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In order to supply an initial vertex position, we first calculate
an approximate vertex position by calculating the intersection
in the *xy* plane of
two helices arbitrarily chosen from the given set of tracks.
There are two such intersections in general.
We then compares the differences of *z* coordinates and take the
intersection corresponding to the smaller distance.
The mid-point in the *z* direction
is our first guess of the common vertex
.We approximate the helices to tangential lines at the
points on the helices that are the closest
in the *xy* plane to the trial vertex position:

| |
(26) |

where
| |
(27) |

and is given by
| |
(28) |

with (*x*_{c}^{i},*y*_{c}^{i}) being the center of the *i*-th helix:
| |
(29) |

Then, what to minimize is the sum of the distance squared
of each tangential line from a new trial vertex :
| |
(30) |

with
| |
(31) |

Notice that the is a quadratic funciton of so
that the minimization condition
| |
(32) |

is a linear equation which can be solved by a single matrix inversion.
The solution is our improved guess of the common vertex.
We repeat this process until the vertex position converges.

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** Up:** Vertex Fitting
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*Keisuke Fujii*

*12/4/1998*