qwt_spline.cpp

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/******************************************************************************
 * Qwt Widget Library
 * Copyright (C) 1997   Josef Wilgen
 * Copyright (C) 2002   Uwe Rathmann
 *
 * This library is free software; you can redistribute it and/or
 * modify it under the terms of the Qwt License, Version 1.0
 *****************************************************************************/
 
#include "qwt_spline.h"
#include "qwt_spline_parametrization.h"
#include "qwt_spline_polynomial.h"
#include "qwt_bezier.h"
 
#include <qpainterpath.h>
 
namespace QwtSplineC1P
{
    struct param
    {
        param( const QwtSplineParametrization* p ):
            parameter( p )
        {
        }
 
        inline double operator()( const QPointF& p1, const QPointF& p2 ) const
        {
            return parameter->valueIncrement( p1, p2 );
        }
 
        const QwtSplineParametrization* parameter;
    };
 
    struct paramY
    {
        inline double operator()( const QPointF& p1, const QPointF& p2 ) const
        {
            return QwtSplineParametrization::valueIncrementY( p1, p2 );
        }
    };
 
    struct paramUniform
    {
        inline double operator()( const QPointF& p1, const QPointF& p2 ) const
        {
            return QwtSplineParametrization::valueIncrementUniform( p1, p2 );
        }
    };
 
    struct paramCentripetal
    {
        inline double operator()( const QPointF& p1, const QPointF& p2 ) const
        {
            return QwtSplineParametrization::valueIncrementCentripetal( p1, p2 );
        }
    };
 
    struct paramChordal
    {
        inline double operator()( const QPointF& p1, const QPointF& p2 ) const
        {
            return QwtSplineParametrization::valueIncrementChordal( p1, p2 );
        }
    };
 
    struct paramManhattan
    {
        inline double operator()( const QPointF& p1, const QPointF& p2 ) const
        {
            return QwtSplineParametrization::valueIncrementManhattan( p1, p2 );
        }
    };
 
    class PathStore
    {
      public:
        inline void init( int size )
        {
            Q_UNUSED(size);
        }
 
        inline void start( double x1, double y1 )
        {
            path.moveTo( x1, y1 );
        }
 
        inline void addCubic( double cx1, double cy1,
            double cx2, double cy2, double x2, double y2 )
        {
            path.cubicTo( cx1, cy1, cx2, cy2, x2, y2 );
        }
 
        inline void end()
        {
            path.closeSubpath();
        }
 
        QPainterPath path;
    };
 
    class ControlPointsStore
    {
      public:
        inline ControlPointsStore():
            m_cp( NULL )
        {
        }
 
        inline void init( int size )
        {
            controlPoints.resize( size );
            m_cp = controlPoints.data();
        }
 
        inline void start( double x1, double y1 )
        {
            Q_UNUSED( x1 );
            Q_UNUSED( y1 );
        }
 
        inline void addCubic( double cx1, double cy1,
            double cx2, double cy2, double x2, double y2 )
        {
            Q_UNUSED( x2 );
            Q_UNUSED( y2 );
 
            QLineF& l = *m_cp++;
            l.setLine( cx1, cy1, cx2, cy2 );
        }
 
        inline void end()
        {
        }
 
        QVector< QLineF > controlPoints;
 
      private:
        QLineF* m_cp;
    };
 
    double slopeBoundary( int boundaryCondition, double boundaryValue,
        const QPointF& p1, const QPointF& p2, double slope1 )
    {
        const double dx = p2.x() - p1.x();
        const double dy = p2.y() - p1.y();
 
        double m = 0.0;
 
        switch( boundaryCondition )
        {
            case QwtSpline::Clamped1:
            {
                m = boundaryValue;
                break;
            }
            case QwtSpline::Clamped2:
            {
                const double c2 = 0.5 * boundaryValue;
                const double c1 = slope1;
 
                m = 0.5 * ( 3.0 * dy / dx - c1 - c2 * dx );
                break;
            }
            case QwtSpline::Clamped3:
            {
                const double c3 = boundaryValue / 6.0;
                m = c3 * dx * dx + 2 * dy / dx - slope1;
                break;
            }
            case QwtSpline::LinearRunout:
            {
                const double s = dy / dx;
                const double r = qBound( 0.0, boundaryValue, 1.0 );
 
                m = s - r * ( s - slope1 );
                break;
            }
            default:
            {
                m = dy / dx; // something
            }
        }
 
        return m;
    }
}
 
template< class SplineStore >
static inline SplineStore qwtSplineC1PathParamX(
    const QwtSplineC1* spline, const QPolygonF& points )
{
    const int n = points.size();
 
    const QVector< double > m = spline->slopes( points );
    if ( m.size() != n )
        return SplineStore();
 
    const QPointF* pd = points.constData();
    const double* md = m.constData();
 
