Determining wire-phi correspondance

xdist - distance between the line of intersection of the two end-plate planes and the target position =8.298 cm th_min=4.694

Using Mathematica, we can produce a 3D rendering of how the sectors for Level 1 would have to interact with a steady angle theta with respect to the beam line, as angle phi is rotated through 360 degrees.

Looking just at sector 1, we can see that the intersection of level 1 and the cone of constant angle theta forms a conic section.

Following the rules of conic sections we know that the eccentricity of the conic is given by:

$e=\frac{\sin \beta}{\sin\alpha}$

Where β is the angle of the plane, which in our case is 25 degrees for the sectors with respect to the beam line (axis of rotation of the cone).

This leaves α is the slant of the cone, which is the angle theta that the particle must be traveling with respect to the beamline.

If the conic is an circle, e=0

If the conic is an parabola, e=1

If the conic is an ellipse, $e=\sqrt{1-\frac{b^2}{a^2}}$

For ellipses centered at (h,k):

$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$

where

$a = major\ radius\ (= \frac{1}{2}\ length\ major\ axis)$

$b = minor\ radius\ (= \frac{1}{2}\ length\ minor\ axis)$

Where

$c^2=a^2-b^2$

For a parabola:

$4px = y^2$

where

p = distance from vertex to focus (or directrix)