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<!DOCTYPE html
PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN">
<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Crystal Geometry (DocHelp Toolbox)
</title>
<!-- DOCNAME: DocHelp Toolbox -->
<meta name="chunktype" content="refpage">
<!-- CHUNKNAME: CrystalGeometry -->
<!-- HEADSTUFF: CrystalGeometry -->
<!-- HEADSTUFF -->
<meta name="refentity" content="method:CrystalGeometry">
<meta http-equiv="Content-Script-Type" content="text/javascript">
<meta name="toctype" content="fcn">
<link rel="stylesheet" href="up.css"><script language="JavaScript" src="docscripts.js"></script></head>
<body><a name="top_of_page"></a><div class="navbar navbar-inverse navbar-fixed-top">
<div class="navbar-inner">
<div class="container"><a class="brand" href="/">MTEX Toolbox</a><ul class="nav">
<li><a href="/download.html">Downloads</a></li>
</ul>
<ul class="nav">
<li><a href="/documentation.html">Documentation</a></li>
</ul>
<ul class="nav">
<li><a href="/team.html">People</a></li>
</ul>
<ul class="nav">
<li><a href="/publications.html">Publications</a></li>
</ul>
<ul class="nav">
<li><a href="/support.html">Support</a></li>
</ul>
</div>
</div>
</div>
<div class="container">
<h1>Crystal Geometry</h1>
<div class="subheading">
<p>Introduces key concepts about the MTEX representation of specimen directions, crystal directions, crystal symmetries, rotations
and orientations.
</p>
</div>
<p>
<table class="ref" width="90%">
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('SpecimenDirections_block','SpecimenDirections_expandable_text');"
valign="top"
width="15">
<img id="SpecimenDirections_expandable_text" src="arrow_right.gif"
style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="SpecimenDirections.html">Specimen Directions<td valign="top" width="75%">How to represent directions with respect
to the sample or specimen reference system.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="SpecimenDirections_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="SpecimenDirections.html#4">Cartesian Coordinates</a>
</td>
</tr>
<tr>
<td>
<a href="SpecimenDirections.html#9">Polar Coordinates</a>
</td>
</tr>
<tr>
<td>
<a href="SpecimenDirections.html#11">Calculating with Specimen Directions</a>
</td>
</tr>
<tr>
<td>
<a href="SpecimenDirections.html#14">Lists of vectors</a>
</td>
</tr>
<tr>
<td>
<a href="SpecimenDirections.html#19">Indexing lists of vectors</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('Rotations_block','Rotations_expandable_text');"
valign="top"
width="15">
<img id="Rotations_expandable_text" src="arrow_right.gif" style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="Rotations.html">Rotations<td valign="top" width="75%">Rotations are the basic concept to understand crystal orientations
and crystal symmetries.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="Rotations_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="Rotations.html#4">Euler Angle Conventions</a>
</td>
</tr>
<tr>
<td>
<a href="Rotations.html#8">Other Ways of Defining a Rotation</a>
</td>
</tr>
<tr>
<td>
<a href="Rotations.html#15">Calculating with Rotations</a>
</td>
</tr>
<tr>
<td>
<a href="Rotations.html#23">Improper Rotations</a>
</td>
</tr>
<tr>
<td>
<a href="Rotations.html#28">Conversion into Euler Angles and Rodrigues Parametrisation</a>
</td>
</tr>
<tr>
<td>
<a href="Rotations.html#29">Plotting Rotations</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('CrystalSymmetries_block','CrystalSymmetries_expandable_text');"
valign="top"
width="15">
<img id="CrystalSymmetries_expandable_text" src="arrow_right.gif" style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="CrystalSymmetries.html">Crystal Symmetries<td valign="top" width="75%">This section covers the unit cell of a crystal,
its space, point and Laue groups as well as alignments of the crystal coordinate system.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="CrystalSymmetries_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="CrystalSymmetries.html#3">Crystallographic Space, Point and Laue Groups</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalSymmetries.html#14">The Crystal Coordinate System</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalSymmetries.html#17">Calculations</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalSymmetries.html#18">Plotting symmetries</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('CrystalDirections_block','CrystalDirections_expandable_text');"
valign="top"
width="15">
<img id="CrystalDirections_expandable_text" src="arrow_right.gif" style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="CrystalDirections.html">Crystal Directions<td valign="top" width="75%">Crystal directions are directions relative
to a crystal reference frame and are usually defined in terms of Miller indices. This sections explains how to calculate with
crystal directions in MTEX.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="CrystalDirections_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="CrystalDirections.html#3">Definition</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalDirections.html#8">Trigonal and Hexagonal Convention</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalDirections.html#9">Symmetrically Equivalent Crystal Directions</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalDirections.html#14">Angles</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalDirections.html#16">Conversions</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalDirections.html#18">Calculations</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('CrystalOrientations_block','CrystalOrientations_expandable_text');"
valign="top"
width="15">
<img id="CrystalOrientations_expandable_text" src="arrow_right.gif"
style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="CrystalOrientations.html">Crystal Orientations<td valign="top" width="75%">Explains how to define crystal orientations,
how to switch between different convention and how to compute crystallographic equivalent orientations.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="CrystalOrientations_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="CrystalOrientations.html#5">Defining a crystal orientation</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#7">SUB: Bunge Euler angle convention</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#8">SUB: Matthies Euler angle convention</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#9">SUB: Axis angle parametrisation</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#10">SUB: Miller indice</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#11">SUB: Four vectors defining a rotation</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#12">SUB: Defining an orientation by a 3 times 3 matrix</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#13">SUB: Predefined orientations</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#15">SUB: Random orientations</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#16">Coordinate transformations</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#23">Symmetric equivalence</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#25">Conversion into Euler angles, matrix, quaternion or Rodrigues vector</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#26">Plotting Orientations</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#27">SUB: in Euler angle space</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#29">SUB: in axis angle space</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#31">SUB: in (inverse) pole figures</a>
