qgsinternalgeometryengine.cpp

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/***************************************************************************
  qgsinternalgeometryengine.cpp - QgsInternalGeometryEngine
 
 ---------------------
 begin                : 13.1.2016
 copyright            : (C) 2016 by Matthias Kuhn
 email                : [email protected]
 ***************************************************************************
 *                                                                         *
 *   This program is free software; you can redistribute it and/or modify  *
 *   it under the terms of the GNU General Public License as published by  *
 *   the Free Software Foundation; either version 2 of the License, or     *
 *   (at your option) any later version.                                   *
 *                                                                         *
 ***************************************************************************/
 
#include "qgsinternalgeometryengine.h"
 
#include "qgslinestring.h"
#include "qgsmultipolygon.h"
#include "qgspolygon.h"
#include "qgsmulticurve.h"
#include "qgscircularstring.h"
#include "qgsgeometry.h"
#include "qgsgeometryutils.h"
#include "qgslinesegment.h"
#include "qgscircle.h"
#include "qgslogger.h"
#include "qgstessellator.h"
#include "qgsfeedback.h"
#include "qgsgeometryengine.h"
#include <QTransform>
#include <functional>
#include <memory>
#include <queue>
#include <random>
 
QgsInternalGeometryEngine::QgsInternalGeometryEngine( const QgsGeometry &geometry )
  : mGeometry( geometry.constGet() )
{
 
}
 
QString QgsInternalGeometryEngine::lastError() const
{
  return mLastError;
}
 
/***************************************************************************
 * This class is considered CRITICAL and any change MUST be accompanied with
 * full unit tests.
 * See details in QEP #17
 ****************************************************************************/
 
QgsGeometry QgsInternalGeometryEngine::extrude( double x, double y ) const
{
  mLastError.clear();
  QVector<QgsLineString *> linesToProcess;
 
  const QgsMultiCurve *multiCurve = qgsgeometry_cast< const QgsMultiCurve * >( mGeometry );
  if ( multiCurve )
  {
    linesToProcess.reserve( multiCurve->partCount() );
    for ( int i = 0; i < multiCurve->partCount(); ++i )
    {
      linesToProcess << static_cast<QgsLineString *>( multiCurve->geometryN( i )->clone() );
    }
  }
 
  const QgsCurve *curve = qgsgeometry_cast< const QgsCurve * >( mGeometry );
  if ( curve )
  {
    linesToProcess << static_cast<QgsLineString *>( curve->segmentize() );
  }
 
  std::unique_ptr<QgsMultiPolygon> multipolygon( linesToProcess.size() > 1 ? new QgsMultiPolygon() : nullptr );
  QgsPolygon *polygon = nullptr;
 
  if ( !linesToProcess.empty() )
  {
    std::unique_ptr< QgsLineString > secondline;
    for ( QgsLineString *line : qgis::as_const( linesToProcess ) )
    {
      QTransform transform = QTransform::fromTranslate( x, y );
 
      secondline.reset( line->reversed() );
      secondline->transform( transform );
 
      line->append( secondline.get() );
      line->addVertex( line->pointN( 0 ) );
 
      polygon = new QgsPolygon();
      polygon->setExteriorRing( line );
 
      if ( multipolygon )
        multipolygon->addGeometry( polygon );
    }
 
    if ( multipolygon )
      return QgsGeometry( multipolygon.release() );
    else
      return QgsGeometry( polygon );
  }
 
  return QgsGeometry();
}
 
 
 
// polylabel implementation
// ported from the original JavaScript implementation developed by Vladimir Agafonkin
// originally licensed under the ISC License
 
class Cell
{
  public:
    Cell( double x, double y, double h, const QgsPolygon *polygon )
      : x( x )
      , y( y )
      , h( h )
      , d( polygon->pointDistanceToBoundary( x, y ) )
      , max( d + h * M_SQRT2 )
    {}
 
    double x;
    double y;
    double h;
    double d;
    double max;
};
 
struct GreaterThanByMax
{
  bool operator()( const Cell *lhs, const Cell *rhs )
  {
    return rhs->max > lhs->max;
  }
};
 
Cell *getCentroidCell( const QgsPolygon *polygon )
{
  double area = 0;
  double x = 0;
  double y = 0;
 
  const QgsLineString *exterior = static_cast< const QgsLineString *>( polygon->exteriorRing() );
  int len = exterior->numPoints() - 1; //assume closed
  for ( int i = 0, j = len - 1; i < len; j = i++ )
  {
    double aX = exterior->xAt( i );
    double aY = exterior->yAt( i );
    double bX = exterior->xAt( j );
    double bY = exterior->yAt( j );
    double f = aX * bY - bX * aY;
    x += ( aX + bX ) * f;
    y += ( aY + bY ) * f;
    area += f * 3;
  }
  if ( area == 0 )
    return new Cell( exterior->xAt( 0 ), exterior->yAt( 0 ), 0, polygon );
  else
    return new Cell( x / area, y / area, 0.0, polygon );
}
 
QgsPoint surfacePoleOfInaccessibility( const QgsSurface *surface, double precision, double &distanceFromBoundary )
{
  std::unique_ptr< QgsPolygon > segmentizedPoly;
  const QgsPolygon *polygon = qgsgeometry_cast< const QgsPolygon * >( surface );
  if ( !polygon )
  {
    segmentizedPoly.reset( static_cast< QgsPolygon *>( surface->segmentize() ) );
    polygon = segmentizedPoly.get();
  }
 
  // start with the bounding box
  QgsRectangle bounds = polygon->boundingBox();
 
  // initial parameters
  double cellSize = std::min( bounds.width(), bounds.height() );
 
  if ( qgsDoubleNear( cellSize, 0.0 ) )
    return QgsPoint( bounds.xMinimum(), bounds.yMinimum() );
 
  double h = cellSize / 2.0;
  std::priority_queue< Cell *, std::vector<Cell *>, GreaterThanByMax > cellQueue;
 
  // cover polygon with initial cells
  for ( double x = bounds.xMinimum(); x < bounds.xMaximum(); x += cellSize )
  {
    for ( double y = bounds.yMinimum(); y < bounds.yMaximum(); y += cellSize )
    {
      cellQueue.push( new Cell( x + h, y + h, h, polygon ) );
    }
  }
 
  // take centroid as the first best guess
  std::unique_ptr< Cell > bestCell( getCentroidCell( polygon ) );
 
