Helix

Let us write some the simplest definitions:
the distance along $Z$ is $z = \Theta b$ and distance along 2-dim circle is $ a = \Theta R$ from these two equation and taken into account that the same point has the same time to reach it one can derive in some approximation:
$t = D \Theta R m/p_{xy} = D \Theta b m/p_z$; where: $m$ is particle mass:
then we have $b = R (p_z/p_{xy})$

Some definitions:

\begin{displaymath} \mathit{p_{xy}} = \sqrt{\mathit{p_x}^{2} + \mathit{p_y}^{2}}... ...c {\mathit{p}_{z}}{ \left\vert \! \,q\,B\, \! \right\vert }} ; \end{displaymath}


\begin{displaymath} \mathit{x}_{c} = \mathit{x}_{p} + {\displaystyle \frac {\mat... ...\mathit{y}_{p} - {\displaystyle \frac {\mathit{p}_{x}}{q\,B}} \end{displaymath}


\begin{displaymath} \mathit{R^2+b^2} = {\displaystyle \frac {\mathit{px}^{2} + ... ...thit{pz}^{2}}{ \left\vert \! \,q\,B\, \! \right\vert ^{2}}} \end{displaymath}

Helix equation with axes along Z and with zero $\Theta$ at the point $x_p, y_p, z_p$ exactly:

\begin{displaymath} \left\{ \begin{array}{l} x = \cos{(q\; \Theta)}(x_p-x_c) - \... ...p-x_c) + y_c; \\ z = b\; \Theta + z_p; \\ \end{array}\right. \end{displaymath}


\begin{displaymath} dx = \frac{\partial x}{\partial \Theta}; \; \; dy = \frac{\p... ... \Theta}; \; \; dz = \frac{\partial z}{\partial \Theta}; \; \; \end{displaymath}

Equation of normal plane to helix through the hit point:

\begin{displaymath} (x_h-x)dx/(R^2+b^2) + (y_h-y)dy/(R^2+b^2) + (z_h-z)dz/(R^2+b^2)=0 \end{displaymath}

After substitutions:

\begin{displaymath} \begin{array}{l} - p_y\; q\; B x_h \sin(q\; \Theta)\; \vert ... ...\Theta + p_z\; q\; B^2 z_p\; \vert q\; B\vert = 0 \end{array}\end{displaymath}

The second order approximation (Tailor raw) of equation of normal plane to helix passes through the hit point near by $\Theta=0$ is

\begin{displaymath} \begin{array}{l} \left( {\displaystyle \frac {( - {\displays... ..._x}^{2} + \mathit{p_y} ^{2} + \mathit{p_z}^{2}}} =0 \end{array}\end{displaymath}

The solution of this equation gives the 3-D nearest to the hit point at the helix with rather good accuracy.

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Subroutine Mom_Helix_Dist_Array (xp,yp,zp,cx,cy,cz,p,qq,nh,xhi,yhi,zhi,dist)

The same as Mom_Helix_Dist but for array of hits.

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Subroutine Helix_dist

(xs,ys,zs,xh,yh,zh,x0,y0,rad,b,phi0,dphi,xm,ym,zm,dist)

Yuri Fisyak
Last modified: Sun Nov 28 11:27:41 EST 2004