1. Perspective projection
Prepared by Cindy Grimm and Ken Goldman
for Washington University CSE131
based on slides to accompany Fundamentals of Interactive Computer Graphics by Foley and Van Dam

2. Properties of perspective projection
edges same size,
with farther ones smaller
Straight lines go to straight lines
Closer objects are bigger
Parallel lines
Stay parallel (lines are perpendicular to viewer)
Converge to a vanishing point (lines are not perpendicular to viewer)
parallel edges converging
Perspective projection

3. Perspective projection
View volume
What you can see
View point (eye)
View direction (projectors)
projectors
eye, or
Center of Projection
(COP)
projectors
picture plane
Perspective projection

4. Perspective projection
Specifying a camera
Eye position (type: Point)
Look direction (type: Vector)
Up direction (type: Vector)
Near/far (percentage along Look)
(0.1, 1,000)
Width
Up vector
Look vector y x
Height
Position z
Near
Far
Perspective projection

5. Perspective projection
Our camera
Eye point is point on sphere. Given d (distance), q (latitude) and f (longitude), eye point is:
Eyex = d cos(q) cos(f)
Eyey = d sin(q)
Eyez = d sin(f) cos(q)
Up vector will always be (0,1,0)
Note that sphere is oriented with the north/south poles in the y direction
Look vector: looks at origin (Eye – (0,0,0))
Perspective projection

6. Perspective projection
Mathematically…
Projection is a 4x4 matrix
Build from four component matrices, PSRT
P: Perspective matrix
Responsible for fore-shortening
S: Scale matrix
Adjusts for aspect ratio
R: Rotation matrix
Which direction to look at
T: Translation matrix
Where the camera is
Perspective projection

7. Perspective projection
near (0.1) far (1000)
k = near/far
Scales x,y by z P
Flip the z-axis
and unhinge
Note: not 1!
Perspective projection

8. Perspective projection
Scale
Pre-adjust for aspect ratio
W: width of final image
H: height of final image
Aspect ratio: W/H
Bring the far plane up
Note: can put W/H in S(1,1) instead
Perspective projection

9. Perspective projection
Rotation
Make the camera point at the origin
Given: Eye (point) and Up (vector) = (0,1,0)
r,u,n are orthogonal, unit length vectors
Dot product of any pair is zero
Length of each is one
Perspective projection

10. Perspective projection
Translation
Move camera to position on sphere
Why negative? Matrix needs to undo translation to bring camera to origin
Perspective projection

11. Putting it all together
Convert point (x,y,z) to 4x1 matrix by adding a one (homogeneous points)
Note: Pre-multiply PSRT to make one 4x4 matrix which is applied to all points
Perspective projection

12. Perspective projection
To image space
Divide by w to get point in camera space
Anything with u/w,v/w in [-1,-1]x[1,1] and d/w in [0,1] is visible and in front of the camera
If w=0 then point is “smeared” over all of the z=0 plane; one fix is to set u/w and v/w to be zero.
Convert to image space by scaling by width and height
Camera to image transformation
Perspective projection