  // special case for rectangular polygons
  std::unique_ptr< Cell > bboxCell( new Cell( bounds.xMinimum() + bounds.width() / 2.0,
                                    bounds.yMinimum() + bounds.height() / 2.0,
                                    0, polygon ) );
  if ( bboxCell->d > bestCell->d )
  {
    bestCell = std::move( bboxCell );
  }
 
  while ( !cellQueue.empty() )
  {
    // pick the most promising cell from the queue
    std::unique_ptr< Cell > cell( cellQueue.top() );
    cellQueue.pop();
    Cell *currentCell = cell.get();
 
    // update the best cell if we found a better one
    if ( currentCell->d > bestCell->d )
    {
      bestCell = std::move( cell );
    }
 
    // do not drill down further if there's no chance of a better solution
    if ( currentCell->max - bestCell->d <= precision )
      continue;
 
    // split the cell into four cells
    h = currentCell->h / 2.0;
    cellQueue.push( new Cell( currentCell->x - h, currentCell->y - h, h, polygon ) );
    cellQueue.push( new Cell( currentCell->x + h, currentCell->y - h, h, polygon ) );
    cellQueue.push( new Cell( currentCell->x - h, currentCell->y + h, h, polygon ) );
    cellQueue.push( new Cell( currentCell->x + h, currentCell->y + h, h, polygon ) );
  }
 
  distanceFromBoundary = bestCell->d;
 
  return QgsPoint( bestCell->x, bestCell->y );
}
 
 
QgsGeometry QgsInternalGeometryEngine::poleOfInaccessibility( double precision, double *distanceFromBoundary ) const
{
  mLastError.clear();
  if ( distanceFromBoundary )
    *distanceFromBoundary = std::numeric_limits<double>::max();
 
  if ( !mGeometry || mGeometry->isEmpty() )
    return QgsGeometry();
 
  if ( precision <= 0 )
    return QgsGeometry();
 
  if ( const QgsGeometryCollection *gc = qgsgeometry_cast< const QgsGeometryCollection *>( mGeometry ) )
  {
    int numGeom = gc->numGeometries();
    double maxDist = 0;
    QgsPoint bestPoint;
    bool found = false;
    for ( int i = 0; i < numGeom; ++i )
    {
      const QgsSurface *surface = qgsgeometry_cast< const QgsSurface * >( gc->geometryN( i ) );
      if ( !surface )
        continue;
 
      found = true;
      double dist = std::numeric_limits<double>::max();
      QgsPoint p = surfacePoleOfInaccessibility( surface, precision, dist );
      if ( dist > maxDist )
      {
        maxDist = dist;
        bestPoint = p;
      }
    }
 
    if ( !found )
      return QgsGeometry();
 
    if ( distanceFromBoundary )
      *distanceFromBoundary = maxDist;
    return QgsGeometry( new QgsPoint( bestPoint ) );
  }
  else
  {
    const QgsSurface *surface = qgsgeometry_cast< const QgsSurface * >( mGeometry );
    if ( !surface )
      return QgsGeometry();
 
    double dist = std::numeric_limits<double>::max();
    QgsPoint p = surfacePoleOfInaccessibility( surface, precision, dist );
    if ( distanceFromBoundary )
      *distanceFromBoundary = dist;
    return QgsGeometry( new QgsPoint( p ) );
  }
}
 
 
// helpers for orthogonalize
// adapted from original code in potlatch/id osm editor
 
bool dotProductWithinAngleTolerance( double dotProduct, double lowerThreshold, double upperThreshold )
{
  return lowerThreshold > std::fabs( dotProduct ) || std::fabs( dotProduct ) > upperThreshold;
}
 
double normalizedDotProduct( const QgsPoint &a, const QgsPoint &b, const QgsPoint &c )
{
  QgsVector p = a - b;
  QgsVector q = c - b;
 
  if ( p.length() > 0 )
    p = p.normalized();
  if ( q.length() > 0 )
    q = q.normalized();
 
  return p * q;
}
 
double squareness( QgsLineString *ring, double lowerThreshold, double upperThreshold )
{
  double sum = 0.0;
 
  bool isClosed = ring->isClosed();
  int numPoints = ring->numPoints();
  QgsPoint a;
  QgsPoint b;
  QgsPoint c;
 
  for ( int i = 0; i < numPoints; ++i )
  {
    if ( !isClosed && i == numPoints - 1 )
      break;
    else if ( !isClosed && i == 0 )
    {
      b = ring->pointN( 0 );
      c = ring->pointN( 1 );
    }
    else
    {
      if ( i == 0 )
      {
        a = ring->pointN( numPoints - 1 );
        b = ring->pointN( 0 );
      }
      if ( i == numPoints - 1 )
        c = ring->pointN( 0 );
      else
        c = ring->pointN( i + 1 );
 
      double dotProduct = normalizedDotProduct( a, b, c );
      if ( !dotProductWithinAngleTolerance( dotProduct, lowerThreshold, upperThreshold ) )
        continue;
 
      sum += 2.0 * std::min( std::fabs( dotProduct - 1.0 ), std::min( std::fabs( dotProduct ), std::fabs( dotProduct + 1 ) ) );
    }
    a = b;
    b = c;
  }
 
  return sum;
}
 
QgsVector calcMotion( const QgsPoint &a, const QgsPoint &b, const QgsPoint &c,
                      double lowerThreshold, double upperThreshold )
{
  QgsVector p = a - b;
  QgsVector q = c - b;
 
  if ( qgsDoubleNear( p.length(), 0.0 ) || qgsDoubleNear( q.length(), 0.0 ) )
    return QgsVector( 0, 0 );
 
  // 2.0 is a magic number from the original JOSM source code
  double scale = 2.0 * std::min( p.length(), q.length() );
 
  p = p.normalized();
  q = q.normalized();
  double dotProduct = p * q;
 
  if ( !dotProductWithinAngleTolerance( dotProduct, lowerThreshold, upperThreshold ) )
  {
    return QgsVector( 0, 0 );
  }
 
  // wonderful nasty hack which has survived through JOSM -> id -> QGIS
  // to deal with almost-straight segments (angle is closer to 180 than to 90/270).
  if ( dotProduct < -M_SQRT1_2 )
    dotProduct += 1.0;
 
  QgsVector new_v = p + q;
  // 0.1 magic number from JOSM implementation - think this is to limit each iterative step
  return new_v.normalized() * ( 0.1 * dotProduct * scale );
}
 
QgsLineString *doOrthogonalize( QgsLineString *ring, int iterations, double tolerance, double lowerThreshold, double upperThreshold )
{
  double minScore = std::numeric_limits<double>::max();
 
  bool isClosed = ring->isClosed();
  int numPoints = ring->numPoints();
 
  std::unique_ptr< QgsLineString > best( ring->clone() );
 
