Fix incorrect area calculation in corner cases (fix #16820)

In certain circumstances very proximal nodes could cause instability
in the ellipsoidal area calculation.

Port (slightly tweaked) fix from grass changeset 71167 for same issue,
and add a unit test

(cherry-picked from d850393)

Author: nyalldawson

src/core/qgsdistancearea.cpp

@@ -897,6 +897,14 @@ double QgsDistanceArea::computePolygonArea( const QList<QgsPoint>& points ) cons
   double Qbar1, Qbar2;
   double area;
 
+  /* GRASS comment: threshold for dy, should be between 1e-4 and 1e-7
+   * QGIS note: while the grass comment states that thres should be between 1e-4->1e-7,
+   * a value of 1e-7 caused TestQgsDistanceArea::regression14675() to regress
+   * The maximum threshold possible which permits regression14675() to pass
+   * was found to be ~0.7e-7.
+  */
+  const double thresh = 0.7e-7;
+
   QgsDebugMsgLevel( "Ellipsoid: " + mEllipsoid, 3 );
   if (( ! mEllipsoidalMode ) || ( mEllipsoid == GEO_NONE ) )
   {
@@ -927,11 +935,28 @@ double QgsDistanceArea::computePolygonArea( const QList<QgsPoint>& points ) cons
         x1 += m_TwoPI;
 
     dx = x2 - x1;
-    area += dx * ( m_Qp - getQ( y2 ) );
-
     dy = y2 - y1;
-    if ( !qgsDoubleNear( dy, 0.0 ) )
-      area += dx * getQ( y2 ) - ( dx / dy ) * ( Qbar2 - Qbar1 );
+    if ( qAbs( dy ) > thresh )
+    {
+      /* account for different latitudes y1, y2 */
+      area += dx * ( m_Qp - ( Qbar2 - Qbar1 ) / dy );
+    }
+    else
+    {
+      /* latitudes y1, y2 are (nearly) identical */
+
+      /* if y2 becomes similar to y1, i.e. y2 -> y1
+       * Qbar2 - Qbar1 -> 0 and dy -> 0
+       * (Qbar2 - Qbar1) / dy -> ?
+       * (Qbar2 - Qbar1) / dy should approach Q((y1 + y2) / 2)
+       * Metz 2017
+       */
+      area += dx * ( m_Qp - getQ( y2 ) );
+
+      /* original:
+       * area += dx * getQ( y2 ) - ( dx / dy ) * ( Qbar2 - Qbar1 );
+       */
+    }
   }
   if (( area *= m_AE ) < 0.0 )
     area = -area;

tests/src/core/testqgsdistancearea.cpp

@@ -45,6 +45,7 @@ class TestQgsDistanceArea: public QObject
     void measureAreaAndUnits();
     void emptyPolygon();
     void regression14675();
+    void regression16820();
 
 };
 
@@ -370,6 +371,16 @@ void TestQgsDistanceArea::regression14675()
   QGSCOMPARENEAR( calc.measureArea( &geom ), 0.83301, 0.02 );
 }
 
+void TestQgsDistanceArea::regression16820()
+{
+  QgsDistanceArea calc;
+  calc.setEllipsoid( "WGS84" );
+  calc.setSourceCrs( QgsCoordinateReferenceSystem( "EPSG:32634" ) );
+  QgsGeometry geom( QgsGeometryFactory::geomFromWkt( "Polygon ((110250.54038314701756462 5084495.57398066483438015, 110243.46975068224128336 5084507.17200060561299324, 110251.23908144699817058 5084506.68309532757848501, 110251.2394439501222223 5084506.68307251576334238, 110250.54048078990308568 5084495.57553235255181789, 110250.54038314701756462 5084495.57398066483438015))" ) );
+  //lots of tolerance here - the formulas get quite unstable with small areas due to division by very small floats
+  QGSCOMPARENEAR( calc.measureArea( &geom ), 43.183369, 0.2 );
+}
+
 QTEST_MAIN( TestQgsDistanceArea )
 #include "testqgsdistancearea.moc"