CS 445 / 645
Introduction to Computer Graphics
Lecture 10
Camera Models
Where are we?
Rendering Pipeline
Modeling
Transformation
Display
Illumination
Rasterization
Viewing
Transformation
Projection
Clipping
Pinhole Camera
Ingredients
• Box
• Film
• Hole Punch
www.kodak.com
Results
www.debevec.org
• Pictures!
www.pinhole.org
Pinhole Camera
Non-zero sized hole
Pinhole
Film Plane
Multiple rays
of projection
Pinhole Camera
Theoretical Pinhole
One ray
of projection
Pinhole
Film Plane
Pinhole Camera
Field of View
Focal
Length
Film Plane
Pinhole
Field of View
Pinhole Camera
Field of View
Focal
Length
Film Plane
Pinhole
Field of View
Moving the Film Plane
Varying distance to film plane
What does this do?
Pinhole
d1
Field of View
d2
Film Plane
Adding a Lens
• Pinhole camera has small aperture (lens opening)
– It’s hard to get enough light to expose the film
• Lens permits larger apertures
• Lens permits changing distance to film plane without actually
moving the film plane
Computer Graphic Camera
We use
• Center of Projection (COP)
• Projection Plane
COP
Projection
Plane
Moving the COP
Perspective vs. Orthographic Views
Perspective
When COP at infinity, Orthographic View
Multi-point Perspective
One-point Perpective
• One Vanishing Point
Two-point Perspective
• Two Vanishing Points
http://www.sanford-artedventures.com/create/tech_2pt_perspective.html
Perspective Projection
Our camera must model perspective
Perspective Projection
Projection
Plane
How tall should
this bunny be?
COP
Perspective Projection
The geometry of the situation is that of similar triangles. View
from above:
View
plane
X
x’ = ?
(0,0,0)
Z
d
What is x’ ?
P (x, y, z)
Perspective Projection
Desired result for a point [x, y, z, 1]T projected onto the
view plane:
x' x
 ,
d
z
dx
x
x' 

,
z
z d
y' y

d
z
dy
y
y' 

,
z
z d
What could a matrix look like to do this?
zd
A Perspective Projection Matrix
Answer:
1
0
Mperspective  
0

0
0
1
0
0
0
1
0 1d
0
0
0

0
A Perspective Projection Matrix
Example:
 x  1
 y  0


 z  0

 
 z d  0
0
1
0
0
0
1
0 1d
Or, in 3-D coordinates:
0  x 



0  y 
0  z 
 
0  1 
 x

,
z d
y
,
z d

d 

Homogeneous Coordinates
The role of w in (x, y, z, w)
• All 3-D points are described with a four vector
• All 3-D tranformations are represented with 4x4 matrix
• When projected to screen coordinates (rasterization)
– x, y, and z are divided by point’s w value
• This allows us to perform perspective foreshortening while
preserving the reversibility of the mapping
– We can retrieve x, y, and z by multiplying by w
Perspective Projection
• Perspective projection matrix is not affine
– Parallel lines not preserved
• Perspective projection is irreversible
– Many 3-D points can be mapped to same (x, y, d) on the
projection plane
– No way to retrieve the unique z values
Orthographic Camera Projection
• Camera’s back plane parallel to
lens
 xp   x
   
 y p    y
 z p   0 
 
• Infinite focal length
• No perspective convergence
 x p  1
 y  0
 p  
 z p  0
  
 1  0
0 0 0  x 



1 0 0  y 
0 0 0  z 
 
0 0 1  1 
Pipeline
Modelview
Projection
Perspective
Division
Clip
Rasterize
OpenGL Pipeline
• Projection matrix is stored in GL_PROJECTION stack
– This controls ‘type’ of camera
– All vertices are multiplied by this matrix
• GL_MODELVIEW controls camera location
– All vertices are multiplied by this matrix
Making GL_PROJECTION
glFrustum – for perspective projections
• xmin
• xmax
• Camera looks along –z
• ymin
• min/max need not be symmetric
about any axis
• ymax
• near
• far
• near and far planes are parallel to
plane z=0
Making GL_PROJECTION
gluPerspective – for perspective projections
• fovy
• aspect
• near
• far
• fovy is the angle between top and
bottom of viewing volume
• aspect is ratio of width over height
• This volume is symmetrical
• View plane is parallel to camera
Making GL_PROJECTION
glOrtho – for orthographic projections
• left
• bottom
• (left, bottom) and (right, top)
define dimensions of projection
plane
• top
• near and far used to clip
• right
• near
• far
Making GL_PROJECTION
It’s like any other matrix
• These OpenGL commands just build a matrix for you
• You could build the matrix yourself
• You can multiply the GL_PROJECTION matrix by any affine
transformation you wish
– Not typically needed