  QVector< QgsVector > /* yep */ motions;
  motions.reserve( numPoints );
 
  for ( int it = 0; it < iterations; ++it )
  {
    motions.resize( 0 ); // avoid re-allocations
 
    // first loop through an calculate all motions
    QgsPoint a;
    QgsPoint b;
    QgsPoint c;
    for ( int i = 0; i < numPoints; ++i )
    {
      if ( isClosed && i == numPoints - 1 )
        motions << motions.at( 0 ); //closed ring, so last point follows first point motion
      else if ( !isClosed && ( i == 0 || i == numPoints - 1 ) )
      {
        b = ring->pointN( 0 );
        c = ring->pointN( 1 );
        motions << QgsVector( 0, 0 ); //non closed line, leave first/last vertex untouched
      }
      else
      {
        if ( i == 0 )
        {
          a = ring->pointN( numPoints - 1 );
          b = ring->pointN( 0 );
        }
        if ( i == numPoints - 1 )
          c = ring->pointN( 0 );
        else
          c = ring->pointN( i + 1 );
 
        motions << calcMotion( a, b, c, lowerThreshold, upperThreshold );
      }
      a = b;
      b = c;
    }
 
    // then apply them
    for ( int i = 0; i < numPoints; ++i )
    {
      ring->setXAt( i, ring->xAt( i ) + motions.at( i ).x() );
      ring->setYAt( i, ring->yAt( i ) + motions.at( i ).y() );
    }
 
    double newScore = squareness( ring, lowerThreshold, upperThreshold );
    if ( newScore < minScore )
    {
      best.reset( ring->clone() );
      minScore = newScore;
    }
 
    if ( minScore < tolerance )
      break;
  }
 
  delete ring;
 
  return best.release();
}
 
 
QgsAbstractGeometry *orthogonalizeGeom( const QgsAbstractGeometry *geom, int maxIterations, double tolerance, double lowerThreshold, double upperThreshold )
{
  std::unique_ptr< QgsAbstractGeometry > segmentizedCopy;
  if ( QgsWkbTypes::isCurvedType( geom->wkbType() ) )
  {
    segmentizedCopy.reset( geom->segmentize() );
    geom = segmentizedCopy.get();
  }
 
  if ( QgsWkbTypes::geometryType( geom->wkbType() ) == QgsWkbTypes::LineGeometry )
  {
    return doOrthogonalize( static_cast< QgsLineString * >( geom->clone() ),
                            maxIterations, tolerance, lowerThreshold, upperThreshold );
  }
  else
  {
    // polygon
    const QgsPolygon *polygon = static_cast< const QgsPolygon * >( geom );
    QgsPolygon *result = new QgsPolygon();
 
    result->setExteriorRing( doOrthogonalize( static_cast< QgsLineString * >( polygon->exteriorRing()->clone() ),
                             maxIterations, tolerance, lowerThreshold, upperThreshold ) );
    for ( int i = 0; i < polygon->numInteriorRings(); ++i )
    {
      result->addInteriorRing( doOrthogonalize( static_cast< QgsLineString * >( polygon->interiorRing( i )->clone() ),
                               maxIterations, tolerance, lowerThreshold, upperThreshold ) );
    }
 
    return result;
  }
}
 
QgsGeometry QgsInternalGeometryEngine::orthogonalize( double tolerance, int maxIterations, double angleThreshold ) const
{
  mLastError.clear();
  if ( !mGeometry || ( QgsWkbTypes::geometryType( mGeometry->wkbType() ) != QgsWkbTypes::LineGeometry
                       && QgsWkbTypes::geometryType( mGeometry->wkbType() ) != QgsWkbTypes::PolygonGeometry ) )
  {
    return QgsGeometry();
  }
 
  double lowerThreshold = std::cos( ( 90 - angleThreshold ) * M_PI / 180.00 );
  double upperThreshold = std::cos( angleThreshold * M_PI / 180.0 );
 
  if ( const QgsGeometryCollection *gc = qgsgeometry_cast< const QgsGeometryCollection *>( mGeometry ) )
  {
    int numGeom = gc->numGeometries();
    QVector< QgsAbstractGeometry * > geometryList;
    geometryList.reserve( numGeom );
    for ( int i = 0; i < numGeom; ++i )
    {
      geometryList << orthogonalizeGeom( gc->geometryN( i ), maxIterations, tolerance, lowerThreshold, upperThreshold );
    }
 
    QgsGeometry first = QgsGeometry( geometryList.takeAt( 0 ) );
    for ( QgsAbstractGeometry *g : qgis::as_const( geometryList ) )
    {
      first.addPart( g );
    }
    return first;
  }
  else
  {
    return QgsGeometry( orthogonalizeGeom( mGeometry, maxIterations, tolerance, lowerThreshold, upperThreshold ) );
  }
}
 
// if extraNodesPerSegment < 0, then use distance based mode
QgsLineString *doDensify( const QgsLineString *ring, int extraNodesPerSegment = -1, double distance = 1 )
{
  QVector< double > outX;
  QVector< double > outY;
  QVector< double > outZ;
  QVector< double > outM;
  double multiplier = 1.0 / double( extraNodesPerSegment + 1 );
 
  int nPoints = ring->numPoints();
  outX.reserve( ( extraNodesPerSegment + 1 ) * nPoints );
  outY.reserve( ( extraNodesPerSegment + 1 ) * nPoints );
  bool withZ = ring->is3D();
  if ( withZ )
    outZ.reserve( ( extraNodesPerSegment + 1 ) * nPoints );
  bool withM = ring->isMeasure();
  if ( withM )
    outM.reserve( ( extraNodesPerSegment + 1 ) * nPoints );
  double x1 = 0;
  double x2 = 0;
  double y1 = 0;
  double y2 = 0;
  double z1 = 0;
  double z2 = 0;
  double m1 = 0;
  double m2 = 0;
  double xOut = 0;
  double yOut = 0;
  double zOut = 0;
  double mOut = 0;
  int extraNodesThisSegment = extraNodesPerSegment;
  for ( int i = 0; i < nPoints - 1; ++i )
  {
    x1 = ring->xAt( i );
    x2 = ring->xAt( i + 1 );
    y1 = ring->yAt( i );
    y2 = ring->yAt( i + 1 );
    if ( withZ )
    {
      z1 = ring->zAt( i );
      z2 = ring->zAt( i + 1 );
    }
    if ( withM )
    {
      m1 = ring->mAt( i );
      m2 = ring->mAt( i + 1 );
    }
 
