Welcome to the Corneal Topography
and Ray-trace page (and it needs some serious updating)

Try these links if you cannot see the table above:

Without Spokes:

• f8a0125 (ampl=0.0125; freq=8; 512 x 512 pixels; 6.4 x 6.4 mm)
• f8a0125l (ampl=0.0125; freq=8; 1024 x 1024 pixels; 10.8 x 10.8 mm)
• f8a025 (ampl=0.025; freq=8; 512 x 512 pixels; 6.4 x 6.4 mm)
• f8a025l (ampl=0.025; freq=8; 1024 x 1024 pixels; 10.8 x 10.8 mm)

With Spokes:

• f8a0125s (ampl=0.0125; freq=8; 1024 x 1024 pixels; 7.0 x 7.0mm)
• f8a0125x (ampl=0.0125; freq=8; 512 x 512 pixels; 7.0 x 7.0mm)
• f8a0125w (ampl=0.0125; freq=8; 512 x 512 pixels; 10.0 x 10.0mm)
• f7a0125s (ampl=0.0125; freq=7; 1024 x 1024 pixels; 7.0 x 7.0mm)
• f7a0125x (ampl=0.0125; freq=7; 512 x 512 pixels; 7.0 x 7.0mm)
• f7a0125w (ampl=0.0125; freq=7; 512 x 512 pixels; 10.0 x 10.0mm)

Ray-trace Settings:

z0 = 120
R = 8
f(x,y) = -sqrt(R2 - x2 - y2) + R + z0 + ampl * cos(freq * atan(y/x))
Where the addition of the cos() function only applies for sqrt(x2 + y2) > 2
dz/dx = (f(x+0.001, y) - f(x-0.001, y)) / 0.002
dz/dy = (f(x, y+0.001) - f(x, y-0.001)) / 0.002

l(xl,yl) = sqrt(xl2 + yl2)

Trace equations:

Reflection vector vR:   [0, 0, -1]T (this is a unit vector)
Normal vector vN:       [-dz/dx, -dz/dy, 1]T
Normal unit vector vNu: vN / |vN|

Surface to Placido disc unit vector vPL_u

Reflection equation:    vPLu = 2 * vNu * (vNu * vR) - vR

Placido disk has rotational symetry which simplifies the intersection calculation of the source vector (vPL_u) and the Placido surface (vOL). Mapping the placido disc to a 2D coordinate system and then determine x_l (the x-coordinate of intersection on the placido disc):

x(vector) = x ordinate of 'vector'

rplu = sqrt(x(vPLu)2 + y(vPLu)2)
r = sqrt(x2 + y2)
k = (z - r) / (rplu - z(vPLu)
xl = r + k * rplu

The black/white rings are defined at an interval (RINGINTERVAL) of 4.0

int(xl / RINGINTERVAL) is odd  --> white ring
int(xl / RINGINTERVAL) is even --> black ring

The spokes are generated as an inversion of the ring's black/white pattern in a 0.28 (9/32) degree angle with and 64 spokes over the full 360 degrees.

The alternative for using the 2D shortcut is to solve:

vOP + k * vPLu = vOL

with

vOP = [x, y, z]T
vOL = [xl, yl, sqrt(xl2 + yl2)]T

This yields a quadratic function in k. Using the 2D shortcut is prefered since solving a quadratic is more expensive than a simple linear equation. Secondly, the 2D case only yields one solution for k whereas the quadratic results in two.