</td>
</tr>
<tr>
<td>
<a href="CrystalOrientations.html#33">SUB: in sections of the orientations space</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('crystalShape_index_block','crystalShape_index_expandable_text');"
valign="top"
width="15">
<img id="crystalShape_index_expandable_text" src="arrow_right.gif"
style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="crystalShape_index.html">Crystal Shapes (The Class @crystalShape)<td valign="top" width="75%">How to draw threedimensional
representations of crystals.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="crystalShape_index_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="crystalShape_index.html#3">Simple Crystal Shapes</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#5">Calculating with crystal shapes</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#7">Plotting crystal shapes</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#10">Twinning relationships</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#11">Defining complicated crystal shapes</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#18">Marking crystal faces</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#20">Defining complicated crystals more simple</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#23">Select Faces</a>
</td>
</tr>
<tr>
<td>
<a href="crystalShape_index.html#24">Gallery of hardcoded crystal shapes</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('Misorientations_block','Misorientations_expandable_text');"
valign="top"
width="15">
<img id="Misorientations_expandable_text" src="arrow_right.gif" style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="Misorientations.html">Misorientations<td valign="top" width="75%">Misorientation describes the relative orientation
of two grains with respect to each other. Important concepts are twinnings and CSL (coincidence site lattice),</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="Misorientations_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="Misorientations.html#3"> The misorientation angle</a>
</td>
</tr>
<tr>
<td>
<a href="Misorientations.html#9">Misorientations</a>
</td>
</tr>
<tr>
<td>
<a href="Misorientations.html#12">Coincident lattice planes</a>
</td>
</tr>
<tr>
<td>
<a href="Misorientations.html#14">Twinning misorientations</a>
</td>
</tr>
<tr>
<td>
<a href="Misorientations.html#16">Highlight twinning boundaries</a>
</td>
</tr>
<tr>
<td>
<a href="Misorientations.html#20">Phase transitions</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('fibre_index_block','fibre_index_expandable_text');"
valign="top"
width="15">
<img id="fibre_index_expandable_text" src="arrow_right.gif" style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="fibre_index.html">Fibres<td valign="top" width="75%">This sections describes the class <fibre_index.html fibre>
and gives an overview how to work with fibres in MTEX.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="fibre_index_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="fibre_index.html#2">Defining a Fibre</a>
</td>
</tr>
<tr>
<td>
<a href="fibre_index.html#4">SUB: by two orientations</a>
</td>
</tr>
<tr>
<td>
<a href="fibre_index.html#8">SUB: by two directions</a>
</td>
</tr>
<tr>
<td>
<a href="fibre_index.html#10">SUB: by one orientation and an orientation gradient</a>
</td>
</tr>
<tr>
<td>
<a href="fibre_index.html#11">SUB: predefined fibres</a>
</td>
</tr>
<tr>
<td>
<a href="fibre_index.html#13">Visualize an ODF along a fibre</a>
</td>
</tr>
<tr>
<td>
<a href="fibre_index.html#14">Compute volume of fibre portions</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
<div align="center">
<a href="#"
onClick="return toggleexpander('AxialDirectional_block','AxialDirectional_expandable_text');"
valign="top"
width="15">
<img id="AxialDirectional_expandable_text" src="arrow_right.gif" style="border:0px"/></a>
</div>
</td>
<td valign="top" width="250px">
<a href="AxialDirectional.html">Antipodal Symmetry<td valign="top" width="75%">MTEX allows to identify antipodal directions
to model axes and to identify misorientations with opposite rotational angle. The later is required when working with misorientations
between grains of the same phase and the order of the grains is arbitrary.</td></a>
</td>
</tr>
<tr>
<td/>
<td colspan="2">
<div class="expander" id="AxialDirectional_block"
style="display:none; background:#e7ebf7; padding-left:2ex;">
<table>
<tr>
<td>
<a href="AxialDirectional.html#3">Directions vs. Axes</a>
</td>
</tr>
<tr>
<td>
<a href="AxialDirectional.html#7">The Angle between Directions and Axes</a>
</td>
</tr>
<tr>
<td>
<a href="AxialDirectional.html#9">Antipodal Symmetry in Experimental Pole Figures</a>
</td>
</tr>
<tr>
<td>
<a href="AxialDirectional.html#10">Antipodal Symmetry in Recalculated Pole Figures</a>
</td>
</tr>
<tr>
<td>
<a href="AxialDirectional.html#12">Antipodal Symmetry in Inverse Pole Figures</a>
</td>
</tr>
<tr>
<td>
<a href="AxialDirectional.html#16">EBSD Colocoding</a>
</td>
</tr>
</table>
</div>
</td>
</tr>
<tr>
<td valign="top" width="15">
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<a href="MorawiecCell.html">Fundamental Regions <td valign="top" width="75%">Thanks to crystal symmetry the orientation space
can be reduced to the so called fundamental or asymmetric region. Those regions play an important role for the computation
of axis and angle distributions of misorientations.</td></a>
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<a href="MorawiecCell.html#3">The complete, symmetry free, orientation space</a>
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<a href="MorawiecCell.html#6">Crystal Symmetries</a>
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<a href="MorawiecCell.html#10">Change the center of the fundamental region</a>
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<a href="MorawiecCell.html#11">Fundamental regions of misorientations</a>
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<a href="MorawiecCell.html#13">Fundamental regions of misorientations with antipodal symmetry</a>
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<a href="MorawiecCell.html#16">Axis angle sections</a>
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