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1<head>
2<title>Local Reference System</title>
3<!-- Changed by: Chris ISELIN, 17-Jul-1997 -->
4<!-- Changed by: Hans Grote, 25-Sep-2002 -->
5</head>
6
7<body bgcolor="#ffffff">
8
9<center>
10EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
11<IMG SRC="http://cern.ch/madx/icons/mx7_25.gif" align=right>
12<h2>Local Reference Systems</h2>
13</center>
14
15<h2><a name=straight>Reference System for Straight Beam Elements</a></h2>
16In straight elements the local reference system is simply translated
17by the length of the element along the local <i>s</i>-axis.
18This is true for
19<ul>
20<li><a href="drift.html">Drift space</a>,
21<li><a href="quadrupole.html">Quadrupole</a>,
22<li><a href="sextupole.html">Sextupole</a>,
23<li><a href="octupole.html">Octupole</a>,
24<li><a href="solenoid.html">Solenoid</a>,
25<li><a href="crabcavity.html">CRABCAVITY</a>,
26<li><a href="cavity.html">RF cavity</a>,
27<li><a href="separator.html">Electrostatic separator</a>,
28<li><a href="kickers.html">Closed orbit corrector</a>,
29<li><a href="monitors.html">Beam position monitor</a>.
30</ul>
31The corresponding <i>R</i>, <i>S</i> are
32<p>
33<img align=bottom src="../equations/RSstraight.gif">
34<p>
35A rotation of the element about the <i>S</i>-axis has no effect
36on <i>R</i> and <i>S</i>, since the rotations of the reference system
37before and after the element cancel.
38<center>
39<img align=bottom src="../figures/ref_straight.gif">
40<p>
41<b>Figure 1:</b> Reference System for Straight Beam Elements
42</center>
43
44<h2><a name=rbend>Reference System for Bending Magnets</a></h2>
45<a href="bend.html">Bending magnets</a> have a curved reference orbit.
46For both rectangular and sector bending magnets
47<p>
48<img align=bottom src="../equations/RSbend.gif">
49<p>
50where alphais the bend angle.
51A positive bend angle represents a bend to the right,
52i.e. towards negative <i>x</i> values.
53For sector bending magnets,
54the bend radius is given by rho,
55and for rectangular bending magnets it has the value
56<p>
57rho = <i>L</i> / 2 sin(alpha/2).
58<p>
59If the magnet is rotated about the <i>s</i>-axis by an angle psi,
60<i>R</i> and <i>S</i> are transformed by
61<p>
62<i>R</i>* = <i>T R</i>, <i>S</i>* = <i>T S T<sup>-1</sup></i>.
63<p>
64where <i>T</i> is the orthogonal rotation matrix
65<p>
66<img align=bottom src="../equations/Trot.gif">
67<p>
68The special value psi = pi/2 represents a bend down.
69
70<center>
71<a name=rbend><img align=bottom src="../figures/ref_rbend.gif"></a>
72<p>
73<b>Figure 2:</b> Reference System for Rectangular Bends;
74The signs of the pole-face rotations are positive as shown.
75</center>
76
77<center>
78<a name=sbend><img align=bottom src="../figures/ref_sbend.gif"></a>
79<p>
80<b>Figure 3:</b>  Reference System for Sector Bends;
81The signs of the pole-face rotations are positive as shown.
82</center>
83
84<h2>Elements which do not Change the Local Reference</h2>
85<a href="marker.html">MARKER</a> elements do not
86affect the reference orbit.
87They are ignored for geometry calculations.
88
89<p>
90<address>
91<a href="http://www.cern.ch/Hans.Grote/hansg_sign.html">hansg</a>,
92January 24, 1997
93</address>
94
95</body>
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