This document discusses different types of projections used in virtual reality, including orthographic and perspective projections. It explains that perspective projection takes the position of the eye into account by projecting 3D points onto a virtual screen through the origin. The transformations applied to 3D models include: rigid body transforms, eye coordinate transforms, canonical view transforms, additional translations and scaling, and final viewport transforms to map onto a physical display. Having separate left and right eye views is also discussed.

1. Geometry
Deck.
From Virtual Reality of Steven M. LaValle

2. Contents
4. Perspective ( Virtual Camera )
5. Put it all together ( Chaining the
transformations )

3. 4. Viewing Transformation
1. How the models transform to “appear” in
a virtual world.
2. Virtual Camera should be taken into
consideration.

4. 4. An eye's view
A flat rectangular shape is chosen for a Virtual Screen for engineering and
historical reasons, not matching our retina though.

5. 4. An eye's view
Generally eye is assumed to look down the negative z axis.
This way the view can have the default perspective set to
x = horizontal, y = vertical and right-handed Cartesian Coord.

6. 4. An eye's view
Assume we have Reye, to behold models appear in VR,
Reye should move as inverted to its input.
Moving the world instead of the eye object is more plausible this time. ( Fig 3.4(c) )

7. 4. Starting from a look-at
Components of look-at
1. Position of the eye : e
2. Central looking direction of the eye : ĉ
3. Up direction : û
ĉ is a vector from p to e, in short.
Translation of look-at : e
Rotation of look-at :

8. 4. Orthographic projection
Project 3D to xy plane with z = 0, it's orthographic projection.
Problem arises as follows :
1. Jumbled objects superimposing onto each other.
2. The display would extend infinitely except z axis
3. Objects that are closer should appear larger than those further away

9. 4. Perspective projection
Consider each line through the origin for each point.
Assume λz = -1, then x' = -x/z, y' = -y/z

10. 4. Perspective projection
Vertical Plane is Virtual Screen where the model is projected onto.
x' = dx/z
y' = dy/z.

11. 5. Chaining the Transformations
The Transform Chain appears generally as follows :
T is a point ( x, y, z, 1 )
Chain operations are applied from right to left.
Trb = Rigid Body Transform
Teye = Coordinate frame of the eye

12. 5. Canonical View Transform

13. 5. Keeping track of depth
z is to keep tracking of the distance of each point from the eye
When divided by z, the formula doesn't preserve its exact distance,
Still the distance ordering is preserved.

14. 5. Additional Translation and Scaling
The matrix performs a simple translation of the box along the z axis.
On condition the frustrum is centered perfectly in the xy plane,
then -(r+l)/(r-l) = 0, -(t+b)/(t-b) = 0
So,

15. 5. Viewport transform
The last tranform is viewport transform
Suppose we need to project all these points to the physical display such as
1980(m) x 1080(n)
m is horizontal ranging from 0 to m-1,
n is vertical ranging from 0 to n-1.

16. 5. Left and Right eyes
t is the distance value between left and right eyes
t varies among people, however the average is t = 0.064
Simply move cyclopean eye horizontally to the left, which means models need moving to
the right.
FYI, right eye transform is,