r.fuzzy.set: correction of sshape formula in fuzzy.c (#931)

This PR corrects two issues with how the S-shape rescaling was implemented for negative values of the shape parameter. This includes the issue reported in #931.

In addition, in the manual page r.fuzzy.set.html, the description of the equations were updated to ensure that they match the equations as implemented in the code. And an issue with a wrongly formatted URL was corrected.

src/raster/r.fuzzy.set/fuzzy.c

@@ -1,5 +1,6 @@
 #include <grass/glocale.h>
 #include "local_proto.h"
+#include <math.h>
 
 float fuzzy(FCELL cell)
 {
@@ -42,8 +43,8 @@ float fuzzy(FCELL cell)
     }
 
     if (type == SSHAPE) {
-        m = pow(2, exp(2 * abs(shape)));
-        return (shape < 0) ? pow(1 - cos(x * PI2), m) : pow(sin(x * PI2), m);
+        m = pow(2, exp(2 * fabs(shape)));
+        return (shape < 0) ? 1 - pow(cos(x * PI2), m) : pow(sin(x * PI2), m);
     }
 
     return -1; /* error */

src/raster/r.fuzzy.set/r.fuzzy.set.html

@@ -75,7 +75,7 @@ <h4>Calculation of boundary shape</h4>
 Depending on type of the boundary different equation are used to determine its
 shape:
 <p>
-<b>Linear:</b> the membership is calculated according following equation:<br>
+<b>Linear:</b> the membership is calculated according following equation:
 <pre><code>
 value  <=  A -> x = 0
 A < value > B -> x = (value-A)/(B-A)
@@ -87,33 +87,38 @@ <h4>Calculation of boundary shape</h4>
 </code></pre>
 
 <p>
-<b>S-shaped:</b> it use following equation:
+<b>S-shaped:</b> the membership is calculated according following equation:
 <pre><code>
 sin(x * Pi/2)^m (for positive shape parameter)
 1-cos(x * Pi/2)^m (for negative shape parameter)
 
 where x: membership, and
-m = 2^exp(2,shape) (for positive shape parameter)
-m = 2^(1+shape) (for negative shape parameter)
-where m: shape parameter.
+m = 2^exp(2 * |shape|) 
+For default shape=0, m = 2 (most common parameter for that equation).
 </code></pre>
 
-For default shape parameter = 0 m is = 2 which is most common parameter for
-that equation.
+<p>
+<b>G-shaped:</b> the membership is calculated according following equation:
+<pre><code>
+tan(x * Pi/4)^1/m
+
+where x: membership, and
+m = 2^exp(-2 * shape) (for negative shape parameter)
+m = 2^(1-shape) (for positive shape parameter)
+For default shape=0, m = 2 (most common parameter for that equation).
+</code></pre>
 
 <p>
-<b>G-shaped and J shaped:</b> it use following equations:
+<b>J shaped:</b> it use following equations:
 <pre><code>
-tan(x * Pi/4)^m (for J-shaped)
-tan(x * Pi/4)^1/m (for G-shaped)
+tan(x * Pi/4)^m
 
 where x: membership, and
-m = 2^exp(2,shape) (for positive shape parameter)
+m = 2^exp(2 * shape) (for positive shape parameter)
 m = 2^(1+shape) (for negative shape parameter)
-where m: shape parameter.
+For default shape=0, m = 2 (most common parameter for that equation).
 </code></pre>
 
-
 <h2>SEE ALSO</h2>
 
 <em>
@@ -130,7 +135,7 @@ <h2>REFERENCES</h2>
 Computers & Geosciences (37) 1525-1531. DOI <a href=https://doi.org/10.1016/j.cageo.2010.09.008>https://doi.org/10.1016/j.cageo.2010.09.008</a>
 
 <li>Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353.
-<a href="https://doi.org/10.1016/S0019-9958(65)90241-X"><a href="https://doi.org/10.1016/S0019-9958(65)90241-X"</a>.
+<a href="https://doi.org/10.1016/S0019-9958(65)90241-X">https://doi.org/10.1016/S0019-9958(65)90241-X</a>.
 
 <li>Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger.
 ISBN 0-85274-583-4.