    outX << x1;
    outY << y1;
    if ( withZ )
      outZ << z1;
    if ( withM )
      outM << m1;
 
    if ( extraNodesPerSegment < 0 )
    {
      // distance mode
      extraNodesThisSegment = std::floor( std::sqrt( ( x2 - x1 ) * ( x2 - x1 ) + ( y2 - y1 ) * ( y2 - y1 ) ) / distance );
      if ( extraNodesThisSegment >= 1 )
        multiplier = 1.0 / ( extraNodesThisSegment + 1 );
    }
 
    for ( int j = 0; j < extraNodesThisSegment; ++j )
    {
      double delta = multiplier * ( j + 1 );
      xOut = x1 + delta * ( x2 - x1 );
      yOut = y1 + delta * ( y2 - y1 );
      if ( withZ )
        zOut = z1 + delta * ( z2 - z1 );
      if ( withM )
        mOut = m1 + delta * ( m2 - m1 );
 
      outX << xOut;
      outY << yOut;
      if ( withZ )
        outZ << zOut;
      if ( withM )
        outM << mOut;
    }
  }
  outX << ring->xAt( nPoints - 1 );
  outY << ring->yAt( nPoints - 1 );
  if ( withZ )
    outZ << ring->zAt( nPoints - 1 );
  if ( withM )
    outM << ring->mAt( nPoints - 1 );
 
  QgsLineString *result = new QgsLineString( outX, outY, outZ, outM );
  return result;
}
 
QgsAbstractGeometry *densifyGeometry( const QgsAbstractGeometry *geom, int extraNodesPerSegment = 1, double distance = 1 )
{
  std::unique_ptr< QgsAbstractGeometry > segmentizedCopy;
  if ( QgsWkbTypes::isCurvedType( geom->wkbType() ) )
  {
    segmentizedCopy.reset( geom->segmentize() );
    geom = segmentizedCopy.get();
  }
 
  if ( QgsWkbTypes::geometryType( geom->wkbType() ) == QgsWkbTypes::LineGeometry )
  {
    return doDensify( static_cast< const QgsLineString * >( geom ), extraNodesPerSegment, distance );
  }
  else
  {
    // polygon
    const QgsPolygon *polygon = static_cast< const QgsPolygon * >( geom );
    QgsPolygon *result = new QgsPolygon();
 
    result->setExteriorRing( doDensify( static_cast< const QgsLineString * >( polygon->exteriorRing() ),
                                        extraNodesPerSegment, distance ) );
    for ( int i = 0; i < polygon->numInteriorRings(); ++i )
    {
      result->addInteriorRing( doDensify( static_cast< const QgsLineString * >( polygon->interiorRing( i ) ),
                                          extraNodesPerSegment, distance ) );
    }
 
    return result;
  }
}
 
QgsGeometry QgsInternalGeometryEngine::densifyByCount( int extraNodesPerSegment ) const
{
  mLastError.clear();
  if ( !mGeometry )
  {
    return QgsGeometry();
  }
 
  if ( QgsWkbTypes::geometryType( mGeometry->wkbType() ) == QgsWkbTypes::PointGeometry )
  {
    return QgsGeometry( mGeometry->clone() ); // point geometry, nothing to do
  }
 
  if ( const QgsGeometryCollection *gc = qgsgeometry_cast< const QgsGeometryCollection *>( mGeometry ) )
  {
    int numGeom = gc->numGeometries();
    QVector< QgsAbstractGeometry * > geometryList;
    geometryList.reserve( numGeom );
    for ( int i = 0; i < numGeom; ++i )
    {
      geometryList << densifyGeometry( gc->geometryN( i ), extraNodesPerSegment );
    }
 
    QgsGeometry first = QgsGeometry( geometryList.takeAt( 0 ) );
    for ( QgsAbstractGeometry *g : qgis::as_const( geometryList ) )
    {
      first.addPart( g );
    }
    return first;
  }
  else
  {
    return QgsGeometry( densifyGeometry( mGeometry, extraNodesPerSegment ) );
  }
}
 
QgsGeometry QgsInternalGeometryEngine::densifyByDistance( double distance ) const
{
  mLastError.clear();
  if ( !mGeometry )
  {
    return QgsGeometry();
  }
 
  if ( QgsWkbTypes::geometryType( mGeometry->wkbType() ) == QgsWkbTypes::PointGeometry )
  {
    return QgsGeometry( mGeometry->clone() ); // point geometry, nothing to do
  }
 
  if ( const QgsGeometryCollection *gc = qgsgeometry_cast< const QgsGeometryCollection *>( mGeometry ) )
  {
    int numGeom = gc->numGeometries();
    QVector< QgsAbstractGeometry * > geometryList;
    geometryList.reserve( numGeom );
    for ( int i = 0; i < numGeom; ++i )
    {
      geometryList << densifyGeometry( gc->geometryN( i ), -1, distance );
    }
 
    QgsGeometry first = QgsGeometry( geometryList.takeAt( 0 ) );
    for ( QgsAbstractGeometry *g : qgis::as_const( geometryList ) )
    {
      first.addPart( g );
    }
    return first;
  }
  else
  {
    return QgsGeometry( densifyGeometry( mGeometry, -1, distance ) );
  }
}
 
//
// QgsLineSegmentDistanceComparer
//
 
// adapted for QGIS geometry classes from original work at https://github.com/trylock/visibility by trylock
bool QgsLineSegmentDistanceComparer::operator()( QgsLineSegment2D ab, QgsLineSegment2D cd ) const
{
  Q_ASSERT_X( ab.pointLeftOfLine( mOrigin ) != 0,
              "line_segment_dist_comparer",
              "AB must not be collinear with the origin." );
  Q_ASSERT_X( cd.pointLeftOfLine( mOrigin ) != 0,
              "line_segment_dist_comparer",
              "CD must not be collinear with the origin." );
 
  // flip the segments so that if there are common endpoints,
  // they will be the segment's start points
  if ( ab.end() == cd.start() || ab.end() == cd.end() )
    ab.reverse();
  if ( ab.start() == cd.end() )
    cd.reverse();
 