    SplineStore store;
    store.init( m.size() - 1 );
    store.start( pd[0].x(), pd[0].y() );
 
    for ( int i = 0; i < n - 1; i++ )
    {
        const double dx3 = ( pd[i + 1].x() - pd[i].x() ) / 3.0;
 
        store.addCubic( pd[i].x() + dx3, pd[i].y() + md[i] * dx3,
            pd[i + 1].x() - dx3, pd[i + 1].y() - md[i + 1] * dx3,
            pd[i + 1].x(), pd[i + 1].y() );
    }
 
    return store;
}
 
template< class SplineStore >
static inline SplineStore qwtSplineC1PathParamY(
    const QwtSplineC1* spline, const QPolygonF& points )
{
    const int n = points.size();
 
    QPolygonF pointsFlipped( n );
    for ( int i = 0; i < n; i++ )
    {
        pointsFlipped[i].setX( points[i].y() );
        pointsFlipped[i].setY( points[i].x() );
    }
 
    const QVector< double > m = spline->slopes( pointsFlipped );
    if ( m.size() != n )
        return SplineStore();
 
    const QPointF* pd = pointsFlipped.constData();
    const double* md = m.constData();
 
    SplineStore store;
    store.init( m.size() - 1 );
    store.start( pd[0].y(), pd[0].x() );
 
    QVector< QLineF > lines( n );
    for ( int i = 0; i < n - 1; i++ )
    {
        const double dx3 = ( pd[i + 1].x() - pd[i].x() ) / 3.0;
 
        store.addCubic( pd[i].y() + md[i] * dx3, pd[i].x() + dx3,
            pd[i + 1].y() - md[i + 1] * dx3, pd[i + 1].x() - dx3,
            pd[i + 1].y(), pd[i + 1].x() );
    }
 
    return store;
}
 
template< class SplineStore, class Param >
static inline SplineStore qwtSplineC1PathParametric(
    const QwtSplineC1* spline, const QPolygonF& points, Param param )
{
    const bool isClosing = ( spline->boundaryType() == QwtSpline::ClosedPolygon );
    const int n = points.size();
 
    QPolygonF pointsX, pointsY;
    pointsX.resize( isClosing ? n + 1 : n );
    pointsY.resize( isClosing ? n + 1 : n );
 
    QPointF* px = pointsX.data();
    QPointF* py = pointsY.data();
    const QPointF* p = points.constData();
 
    double t = 0.0;
 
    px[0].rx() = py[0].rx() = t;
    px[0].ry() = p[0].x();
    py[0].ry() = p[0].y();
 
    int numParamPoints = 1;
    for ( int i = 1; i < n; i++ )
    {
        const double td = param( points[i - 1], points[i] );
        if ( td > 0.0 )
        {
            t += td;
 
            px[numParamPoints].rx() = py[numParamPoints].rx() = t;
 
            px[numParamPoints].ry() = p[i].x();
            py[numParamPoints].ry() = p[i].y();
 
            numParamPoints++;
        }
    }
 
    if ( isClosing )
    {
        const double td = param( points[n - 1], points[0] );
 
        if ( td > 0.0 )
        {
            t += td;
 
            px[numParamPoints].rx() = py[numParamPoints].rx() = t;
 
            px[numParamPoints].ry() = p[0].x();
            py[numParamPoints].ry() = p[0].y();
 
            numParamPoints++;
        }
    }
 
    if ( pointsX.size() != numParamPoints )
    {
        pointsX.resize( numParamPoints );
        pointsY.resize( numParamPoints );
    }
 
    const QVector< double > slopesX = spline->slopes( pointsX );
    const QVector< double > slopesY = spline->slopes( pointsY );
 
    const double* mx = slopesX.constData();
    const double* my = slopesY.constData();
 
    // we don't need it anymore
    pointsX.clear();
    pointsY.clear();
 
    SplineStore store;
    store.init( isClosing ? n : n - 1 );
    store.start( points[0].x(), points[0].y() );
 
    int j = 0;
 
    for ( int i = 0; i < n - 1; i++ )
    {
        const QPointF& p1 = p[i];
        const QPointF& p2 = p[i + 1];
 
        const double td = param( p1, p2 );
 
        if ( td != 0.0 )
        {
            const double t3 = td / 3.0;
 
            const double cx1 = p1.x() + mx[j] * t3;
            const double cy1 = p1.y() + my[j] * t3;
 
            const double cx2 = p2.x() - mx[j + 1] * t3;
            const double cy2 = p2.y() - my[j + 1] * t3;
 
