<b>Geometric Progression</b> hypothesis allows splitting edges into
segments with a length that changes in geometric progression (Lk =
-Lk-1 * d) starting from a given <b>Start Length</b> and <b>Common Ratio</b>.
+Lk-1 * d) starting from a given <b>Start Length</b> and with a given
+<b>Common Ratio</b>.
The splitting direction is defined by the orientation of the
underlying geometrical edge.
underlying geometrical edge. <b>"Reverse Edges"</b> list box allows to
specify the edges for which the splitting should be made in the
direction opposing to their orientation. This list box is enabled only
-if the geometry object is selected for the meshing. In this case the
-user can select edges to be reversed either by directly picking them
+if the geometry object is selected for the meshing. In this case it is
+possible to select edges to be reversed either by directly picking them
in the 3D viewer or by selecting the edges or groups of edges in the
Object Browser.
<br><b>Equidistant Distribution</b> - all segments will have the same
length, you define only the <b>Number of Segments</b>.
-<br><b>Scale Distribution</b> - length of segments gradually changes depending on the <b>Scale Factor</b>, which is a ratio of the first segment length to the last segment length.
+<br><b>Scale Distribution</b> - length of segments gradually changes
+depending on the <b>Scale Factor</b>, which is a ratio of the first
+segment length to the last segment length.<br>
+Length of segments changes in geometric progression with the common
+ratio (A) depending on the <b>Scale Factor</b> (S) and <b>Number of
+Segments</b> (N) as follows: <code> A = S**(1/(N-1))</code>. For an
+edge of length L, length of the first segment is
+<code>L * (1 - A)/(1 - A**N)</code>.
-\image html a-nbsegments2.png
-
-<br><b>Distribution with Table Density</b> - you input a number of
-pairs <b>t - F(t)</b>, where \b t ranges from 0 to 1, and the module computes the
-formula, which will rule the change of length of segments and shows
-the curve in the plot. You can select the <b>Conversion mode</b> from
-\b Exponent and <b>Cut negative</b>.
-\image html distributionwithtabledensity.png
+\image html a-nbsegments2.png
<br><b>Distribution with Analytic Density</b> - you input the formula,
which will rule the change of length of segments and the module shows
-the curve in the plot.
+in the plot the density function curve in red and the node
+distribution as blue crosses.
\image html distributionwithanalyticdensity.png
+<br>
+\anchor analyticdensity_anchor
+The node distribution is computed so that to have the density function
+integral on the range between two nodes equal for all segments.
+\image html analyticdensity.png
+
+<br><b>Distribution with Table Density</b> - you input a number of
+pairs <b>t - F(t)</b>, where \b t ranges from 0 to 1, and the module computes the
+formula, which will rule the change of length of segments and shows
+in the plot the density function curve in red and the node
+distribution as blue crosses. The node distribution is computed in the
+same way as for
+\ref analyticdensity_anchor "Distribution with Analytic Density". You
+can select the <b>Conversion mode</b> from\b Exponent and <b>Cut
+negative</b>.
+
+\image html distributionwithtabledensity.png
+
<b>See Also</b> a sample TUI Script of a
\ref tui_deflection_1d "Defining Number of Segments" hypothesis
operation.
+
<br>
\anchor start_and_end_length_anchor
<h2>Start and End Length hypothesis</h2>
The direction of the splitting is defined by the orientation of the
underlying geometrical edge. <b>"Reverse Edges"</b> list box allows to
-specify the edges for which the splitting should be made in the
+specify the edges, for which the splitting should be made in the
direction opposing to their orientation. This list box is enabled only
-if the geometry object is selected for the meshing. In this case the
-user can select edges to be reversed either by directly picking them
+if the geometry object is selected for the meshing. In this case it is
+possible to select edges to be reversed either by directly picking them
in the 3D viewer or by selecting the edges or groups of edges in the
Object Browser.
\anchor automatic_length_anchor
<h2>Automatic Length</h2>
-This hypothesis is automatically applied when you select <b>Assign a
-set of hypotheses</b> option in Create Mesh menu.
-
-\image html automaticlength.png
-
The dialog box prompts you to define the quality of the future mesh by
only one parameter, which is \b Fineness, ranging from 0 (coarse mesh,
low number of elements) to 1 (extremely fine mesh, great number of
-elements). Compare one and the same object (sphere) meshed with
+elements).
+
+\image html automaticlength.png
+
+Compare one and the same object (sphere) meshed with
minimum and maximum value of this parameter.
\image html image147.gif "Example of a very rough mesh. Automatic Length works for 0."
