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<h1>Aspect ratio 3D</h1>
<p><img src="image86.jpg" width="416px" height="385px" border="0" class="img_whs1"></p>
<p> </p>
<p><img src="i_blue.jpg" x-maintain-ratio="TRUE" width="30px" height="30px" border="0" class="img_whs2">This mesh quality criterion resembles to the Aspect ratio
criterion, however, it is applied to 3D mesh elements: tetrahedrons,
pentahedrons, hexahedrons, etc. </p>
<p>There are simple formulas to evaluate directly the value of quality
Qk for simplex in 3D. The formulas are :</p>
<br>
<p>� For tetrahedron :</p>
<p> <img src="image20.gif" width="258px" height="246px" border="0" class="img_whs6"></p>
<p> <img src="image19.gif" width="133px" height="56px" border="0" class="img_whs7"></p>
<p>where :</p>
<p><img src="image18.gif" width="425px" height="192px" border="0" class="img_whs8"> </p>
<p>To calculate Sk , it is needed to sum the area of each of the 3 faces
with the formula given with the triangle.</p>
<p> </p>
<p>For the other types of element like pentahedron and hexahedron
the formula is the following :</p>
<p><img src="image30.gif" width="89px" height="32px" border="0" class="img_whs9"> </p>
<p>Where Qi are the quality of all the possible simplex (of same dimension
as the element) that compose the element.</p>
<p>For example, hexahedron is analysed via 6 tetrahedrons this way:</p>
<p><img src="image139.gif" border="0" class="img_whs10"> </p>
<p> </p>
<p>For 3D elements with quadrangular faces, aspect ratio of such faces
is also analysed and the maximal one over 3D and 2D qualities is taken.</p>
<p> </p>
<p> </p>
<p class="whs10"><span style="font-weight: bold;"><B>See Also</B></span>
a sample TUI Script of an <a href="quality_controls.htm#bookmark11">Aspect
Ratio quality control</a> operation. </p>
<p> </p>
<p> </p>
<p> </p>
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