            store.addCubic( cx1, cy1, cx2, cy2, p2.x(), p2.y() );
 
            j++;
        }
        else
        {
            // setting control points to the ends
            store.addCubic( p1.x(), p1.y(), p2.x(), p2.y(), p2.x(), p2.y() );
        }
    }
 
    if ( isClosing )
    {
        const QPointF& p1 = p[n - 1];
        const QPointF& p2 = p[0];
 
        const double td = param( p1, p2 );
 
        if ( td != 0.0 )
        {
            const double t3 = td / 3.0;
 
            const double cx1 = p1.x() + mx[j] * t3;
            const double cy1 = p1.y() + my[j] * t3;
 
            const double cx2 = p2.x() - mx[0] * t3;
            const double cy2 = p2.y() - my[0] * t3;
 
            store.addCubic( cx1, cy1, cx2, cy2, p2.x(), p2.y() );
        }
        else
        {
            store.addCubic( p1.x(), p1.y(), p2.x(), p2.y(), p2.x(), p2.y() );
        }
 
        store.end();
    }
 
    return store;
}
 
template< QwtSplinePolynomial toPolynomial( const QPointF&, double, const QPointF&, double ) >
static QPolygonF qwtPolygonParametric( double distance,
    const QPolygonF& points, const QVector< double >& values, bool withNodes )
{
    QPolygonF fittedPoints;
 
    const QPointF* p = points.constData();
    const double* v = values.constData();
 
    fittedPoints += p[0];
    double t = distance;
 
    const int n = points.size();
 
    for ( int i = 0; i < n - 1; i++ )
    {
        const QPointF& p1 = p[i];
        const QPointF& p2 = p[i + 1];
 
        const QwtSplinePolynomial polynomial = toPolynomial( p1, v[i], p2, v[i + 1] );
 
        const double l = p2.x() - p1.x();
 
        while ( t < l )
        {
            fittedPoints += QPointF( p1.x() + t, p1.y() + polynomial.valueAt( t ) );
            t += distance;
        }
 
        if ( withNodes )
        {
            if ( qFuzzyCompare( fittedPoints.last().x(), p2.x() ) )
                fittedPoints.last() = p2;
            else
                fittedPoints += p2;
        }
        else
        {
            t -= l;
        }
    }
 
    return fittedPoints;
}
 
class QwtSpline::PrivateData
{
  public:
    PrivateData()
        : boundaryType( QwtSpline::ConditionalBoundaries )
    {
        parametrization = new QwtSplineParametrization(
            QwtSplineParametrization::ParameterChordal );
 
        // parabolic runout at both ends
 
        boundaryConditions[0].type = QwtSpline::Clamped3;
        boundaryConditions[0].value = 0.0;
 
        boundaryConditions[1].type = QwtSpline::Clamped3;
        boundaryConditions[1].value = 0.0;
    }
 
    ~PrivateData()
    {
        delete parametrization;
    }
 
    QwtSplineParametrization* parametrization;
    QwtSpline::BoundaryType boundaryType;
 
    struct
    {
        int type;
        double value;
 
    } boundaryConditions[2];
};
 
QPolygonF QwtSpline::polygon( const QPolygonF& points, double tolerance ) const
{
    if ( tolerance <= 0.0 )
        return QPolygonF();
 
    const QPainterPath path = painterPath( points );
    const int n = path.elementCount();
    if ( n == 0 )
        return QPolygonF();
 
    const QPainterPath::Element el = path.elementAt( 0 );
    if ( el.type != QPainterPath::MoveToElement )
        return QPolygonF();
 
    QPointF p1( el.x, el.y );
 
    QPolygonF polygon;
    QwtBezier bezier( tolerance );
 
    for ( int i = 1; i < n; i += 3 )
    {
        const QPainterPath::Element el1 = path.elementAt( i );
        const QPainterPath::Element el2 = path.elementAt( i + 1 );
        const QPainterPath::Element el3 = path.elementAt( i + 2 );
 
        const QPointF cp1( el1.x, el1.y );
        const QPointF cp2( el2.x, el2.y );
        const QPointF p2( el3.x, el3.y );
 
        bezier.appendToPolygon( p1, cp1, cp2, p2, polygon );
 
        p1 = p2;
    }
 
    return polygon;
}
 
QwtSpline::QwtSpline()
{
    m_data = new PrivateData;
}
 
QwtSpline::~QwtSpline()
{
    delete m_data;
}
 
uint QwtSpline::locality() const
{
    return 0;
}
 
void QwtSpline::setParametrization( int type )
{
    if ( m_data->parametrization->type() != type )
    {
        delete m_data->parametrization;
        m_data->parametrization = new QwtSplineParametrization( type );
    }
}
 
void QwtSpline::setParametrization( QwtSplineParametrization* parametrization )
{
    if ( ( parametrization != NULL ) && ( m_data->parametrization != parametrization ) )
    {
        delete m_data->parametrization;
        m_data->parametrization = parametrization;
    }
}
 
const QwtSplineParametrization* QwtSpline::parametrization() const
{
    return m_data->parametrization;
}
 
void QwtSpline::setBoundaryType( BoundaryType boundaryType )
{
    m_data->boundaryType = boundaryType;
}
 
