diff -u -r1.2 -r1.3
--- vertexing.lyx	18 Jul 2006 03:26:53 -0000	1.2
+++ vertexing.lyx	19 Jul 2006 08:17:03 -0000	1.3
@@ -33,6 +33,10 @@

 Vertexing in org.lcsim
 \end_layout
 

+\begin_layout Author
+Jan Strube
+\end_layout
+

 \begin_layout Section
 Track Parametrization at the point of closest approach to a reference point
 \end_layout

@@ -56,6 +60,70 @@

  element symmetric table.
 \end_layout
 

+\begin_layout Subsection
+Basic Definitions
+\end_layout
+
+\begin_layout Standard
+The inverse of the radius of curvature 
+\begin_inset Formula $R$
+\end_inset
+
+ is called 
+\begin_inset Formula $\omega$
+\end_inset
+
+.
+ Both 
+\begin_inset Formula $R$
+\end_inset
+
+ and 
+\begin_inset Formula $\omega$
+\end_inset
+
+ are signed quantities.
+ The sign of 
+\begin_inset Formula $\omega$
+\end_inset
+
+ is determined by the charge 
+\begin_inset Formula $q$
+\end_inset
+
+ of the particle that traverses a field 
+\begin_inset Formula $\vec{B}=(0,0,B_{z}),\ B_{z}>0$
+\end_inset
+
+: 
+\begin_inset Formula $\text{sign}(\omega)=\text{sign}(q)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+The direction of rotation is given by the sign of 
+\begin_inset Formula $\omega:$
+\end_inset
+
+ 
+\begin_inset Formula \[
+\begin{array}{cc}
+\omega>0 & \text{negative (clockwise) rotation}\\
+\omega<0 & \text{positive (counter-clockwise) rotation}\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+

 \begin_layout Standard
 \begin_inset Graphics
 	filename helixDef.png

@@ -65,6 +133,112 @@

 
 \end_layout
 

+\begin_layout Caption
+The projection of the helix into the x-y plane.
+ As shown, a rotation from 
+\begin_inset Formula $P^{0}$
+\end_inset
+
+ to 
+\begin_inset Formula $P$
+\end_inset
+
+ is in the negative sense, if 
+\begin_inset Formula $\omega$
+\end_inset
+
+>0.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The definition of 
+\begin_inset Formula $d_{0}$
+\end_inset
+
+ is as follows: The circle that is the projection of the helix in the x-y
+ plane can be parametrized as 
+\begin_inset Formula \[
+\vec{f}(\phi)=R(\begin{array}{c}
+\cos(\phi)\\
+\sin(\phi)\end{array})+\vec{R_{\text{c}}}\]
+
+\end_inset
+
+, where 
+\begin_inset Formula $\vec{R_{\text{c}}}$
+\end_inset
+
+ is the position of the center of the circle.
+ With this definition it follows that the equations of the tangent 
+\begin_inset Formula $\vec{t}$
+\end_inset
+
+ and the normal 
+\begin_inset Formula $\vec{n}$
+\end_inset
+
+ are
+\begin_inset Formula \begin{align*}
+\vec{t}(\phi) & =(\begin{array}{c}
+\cos(\phi)\\
+\sin(\phi)\end{array})\\
+\vec{n} & =(\begin{array}{c}
+\cos(\phi+\frac{\pi}{2})\\
+\sin(\phi+\frac{\pi}{2})\end{array})=(\begin{array}{c}
+-\sin(\phi)\\
+\cos(\phi)\end{array})\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This definition contains another subtlety: The direction of the tangent
+ is the same as that of the rotation in the osculation point.
+ The normal vector is defined by a rotation of the tangent by 
+\begin_inset Formula $+\pi/2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Now we can define 
+\begin_inset Formula $\vec{d}=\vec{P^{0}}-\vec{P^{\text{r}}}=d_{0}\vec{n}$
+\end_inset
+
+, where the sign of 
+\begin_inset Formula $d_{0}$
+\end_inset
+
+ now depends on the direction of rotation and whether 
+\begin_inset Formula $\vec{P^{\text{r}}}$
+\end_inset
+
+ is inside or otside the circle.
+ This definition is sensible, because 
+\begin_inset Formula $\vec{P^{0}}$
+\end_inset
+
+ is the point of closest approach to the reference point.
+ Since 
+\begin_inset Formula $|\vec{n}|\equiv1$
+\end_inset
+
+, 
+\begin_inset Formula $d_{0}=(\vec{d},\vec{n})=(\vec{P^{0}}-\vec{P^{\text{r}}},\vec{n})=-(P_{x}^{0}-P_{x}^{\text{r}})\sin(\phi)+(P_{y}^{0}-P_{y}^{\text{r}})\cos(\phi)$
+\end_inset
+
+.
+ 
+\end_layout
+

 \begin_layout Subsection
 From physical quantities to track parameters
 \end_layout

@@ -247,7 +421,7 @@

 \end_inset
 
 , 

-\begin_inset Formula $l=(\phi_{0}-\phi)p_{t}/qaB_{z}$

+\begin_inset Formula $l=(\text{atan2}(\tilde{p}_{y},\tilde{p}_{x})-\phi)p_{t}/qaB_{z}$

 \end_inset
 
 .