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This page describes the formulae used project tracking errors to the ACD

These are written up in greater detail here: formulae.pdf

 General framework

The tracker has a 4x4 representation x_tkr which uses a pair of slope intercepts ( x_0,y_0,m_x, m_y ) to parametrize the position as a function of z.  This is great for a bunch of planes normal to the z axis, but really not so nice for planes normal to the x and y axes.  The first order of business is to transform the track parameterization to a 5x5 representation x_acd which uses a Point at Z=0 and a Vector (x_0,y_0, v_x,v_y,v_z) to parameterize the positions by arclength along the track (s).

We can the transform the covarience matrix by using the 5x4 track representation derivative matrix A = d x_acd / d x_tkr:
  C_acd (5x5) = A C_tkr (4x4) A_T

If we then use the track parameterization to calculate a \[point of closest approach (POCA) or an intersection point p, the covarience on that point can be found using 5x3 solution derivative matrix
  B = d p /  d A_acd
with
  C_p (3x3) = B C_acd (5x5) B_T

In the case where we care about the distance of closest approach to a point or a ray, we can project along the vector of closest approach J (the vector from the POCA to the point or ray in question).   Then we find:
  C_doca (1x1) = J C_p(3x3) J_T

In the case where we care about the intersection of a track with a plane, we can rotate into the plane using the rotation matrix R.
  C_plane (3x3) = R C_p(3x3) R_T
and take only the upper-left 4 entries.

 Errors on POCA between a track and a point

Call the point q, and the track (x,v) the POCA occurs at an arclength 
   s = (x - q) . v
therefore the poca is
   p = x + sv 
and the vector of closest approach is:
   J = q-p = q - x + sv

The full derivative matrix is given in the attached write up. 

 Errors on POCA between a track and a ray

Call the ray (r,q) and the track (x,v), the POCA occurs at an arclength along the track
   s = \[b(r.(x-q) - v.(x-q)\] / \[1-b*b\]
and an arclength along the ray:
  t = \[r.(x-q) - b(v.(x-q))\] / \[1-b*b\]
where b = v.r

Then the poca and vector of closest approach are:
  p = x + sv
  J = (x-q) + sv - tr

 The full derivative matrix is given in the attached write up.

 Errors on point where a track crosses a plane

Given a plane defined by a point q and a rotation matrix R, the intersection occurs at:
  s = \[ (x-q).R_z \] / \[ v. R_z \]

Then the intersection point in the global frame is:
  p = x + sv

Of course we can get the intersection point in the local frame l:
  l = R p + q

The full derivative matrix is given in the attached write up.