  // cases with common endpoints
  if ( ab.start() == cd.start() )
  {
    const int oad = QgsGeometryUtils::leftOfLine( cd.endX(), cd.endY(), mOrigin.x(), mOrigin.y(), ab.startX(), ab.startY() );
    const int oab = ab.pointLeftOfLine( mOrigin );
    if ( ab.end() == cd.end() || oad != oab )
      return false;
    else
      return ab.pointLeftOfLine( cd.end() ) != oab;
  }
  else
  {
    // cases without common endpoints
    const int cda = cd.pointLeftOfLine( ab.start() );
    const int cdb = cd.pointLeftOfLine( ab.end() );
    if ( cdb == 0 && cda == 0 )
    {
      return mOrigin.sqrDist( ab.start() ) < mOrigin.sqrDist( cd.start() );
    }
    else if ( cda == cdb || cda == 0 || cdb == 0 )
    {
      const int cdo = cd.pointLeftOfLine( mOrigin );
      return cdo == cda || cdo == cdb;
    }
    else
    {
      const int abo = ab.pointLeftOfLine( mOrigin );
      return abo != ab.pointLeftOfLine( cd.start() );
    }
  }
}
 
//
// QgsClockwiseAngleComparer
//
 
bool QgsClockwiseAngleComparer::operator()( const QgsPointXY &a, const QgsPointXY &b ) const
{
  const bool aIsLeft = a.x() < mVertex.x();
  const bool bIsLeft = b.x() < mVertex.x();
  if ( aIsLeft != bIsLeft )
    return bIsLeft;
 
  if ( qgsDoubleNear( a.x(), mVertex.x() ) && qgsDoubleNear( b.x(), mVertex.x() ) )
  {
    if ( a.y() >= mVertex.y() || b.y() >= mVertex.y() )
    {
      return b.y() < a.y();
    }
    else
    {
      return a.y() < b.y();
    }
  }
  else
  {
    const QgsVector oa = a - mVertex;
    const QgsVector ob = b - mVertex;
    const double det = oa.crossProduct( ob );
    if ( qgsDoubleNear( det, 0.0 ) )
    {
      return oa.lengthSquared() < ob.lengthSquared();
    }
    else
    {
      return det < 0;
    }
  }
}
 
 
//
// QgsRay2D
//
 
bool QgsRay2D::intersects( const QgsLineSegment2D &segment, QgsPointXY &intersectPoint ) const
{
  const QgsVector ao = origin - segment.start();
  const QgsVector ab = segment.end() - segment.start();
  const double det = ab.crossProduct( direction );
  if ( qgsDoubleNear( det, 0.0 ) )
  {
    const int abo = segment.pointLeftOfLine( origin );
    if ( abo != 0 )
    {
      return false;
    }
    else
    {
      const double distA = ao * direction;
      const double distB = ( origin - segment.end() ) * direction;
 
      if ( distA > 0 && distB > 0 )
      {
        return false;
      }
      else
      {
        if ( ( distA > 0 ) != ( distB > 0 ) )
          intersectPoint = origin;
        else if ( distA > distB ) // at this point, both distances are negative
          intersectPoint = segment.start(); // hence the nearest point is A
        else
          intersectPoint = segment.end();
        return true;
      }
    }
  }
  else
  {
    const double u = ao.crossProduct( direction ) / det;
    if ( u < 0.0 || 1.0 < u )
    {
      return false;
    }
    else
    {
      const double t = -ab.crossProduct( ao ) / det;
      intersectPoint = origin + direction * t;
      return qgsDoubleNear( t, 0.0 ) || t > 0;
    }
  }
}
 
QVector<QgsPointXY> generateSegmentCurve( const QgsPoint &center1, const double radius1, const QgsPoint &center2, const double radius2 )
{
  // ensure that first circle is smaller than second
  if ( radius1 > radius2 )
    return generateSegmentCurve( center2, radius2, center1, radius1 );
 
  QgsPointXY t1;
  QgsPointXY t2;
  QgsPointXY t3;
  QgsPointXY t4;
  QVector<QgsPointXY> points;
  if ( QgsGeometryUtils::circleCircleOuterTangents( center1, radius1, center2, radius2, t1, t2, t3, t4 ) )
  {
    points << t1
           << t2
           << t4
           << t3;
  }
  return points;
}
 
// partially ported from JTS VariableWidthBuffer,
// https://github.com/topobyte/jts/blob/master/jts-lab/src/main/java/com/vividsolutions/jts/operation/buffer/VariableWidthBuffer.java
 
QgsGeometry QgsInternalGeometryEngine::variableWidthBuffer( int segments, const std::function< std::unique_ptr< double[] >( const QgsLineString *line ) > &widthFunction ) const
{
  mLastError.clear();
  if ( !mGeometry )
  {
    return QgsGeometry();
  }
 
  std::vector< std::unique_ptr<QgsLineString > > linesToProcess;
 
  const QgsMultiCurve *multiCurve = qgsgeometry_cast< const QgsMultiCurve * >( mGeometry );
  if ( multiCurve )
  {
    for ( int i = 0; i < multiCurve->partCount(); ++i )
    {
      if ( static_cast< const QgsCurve * >( multiCurve->geometryN( i ) )->nCoordinates() == 0 )
        continue; // skip 0 length lines
 
      linesToProcess.emplace_back( static_cast<QgsLineString *>( multiCurve->geometryN( i )->clone() ) );
    }
  }
 
  const QgsCurve *curve = qgsgeometry_cast< const QgsCurve * >( mGeometry );
  if ( curve )
  {
    if ( curve->nCoordinates() > 0 )
      linesToProcess.emplace_back( static_cast<QgsLineString *>( curve->segmentize() ) );
  }
 
  if ( linesToProcess.empty() )
  {
    QgsGeometry g;
    g.mLastError = QStringLiteral( "Input geometry was not a curve type geometry" );
    return g;
  }
 
  QVector<QgsGeometry> bufferedLines;
  bufferedLines.reserve( linesToProcess.size() );
 
  for ( std::unique_ptr< QgsLineString > &line : linesToProcess )
  {
    QVector<QgsGeometry> parts;
    QgsPoint prevPoint;
    double prevRadius = 0;
    QgsGeometry prevCircle;
 
    std::unique_ptr< double[] > widths = widthFunction( line.get() ) ;
    for ( int i = 0; i < line->nCoordinates(); ++i )
    {
      QgsPoint thisPoint = line->pointN( i );
      QgsGeometry thisCircle;
      double thisRadius = widths[ i ] / 2.0;
      if ( thisRadius > 0 )
      {
        QgsGeometry p = QgsGeometry( thisPoint.clone() );
 
