Commit 09fb0a79 authored by Emmanuel Bertin's avatar Emmanuel Bertin
Browse files

Doc: added WIN error ellipse equations.

parent 2e62b5e8

doc/src/Position.rst

+1 −1
Original line number Diff line number Diff line
@@ -285,7 +285,7 @@ error ellipse are computed from the covariance matrix exactly like in 
   \tan (2{\tt ERRTHETA}) & = & 2 \,\frac{{\rm cov}(\overline{x},\overline{y})}

                       {{\rm var}(\overline{x}) - {\rm var}(\overline{y})}.\end{aligned}



And the ellipse parameters are:

And the error ellipse parameters are:



.. math::

  :label: errellipse

doc/src/PositionWin.rst

+40 −3
Original line number Diff line number Diff line
@@ -173,10 +173,10 @@ to :eq:`xywin` writes: 


   \begin{aligned}

   {\tt ERRX2WIN} & = {\rm var}(\overline{x_{\tt WIN}})

   = & 4\,\frac{\sum_{r_i < r_{\rm max}} w_i^2 \sigma^2_i (x_i-\overline{x})^2}

   = & 4\,\frac{\sum_{r_i < r_{\rm max}} w_i^2 \sigma^2_i (x_i-\overline{x_{\tt WIN}})^2}

   {\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2},\\

   {\tt ERRY2WIN} & = {\rm var}(\overline{y_{\tt WIN}})

   = & 4\,\frac{\sum_{r_i < r_{\rm max}} w_i^2 \sigma^2_i (y_i-\overline{y})^2}

   = & 4\,\frac{\sum_{r_i < r_{\rm max}} w_i^2 \sigma^2_i (y_i-\overline{y_{\tt WIN}})^2}

   {\left(\sum_{r_i < r_{\rm max}} w_i I_i\right)^2},\\

   {\tt ERRXYWIN} & = {\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})

   = & 4\,\frac{\sum_{r_i < r_{\rm max}}
@@ -189,7 +189,44 @@ from the covariance matrix elements 
:math:`{\rm var}(\overline{x_{\tt WIN}})`,

:math:`{\rm var}(\overline{y_{\tt WIN}})`,

:math:`{\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})`,

similarly to the :ref:`isophotal error ellipse <poserr>`.

similarly to the :ref:`isophotal error ellipse <poserr>`:



.. math::

  :label: errabthetawin



   \begin{aligned}

   {\tt ERRAWIN}^2 & = & \frac{{\rm var}(\overline{x_{\tt WIN}})+{\rm var}(\overline{y_{\tt WIN}})}{2}

       + \sqrt{\left(\frac{{\rm var}(\overline{x_{\tt WIN}})-{\rm var}(\overline{y_{\tt WIN}})}{2}\right)^2

       + {\rm cov}^2(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})},\\

   {\tt ERRBWIN}^2 & = & \frac{{\rm var}(\overline{x_{\tt WIN}})+{\rm var}(\overline{y_{\tt WIN}})}{2}

       - \sqrt{\left(\frac{{\rm var}(\overline{x_{\tt WIN}})-{\rm var}(\overline{y_{\tt WIN}})}{2}\right)^2

       + {\rm cov}^2(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})},\\

   \tan (2{\tt ERRTHETAWIN}) & = & 2 \,\frac{{\rm cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})}

                       {{\rm var}(\overline{x_{\tt WIN}}) - {\rm var}(\overline{y_{\tt WIN}})}.

    \end{aligned}



And the error ellipse parameters are:



.. math::

  :label: errellipsewin



   \begin{aligned}

   {\tt ERRCXXWIN} & = & \frac{\cos^2 {\tt ERRTHETAWIN}}{{\tt ERRAWIN}^2} +

   \frac{\sin^2 {\tt ERRTHETAWIN}}{{\tt ERRBWIN}^2} = \frac{{\rm

   var}(\overline{y_{\tt WIN}})}{{\rm var}(\overline{x_{\tt WIN}}) {\rm var}(\overline{y_{\tt WIN}}) -

   {\rm cov}^2(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})},\\

   {\tt ERRCYYWIN} & = & \frac{\sin^2 {\tt ERRTHETAWIN}}{{\tt ERRAWIN}^2} +

   \frac{\cos^2 {\tt ERRTHETAWIN}}{{\tt ERRBWIN}^2} =

   \frac{{\rm var}(\overline{x_{\tt WIN}})}{{\rm var}(\overline{x_{\tt WIN}}) {\rm var}(\overline{y_{\tt WIN}}) -

   {\rm cov}^2(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})},\\

   {\tt ERRCXYWIN} & = & 2 \cos {\tt

   ERRTHETAWIN}\sin {\tt ERRTHETAWIN} \left( \frac{1}{{\tt ERRAWIN}^2} -

   \frac{1}{{\tt ERRBWIN}^2}\right)\\ & = & -2 \frac{{\rm

   cov}(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})}{{\rm var}(\overline{x_{\tt WIN}}) {\rm var}(\overline{y_{\tt WIN}}) -

   {\rm cov}^2(\overline{x_{\tt WIN}},\overline{y_{\tt WIN}})}.

   \end{aligned}







.. [#win_accuracy] See http://www.astromatic.net/forum/showthread.php?tid=581 .