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+<h1>Aspect ratio 3D</h1>\r
+\r
+<p><img src="image86.jpg" width="416px" height="385px" border="0" class="img_whs1"></p>\r
+\r
+<p> </p>\r
+\r
+<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 \r
+ criterion, however, it is also applied to 3D mesh elements: tetrahedrons, \r
+ pentahedrons, hexahedrons, etc. </p>\r
+\r
+<p>There are 2 simple formulas to evaluate directly the value of quality \r
+ Qk for simplex in 2D and 3D. The formulas are :</p>\r
+\r
+<p>\95 For triangle :</p>\r
+\r
+<p> </p>\r
+\r
+<p><img src="image21.gif" width="158px" height="110px" border="0" class="img_whs3"></p>\r
+\r
+<p><img src="image23.gif" width="120px" height="48px" border="0" class="img_whs4"></p>\r
+\r
+<p>where :</p>\r
+\r
+<p> </p>\r
+\r
+<p><img src="image27.gif" width="234px" height="96px" border="0" class="img_whs5"></p>\r
+\r
+<p> </p>\r
+\r
+<p> </p>\r
+\r
+<p>\95 For tetrahedron :</p>\r
+\r
+<p> <img src="image20.gif" width="258px" height="246px" border="0" class="img_whs6"></p>\r
+\r
+<p> <img src="image19.gif" width="133px" height="56px" border="0" class="img_whs7"></p>\r
+\r
+<p>where :</p>\r
+\r
+<p><img src="image18.gif" width="425px" height="192px" border="0" class="img_whs8"> </p>\r
+\r
+<p>To calculate Sk , it is needed to sum the area of each of the 3 faces \r
+ with the formula given with the triangle.</p>\r
+\r
+<p> </p>\r
+\r
+<p>For the other types of element like quadrangle, pentahedron and hexahedron \r
+ the formula is the following :</p>\r
+\r
+<p><img src="image30.gif" width="89px" height="32px" border="0" class="img_whs9"> </p>\r
+\r
+<p>Where Qi are the quality of all the possible simplex (of same dimension \r
+ as the element) that compose the element.</p>\r
+\r
+<p> </p>\r
+\r
+<p class="whs10"><span style="font-weight: bold;"><B>See Also</B></span> \r
+ a sample TUI Script of an <a href="quality_controls.htm#bookmark11">Aspect \r
+ Ratio quality control</a> operation. </p>\r
+\r
+<p> </p>\r
+\r
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