        QgsCircle circ( thisPoint, thisRadius );
        thisCircle = QgsGeometry( circ.toPolygon( segments * 4 ) );
        parts << thisCircle;
      }
      else
      {
        thisCircle = QgsGeometry( thisPoint.clone() );
      }
 
      if ( i > 0 )
      {
        if ( prevRadius > 0 || thisRadius > 0 )
        {
          QVector< QgsPointXY > points = generateSegmentCurve( prevPoint, prevRadius, thisPoint, thisRadius );
          if ( !points.empty() )
          {
            // snap points to circle vertices
 
            int atVertex = 0;
            int beforeVertex = 0;
            int afterVertex = 0;
            double sqrDist = 0;
            double sqrDistPrev = 0;
            for ( int j = 0; j < points.count(); ++j )
            {
              QgsPointXY pA = prevCircle.closestVertex( points.at( j ), atVertex, beforeVertex, afterVertex, sqrDistPrev );
              QgsPointXY pB = thisCircle.closestVertex( points.at( j ), atVertex, beforeVertex, afterVertex, sqrDist );
              points[j] = sqrDistPrev < sqrDist ? pA : pB;
            }
            // close ring
            points.append( points.at( 0 ) );
 
            std::unique_ptr< QgsPolygon > poly = qgis::make_unique< QgsPolygon >();
            poly->setExteriorRing( new QgsLineString( points ) );
            if ( poly->area() > 0 )
              parts << QgsGeometry( std::move( poly ) );
          }
        }
      }
      prevPoint = thisPoint;
      prevRadius = thisRadius;
      prevCircle = thisCircle;
    }
 
    bufferedLines << QgsGeometry::unaryUnion( parts );
  }
 
  return QgsGeometry::collectGeometry( bufferedLines );
}
 
QgsGeometry QgsInternalGeometryEngine::taperedBuffer( double start, double end, int segments ) const
{
  mLastError.clear();
  start = std::fabs( start );
  end = std::fabs( end );
 
  auto interpolateWidths = [ start, end ]( const QgsLineString * line )->std::unique_ptr< double [] >
  {
    // ported from JTS VariableWidthBuffer,
    // https://github.com/topobyte/jts/blob/master/jts-lab/src/main/java/com/vividsolutions/jts/operation/buffer/VariableWidthBuffer.java
    std::unique_ptr< double [] > widths( new double[ line->nCoordinates() ] );
    widths[0] = start;
    widths[line->nCoordinates() - 1] = end;
 
    double lineLength = line->length();
    double currentLength = 0;
    QgsPoint prevPoint = line->pointN( 0 );
    for ( int i = 1; i < line->nCoordinates() - 1; ++i )
    {
      QgsPoint point = line->pointN( i );
      double segmentLength = point.distance( prevPoint );
      currentLength += segmentLength;
      double lengthFraction = lineLength > 0 ? currentLength / lineLength : 1;
      double delta = lengthFraction * ( end - start );
      widths[i] = start + delta;
      prevPoint = point;
    }
    return widths;
  };
 
  return variableWidthBuffer( segments, interpolateWidths );
}
 
QgsGeometry QgsInternalGeometryEngine::variableWidthBufferByM( int segments ) const
{
  mLastError.clear();
  auto widthByM = []( const QgsLineString * line )->std::unique_ptr< double [] >
  {
    std::unique_ptr< double [] > widths( new double[ line->nCoordinates() ] );
    for ( int i = 0; i < line->nCoordinates(); ++i )
    {
      widths[ i ] = line->mAt( i );
    }
    return widths;
  };
 
  return variableWidthBuffer( segments, widthByM );
}
 
QVector<QgsPointXY> QgsInternalGeometryEngine::randomPointsInPolygon( const QgsGeometry &polygon, int count,
    const std::function< bool( const QgsPointXY & ) > &acceptPoint, unsigned long seed, QgsFeedback *feedback, int maxTriesPerPoint )
{
  if ( polygon.type() != QgsWkbTypes::PolygonGeometry || count == 0 )
    return QVector< QgsPointXY >();
 
  // step 1 - tessellate the polygon to triangles
  QgsRectangle bounds = polygon.boundingBox();
  QgsTessellator t( bounds, false, false, false, true );
 
  if ( polygon.isMultipart() )
  {
    const QgsMultiSurface *ms = qgsgeometry_cast< const QgsMultiSurface * >( polygon.constGet() );
    for ( int i = 0; i < ms->numGeometries(); ++i )
    {
      if ( feedback && feedback->isCanceled() )
        return QVector< QgsPointXY >();
 
      if ( QgsPolygon *poly = qgsgeometry_cast< QgsPolygon * >( ms->geometryN( i ) ) )
      {
        t.addPolygon( *poly, 0 );
      }
      else
      {
        std::unique_ptr< QgsPolygon > p( qgsgeometry_cast< QgsPolygon * >( ms->geometryN( i )->segmentize() ) );
        t.addPolygon( *p, 0 );
      }
    }
  }
  else
  {
    if ( const QgsPolygon *poly = qgsgeometry_cast< const QgsPolygon * >( polygon.constGet() ) )
    {
      t.addPolygon( *poly, 0 );
    }
    else
    {
      std::unique_ptr< QgsPolygon > p( qgsgeometry_cast< QgsPolygon * >( polygon.constGet()->segmentize() ) );
      t.addPolygon( *p, 0 );
    }
  }
 
  if ( feedback && feedback->isCanceled() )
    return QVector< QgsPointXY >();
 
  const QVector<float> triangleData = t.data();
  if ( triangleData.empty() )
    return QVector< QgsPointXY >(); //hm
 
  // calculate running sum of triangle areas
  std::vector< double > cumulativeAreas;
  cumulativeAreas.reserve( t.dataVerticesCount() / 3 );
  double totalArea = 0;
  for ( auto it = triangleData.constBegin(); it != triangleData.constEnd(); )
  {
    if ( feedback && feedback->isCanceled() )
      return QVector< QgsPointXY >();
 
    const float aX = *it++;
    ( void )it++; // z
    const float aY = -( *it++ );
    const float bX = *it++;
    ( void )it++; // z
    const float bY = -( *it++ );
    const float cX = *it++;
    ( void )it++; // z
    const float cY = -( *it++ );
 
    const double area = QgsGeometryUtils::triangleArea( aX, aY, bX, bY, cX, cY );
    totalArea += area;
    cumulativeAreas.emplace_back( totalArea );
  }
 
  std::random_device rd;
  std::mt19937 mt( seed == 0 ? rd() : seed );
  std::uniform_real_distribution<> uniformDist( 0, 1 );
 
