/(F-N) z w 0 0
-1
0 1
glFrustum(xmin, xmax, ymin, ymax, N, F) N
near plane, F far plane
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Within OpenGL
glBegin(GL_POLYGON) glVertex3dv(a)
glVertex3dv(b) glVertex3dv( c)glEnd()