Note: click on the title of each slide to advance to the next slide

Positioning the stripes:

 To make the NY3D display work, we need to interleave the left and right images on the display and also to create a corresponding set of opaque/clear stripes on the optical shutter. But how do we figure out where the stripes should go? The key is to keep drawing crossed lines: Starting from the right eye and the left-most point on the display, we draw a straight line, and see where it crosses the shutter. Then we draw a line from the left eye through this point on the shutter, and see where this new line hits the display. If we keep doing this, always starting with that next point over on the display, we'll construct a working stripe pattern for that pair of eye positions. In the next few slides, we work out the mathematical details.

• Start at one side of the display

• Lines crossing at shutter:

• xn on display  > shutter  > Right eye
• xn+1 on display <  shutter <  Left eye

• Iterate

Establishing a coordinate system:

 The two eye positions: L = (Lx,Ly) R = (Rx,Ry) ``` ``` Shutter location: y=1  Display screen location: y=0 Try dragging the red and green dots. Also, try clicking between the two black lines.

Line-of-sight from display to shutter

Fractional distance of shutter from display screen to p:

py-1 = (1 - 0) / (py - 0)
Location on shutter between (x,0) and p:
fp(x) = px py-1 + x (1 - py-1)
Location on display screen between (x,1) and p :
fp-1(x) = (x - px py-1)/(1 - py-1)

Line-of-sight thru shutter from both eyes

Line 1: (xn ,0)  > (xn,1)  > R
Line 2: (xn+1,0) <  (xn,1) <  L

xn+1 = fL-1(fR(xn))
which expands out to:
(Rx Ry-1 + x (1-Ry-1) - Lx Ly-1)/(1-Ly-1)
Express this as a linear equation in xn:
xn+1 = A xn + B

Solve for linear coefficients:

A = x (1 - Ry-1)/(1 - Ly-1)
B = (Rx Ry-1 - Lx Ly-1)/(1 - Ly-1)
Sequence of display stripe locations:
x0 = 0
x1 = B
x2 = AB + B
x3 = A2B + AB + B
:
:
xn = B (An-1 + ... + A + 1)

Conclusions:

• user can move around and turn head freely

• continuous 3D: not discrete zones

• no eyewear

• compatible with existing CRT monitors

## In a nutshell: How to place the stripes on the display screen:

Assume the two eye positions are: p = (px,py) and q = (qx,qy), that the display screen is on the line y=0, and that the shutter is on the line y=1.

Given a location (x,0) on the display screen, we find the line-of-sight location fp(x) on the shutter that lies between display screen location (x,0) and eye position p by linear interpolation:

fp(x) = px py-1 + x (1 - py-1)
Given a location (x,1) on the shutter, we can find the corresponding line-of-sight location on the display screen by inverting the above equation:
fp-1(x) = (x - px py-1)/(1 - py-1)
Therefore, given a location xn on the display screen that is visible through a clear stripe on the shutter from both p and q, the next such location is given first by finding the location on the shutter fp(xn) in the line-of-sight from p, and then finding the corresponding location on the display screen which is in the line-of-sight from q:
xn+1 = fq-1(fp(xn))
which expands out to:
(px py-1 + x (1 - py-1) - qx qy-1)/(1 - qy-1)
This can be expressed as a linear equation xn+1 = A xn + B, where:
A = x (1 - py-1)/(1 - qy-1)
B = (px py-1 - qx qy-1)/(1 - qy-1)
The nth location in the sequence of stripe locations on the display screen can be calculated by iterating xn+1 = A xn + B:
x0 = 0     x1 = B     x2 = AB + B     x3 = A2B + AB + B

xn = B (An-1 + ... + A + 1)