  // selects a random triangle, weighted by triangle area
  auto selectRandomTriangle = [&cumulativeAreas, totalArea]( double random )->int
  {
    int triangle = 0;
    const double target = random * totalArea;
    for ( auto it = cumulativeAreas.begin(); it != cumulativeAreas.end(); ++it, triangle++ )
    {
      if ( *it > target )
        return triangle;
    }
    Q_ASSERT_X( false, "QgsInternalGeometryEngine::randomPointsInPolygon", "Invalid random triangle index" );
    return 0; // no warnings
  };
 
 
  QVector<QgsPointXY> result;
  result.reserve( count );
  int tries = 0;
  for ( int i = 0; i < count; )
  {
    if ( feedback && feedback->isCanceled() )
      return QVector< QgsPointXY >();
 
    const double triangleIndexRnd = uniformDist( mt );
    // pick random triangle, weighted by triangle area
    const int triangleIndex = selectRandomTriangle( triangleIndexRnd );
 
    // generate a random point inside this triangle
    const double weightB = uniformDist( mt );
    const double weightC = uniformDist( mt );
    double x;
    double y;
 
    // get triangle
    const double aX = triangleData.at( triangleIndex * 9 ) + bounds.xMinimum();
    const double aY = -triangleData.at( triangleIndex * 9 + 2 ) + bounds.yMinimum();
    const double bX = triangleData.at( triangleIndex * 9 + 3 ) + bounds.xMinimum();
    const double bY = -triangleData.at( triangleIndex * 9 + 5 ) + bounds.yMinimum();
    const double cX = triangleData.at( triangleIndex * 9 + 6 ) + bounds.xMinimum();
    const double cY = -triangleData.at( triangleIndex * 9 + 8 ) + bounds.yMinimum();
    QgsGeometryUtils::weightedPointInTriangle( aX, aY, bX, bY, cX, cY, weightB, weightC, x, y );
 
    QgsPointXY candidate( x, y );
    if ( acceptPoint( candidate ) )
    {
      result << QgsPointXY( x, y );
      i++;
      tries = 0;
    }
    else if ( maxTriesPerPoint != 0 )
    {
      tries++;
      // Skip this point if maximum tries is reached
      if ( tries == maxTriesPerPoint )
      {
        tries = 0;
        i++;
      }
    }
  }
  return result;
}
 
// ported from PostGIS' lwgeom pta_unstroke
 
std::unique_ptr< QgsCompoundCurve > lineToCurve( const QgsLineString *lineString, double distanceTolerance,
    double pointSpacingAngleTolerance )
{
  std::unique_ptr< QgsCompoundCurve > out = qgis::make_unique< QgsCompoundCurve >();
 
  /* Minimum number of edges, per quadrant, required to define an arc */
  const unsigned int minQuadEdges = 2;
 
  /* Die on null input */
  if ( !lineString )
    return nullptr;
 
  /* Null on empty input? */
  if ( lineString->nCoordinates() == 0 )
    return nullptr;
 
  /* We can't desegmentize anything shorter than four points */
  if ( lineString->nCoordinates() < 4 )
  {
    out->addCurve( lineString->clone() );
    return out;
  }
 
  /* Allocate our result array of vertices that are part of arcs */
  int numEdges = lineString->nCoordinates() - 1;
  QVector< int > edgesInArcs( numEdges + 1, 0 );
 
  auto arcAngle = []( const QgsPoint & a, const QgsPoint & b, const QgsPoint & c )->double
  {
    double abX = b.x() - a.x();
    double abY = b.y() - a.y();
 
    double cbX = b.x() - c.x();
    double cbY = b.y() - c.y();
 
    double dot = ( abX * cbX + abY * cbY ); /* dot product */
    double cross = ( abX * cbY - abY * cbX ); /* cross product */
 
    double alpha = std::atan2( cross, dot );
 
    return alpha;
  };
 
  /* We make a candidate arc of the first two edges, */
  /* And then see if the next edge follows it */
  int i = 0;
  int j = 0;
  int k = 0;
  int currentArc = 1;
  QgsPoint a1;
  QgsPoint a2;
  QgsPoint a3;
  QgsPoint b;
  double centerX = 0.0;
  double centerY = 0.0;
  double radius = 0;
 
  while ( i < numEdges - 2 )
  {
    unsigned int arcEdges = 0;
    double numQuadrants = 0;
    double angle;
 
    bool foundArc = false;
    /* Make candidate arc */
    a1 = lineString->pointN( i );
    a2 = lineString->pointN( i + 1 );
    a3 = lineString->pointN( i + 2 );
    QgsPoint first = a1;
 
    for ( j = i + 3; j < numEdges + 1; j++ )
    {
      b = lineString->pointN( j );
 
      /* Does this point fall on our candidate arc? */
      if ( QgsGeometryUtils::pointContinuesArc( a1, a2, a3, b, distanceTolerance, pointSpacingAngleTolerance ) )
      {
        /* Yes. Mark this edge and the two preceding it as arc components */
        foundArc = true;
        for ( k = j - 1; k > j - 4; k-- )
          edgesInArcs[k] = currentArc;
      }
      else
      {
        /* No. So we're done with this candidate arc */
        currentArc++;
        break;
      }
 
      a1 = a2;
      a2 = a3;
      a3 = b;
    }
    /* Jump past all the edges that were added to the arc */
    if ( foundArc )
    {
      /* Check if an arc was composed by enough edges to be
       * really considered an arc
       * See http://trac.osgeo.org/postgis/ticket/2420
       */
      arcEdges = j - 1 - i;
      if ( first.x() == b.x() && first.y() == b.y() )
      {
        numQuadrants = 4;
      }
      else
      {
        QgsGeometryUtils::circleCenterRadius( first, b, a1, radius, centerX, centerY );
 
        angle = arcAngle( first, QgsPoint( centerX, centerY ), b );
        int p2Side = QgsGeometryUtils::leftOfLine( b.x(), b.y(), first.x(), first.y(), a1.x(), a1.y() );
        if ( p2Side >= 0 )
          angle = -angle;
 
        if ( angle < 0 )
          angle = 2 * M_PI + angle;
        numQuadrants = ( 4 * angle ) / ( 2 * M_PI );
      }
      /* a1 is first point, b is last point */
      if ( arcEdges < minQuadEdges * numQuadrants )
      {
        // LWDEBUGF( 4, "Not enough edges for a %g quadrants arc, %g needed", num_quadrants, min_quad_edges * num_quadrants );
        for ( k = j - 1; k >= i; k-- )
          edgesInArcs[k] = 0;
      }
 
