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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 applied to 3D mesh elements: tetrahedrons, \r
- pentahedrons, hexahedrons, etc. </p>\r
+<p><img src="i_blue.jpg" x-maintain-ratio="TRUE" width="30px" height="30px" border="0" class="img_whs2"> The Aspect Ratio 3D mesh quality criterion calculates \r
+ the same parameter as the <a href="files/aspect_ratio.htm">Aspect ratio</a> \r
+ criterion, but it is applied to 3D mesh elements: tetrahedrons, pentahedrons, \r
+ hexahedrons, etc. </p>\r
\r
-<p>There are simple formulas to evaluate directly the value of quality \r
- Qk for simplex in 3D. The formulas are :</p>\r
-\r
-<br>\r
+<p> </p>\r
\r
-<p>\95 For tetrahedron :</p>\r
+<ul type="disc" class="whs3">\r
+ \r
+ <li class=kadov-p><p><img src="image20.gif" width="258px" height="246px" align="left" border="0" class="img_whs4">The <span style="font-weight: bold;"><B>Aspect \r
+ Ratio</B></span> of a <span style="font-weight: bold;"><B>tetrahedron</B></span> \r
+ 3D element is calculated by the formula:</p></li>\r
+</ul>\r
\r
-<p> <img src="image20.gif" width="258px" height="246px" border="0" class="img_whs6"></p>\r
+<p> </p>\r
\r
-<p> <img src="image19.gif" width="133px" height="56px" border="0" class="img_whs7"></p>\r
+<p> </p>\r
\r
-<p>where :</p>\r
+<p><img src="image19.gif" width="133px" height="56px" border="0" class="img_whs5">,</p>\r
\r
-<p><img src="image18.gif" width="425px" height="192px" border="0" class="img_whs8"> </p>\r
+<p> </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
+<p>where <span style="font-weight: bold; font-style: italic;"><I><B>S</B></I></span><span \r
+ style="font-weight: bold;\r
+ font-style: italic;\r
+ vertical-align: Sub;"><I><B>K</B></I></span> is the sum of surfaces of the faces \r
+ of К and <span style="font-weight: bold; font-style: italic;"><I><B>V</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>K</B></I></span> is the volume of К :</p>\r
\r
<p> </p>\r
\r
-<p>For the other types of element like pentahedron and hexahedron \r
- the formula is the following :</p>\r
+<p><img src="image18.gif" width="425px" height="192px" border="0" class="img_whs6">T</p>\r
\r
-<p><img src="image30.gif" width="89px" height="32px" border="0" class="img_whs9"> </p>\r
+<p> </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
+<ul type="disc" class="whs3">\r
+ \r
+ <li class=kadov-p><p>Other element types like quadrangle, pentahedron \r
+ and hexahedron use the following formula:</p></li>\r
+</ul>\r
+\r
+<p><img src="image30.gif" width="89px" height="32px" border="0" class="img_whs7"> , where <span style="font-weight: bold; font-style: italic;"><I><B>Q</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>i</B></I></span> represents the value of <span style="font-weight: bold; font-style: italic;"><I><B>Q</B></I></span><span \r
+ style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>k</B></I></span> for all possible simplexes (of the \r
+ same dimension as the element) that compose the element.</p>\r
+\r
+<p>For example, a hexahedron is split in 6 tetrahedrons, the <span style="font-weight: bold;"><B>Aspect \r
+ ratio 3D</B></span> is calculated for each of them, then the greatest <span \r
+ style="font-weight: bold; font-style: italic;"><I><B>Q</B></I></span><span style="vertical-align: Sub;\r
+ font-weight: bold;\r
+ font-style: italic;"><I><B>k</B></I></span> is considered to be the <span \r
+ style="font-weight: bold;"><B>Aspect ratio 3D</B></span> \r
+ criterion for the whole hexahedron.</p>\r
+\r
+<p><img src="pics/image139.gif" x-maintain-ratio="TRUE" width="569px" height="386px" border="0" class="img_whs8"></p>\r
\r
-<p>For example, hexahedron is analysed via 6 tetrahedrons this way:</p>\r
+<p> </p>\r
\r
-<p><img src="image139.gif" border="0" class="img_whs10"> </p>\r
+<p class=TODO>To apply the Aspect Ratio 3D quality criterion to your mesh:</p>\r
\r
-<p> </p>\r
+<p class=TODO> </p>\r
\r
-<p>For 3D elements with quadrangular faces, aspect ratio of such faces\r
-is also analysed and the maximal one over 3D and 2D qualities is taken.</p>\r
+<p class="whs9">1. Display your mesh in the viewer.</p>\r
\r
-<p> </p>\r
+<p class="whs9"> </p>\r
+\r
+<p class="whs9">2. Choose <span style="font-weight: bold;"><B>Controls \r
+ > Aspect Ratio 3D </B></span>or click <img src="image144.gif" width="24px" height="26px" border="0" class="img_whs10"> button of the toolbar. \r
+ Your mesh will be displayed in the viewer with its elements colored according \r
+ to the applied mesh quality control criterion:</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
+<p class="whs11"><span style="font-weight: bold;"><B>See Also</B></span> \r
+ a sample TUI Script of an <span style="font-weight: bold;"><B><a href="quality_controls.htm#bookmark11">Aspect \r
+ Ratio 3D</B></span> quality control</a> operation. </p>\r
\r
<p> </p>\r
\r