      i = j - 1;
    }
    else
    {
      /* Mark this edge as a linear edge */
      edgesInArcs[i] = 0;
      i = i + 1;
    }
  }
 
  int start = 0;
  int end = 0;
  /* non-zero if edge is part of an arc */
  int edgeType = edgesInArcs[0];
 
  auto addPointsToCurve = [ lineString, &out ]( int start, int end, int type )
  {
    if ( type == 0 )
    {
      // straight segment
      QVector< QgsPoint > points;
      for ( int j = start; j < end + 2; ++ j )
      {
        points.append( lineString->pointN( j ) );
      }
      std::unique_ptr< QgsCurve > straightSegment = qgis::make_unique< QgsLineString >( points );
      out->addCurve( straightSegment.release() );
    }
    else
    {
      // curved segment
      QVector< QgsPoint > points;
      points.append( lineString->pointN( start ) );
      points.append( lineString->pointN( ( start + end + 1 ) / 2 ) );
      points.append( lineString->pointN( end + 1 ) );
      std::unique_ptr< QgsCircularString > curvedSegment = qgis::make_unique< QgsCircularString >();
      curvedSegment->setPoints( points );
      out->addCurve( curvedSegment.release() );
    }
  };
 
  for ( int i = 1; i < numEdges; i++ )
  {
    if ( edgeType != edgesInArcs[i] )
    {
      end = i - 1;
      addPointsToCurve( start, end, edgeType );
      start = i;
      edgeType = edgesInArcs[i];
    }
  }
 
  /* Roll out last item */
  end = numEdges - 1;
  addPointsToCurve( start, end, edgeType );
 
  return out;
}
 
std::unique_ptr< QgsAbstractGeometry > convertGeometryToCurves( const QgsAbstractGeometry *geom, double distanceTolerance, double angleTolerance )
{
  if ( QgsWkbTypes::geometryType( geom->wkbType() ) == QgsWkbTypes::LineGeometry )
  {
    return lineToCurve( static_cast< const QgsLineString * >( geom ), distanceTolerance, angleTolerance );
  }
  else
  {
    // polygon
    const QgsPolygon *polygon = static_cast< const QgsPolygon * >( geom );
    std::unique_ptr< QgsCurvePolygon > result = qgis::make_unique< QgsCurvePolygon>();
 
    result->setExteriorRing( lineToCurve( static_cast< const QgsLineString * >( polygon->exteriorRing() ),
                                          distanceTolerance, angleTolerance ).release() );
    for ( int i = 0; i < polygon->numInteriorRings(); ++i )
    {
      result->addInteriorRing( lineToCurve( static_cast< const QgsLineString * >( polygon->interiorRing( i ) ),
                                            distanceTolerance, angleTolerance ).release() );
    }
 
    return result;
  }
}
 
QgsGeometry QgsInternalGeometryEngine::convertToCurves( double distanceTolerance, double angleTolerance ) const
{
  mLastError.clear();
  if ( !mGeometry )
  {
    return QgsGeometry();
  }
 
  if ( QgsWkbTypes::geometryType( mGeometry->wkbType() ) == QgsWkbTypes::PointGeometry )
  {
    return QgsGeometry( mGeometry->clone() ); // point geometry, nothing to do
  }
 
  if ( QgsWkbTypes::isCurvedType( mGeometry->wkbType() ) )
  {
    // already curved. In future we may want to allow this, and convert additional candidate segments
    // in an already curved geometry to curves
    return QgsGeometry( mGeometry->clone() );
  }
 
  if ( const QgsGeometryCollection *gc = qgsgeometry_cast< const QgsGeometryCollection *>( mGeometry ) )
  {
    int numGeom = gc->numGeometries();
    QVector< QgsAbstractGeometry * > geometryList;
    geometryList.reserve( numGeom );
    for ( int i = 0; i < numGeom; ++i )
    {
      geometryList << convertGeometryToCurves( gc->geometryN( i ), distanceTolerance, angleTolerance ).release();
    }
 
    QgsGeometry first = QgsGeometry( geometryList.takeAt( 0 ) );
    for ( QgsAbstractGeometry *g : qgis::as_const( geometryList ) )
    {
      first.addPart( g );
    }
    return first;
  }
  else
  {
    return QgsGeometry( convertGeometryToCurves( mGeometry, distanceTolerance, angleTolerance ) );
  }
}
 
QgsGeometry QgsInternalGeometryEngine::orientedMinimumBoundingBox( double &area, double &angle, double &width, double &height ) const
{
  mLastError.clear();
 
  QgsRectangle minRect;
  area = std::numeric_limits<double>::max();
  angle = 0;
  width = std::numeric_limits<double>::max();
  height = std::numeric_limits<double>::max();
 
  if ( !mGeometry || mGeometry->nCoordinates() < 2 )
    return QgsGeometry();
 
  std::unique_ptr< QgsGeometryEngine >engine( QgsGeometry::createGeometryEngine( mGeometry ) );
  QString error;
  std::unique_ptr< QgsAbstractGeometry > hull( engine->convexHull( &mLastError ) );
  if ( !hull )
    return QgsGeometry();
 
  QgsVertexId vertexId;
  QgsPoint pt0;
  QgsPoint pt1;
  QgsPoint pt2;
  // get first point
  hull->nextVertex( vertexId, pt0 );
  pt1 = pt0;
  double totalRotation = 0;
  while ( hull->nextVertex( vertexId, pt2 ) )
  {
    double currentAngle = QgsGeometryUtils::lineAngle( pt1.x(), pt1.y(), pt2.x(), pt2.y() );
    double rotateAngle = 180.0 / M_PI * currentAngle;
    totalRotation += rotateAngle;
 
    QTransform t = QTransform::fromTranslate( pt0.x(), pt0.y() );
    t.rotate( rotateAngle );
    t.translate( -pt0.x(), -pt0.y() );
 
    hull->transform( t );
 
    QgsRectangle bounds = hull->boundingBox();
    double currentArea = bounds.width() * bounds.height();
    if ( currentArea  < area )
    {
      minRect = bounds;
      area = currentArea;
      angle = totalRotation;
      width = bounds.width();
      height = bounds.height();
    }
 
    pt1 = hull->vertexAt( vertexId );
  }
 
  QgsGeometry minBounds = QgsGeometry::fromRect( minRect );
  minBounds.rotate( angle, QgsPointXY( pt0.x(), pt0.y() ) );
 
  // constrain angle to 0 - 180
  if ( angle > 180.0 )
    angle = std::fmod( angle, 180.0 );
 
  return minBounds;
}