QwtSpline::BoundaryType QwtSpline::boundaryType() const
{
    return m_data->boundaryType;
}
 
void QwtSpline::setBoundaryCondition( BoundaryPosition position, int condition )
{
    if ( ( position == QwtSpline::AtBeginning ) || ( position == QwtSpline::AtEnd ) )
        m_data->boundaryConditions[position].type = condition;
}
 
int QwtSpline::boundaryCondition( BoundaryPosition position ) const
{
    if ( ( position == QwtSpline::AtBeginning ) || ( position == QwtSpline::AtEnd ) )
        return m_data->boundaryConditions[position].type;
 
    return m_data->boundaryConditions[0].type; // should never happen
}
 
void QwtSpline::setBoundaryValue( BoundaryPosition position, double value )
{
    if ( ( position == QwtSpline::AtBeginning ) || ( position == QwtSpline::AtEnd ) )
        m_data->boundaryConditions[position].value = value;
}
 
double QwtSpline::boundaryValue( BoundaryPosition position ) const
{
    if ( ( position == QwtSpline::AtBeginning ) || ( position == QwtSpline::AtEnd ) )
        return m_data->boundaryConditions[position].value;
 
    return m_data->boundaryConditions[0].value; // should never happen
}
 
void QwtSpline::setBoundaryConditions(
    int condition, double valueBegin, double valueEnd )
{
    setBoundaryCondition( QwtSpline::AtBeginning, condition );
    setBoundaryValue( QwtSpline::AtBeginning, valueBegin );
 
    setBoundaryCondition( QwtSpline::AtEnd, condition );
    setBoundaryValue( QwtSpline::AtEnd, valueEnd );
}
 
QwtSplineInterpolating::QwtSplineInterpolating()
{
}
 
QwtSplineInterpolating::~QwtSplineInterpolating()
{
}
 
QPainterPath QwtSplineInterpolating::painterPath( const QPolygonF& points ) const
{
    const int n = points.size();
 
    QPainterPath path;
    if ( n == 0 )
        return path;
 
    if ( n == 1 )
    {
        path.moveTo( points[0] );
        return path;
    }
 
    if ( n == 2 )
    {
        path.addPolygon( points );
        return path;
    }
 
    const QVector< QLineF > controlLines = bezierControlLines( points );
    if ( controlLines.size() < n - 1 )
        return path;
 
    const QPointF* p = points.constData();
    const QLineF* l = controlLines.constData();
 
    path.moveTo( p[0] );
    for ( int i = 0; i < n - 1; i++ )
        path.cubicTo( l[i].p1(), l[i].p2(), p[i + 1] );
 
    if ( ( boundaryType() == QwtSpline::ClosedPolygon )
        && ( controlLines.size() >= n ) )
    {
        path.cubicTo( l[n - 1].p1(), l[n - 1].p2(), p[0] );
        path.closeSubpath();
    }
 
    return path;
}
 
QPolygonF QwtSplineInterpolating::polygon(
    const QPolygonF& points, double tolerance ) const
{
    if ( tolerance <= 0.0 )
        return QPolygonF();
 
    const QVector< QLineF > controlLines = bezierControlLines( points );
    if ( controlLines.isEmpty() )
        return QPolygonF();
 
    const bool isClosed = boundaryType() == QwtSpline::ClosedPolygon;
 
    QwtBezier bezier( tolerance );
 
    const QPointF* p = points.constData();
    const QLineF* cl = controlLines.constData();
 
    const int n = controlLines.size();
 
    QPolygonF polygon;
 
    for ( int i = 0; i < n - 1; i++ )
    {
        const QLineF& l = cl[i];
        bezier.appendToPolygon( p[i], l.p1(), l.p2(), p[i + 1], polygon );
    }
 
    const QPointF& pn = isClosed ? p[0] : p[n];
    const QLineF& l = cl[n - 1];
 
    bezier.appendToPolygon( p[n - 1], l.p1(), l.p2(), pn, polygon );
 
    return polygon;
}
 
QPolygonF QwtSplineInterpolating::equidistantPolygon( const QPolygonF& points,
    double distance, bool withNodes ) const
{
    if ( distance <= 0.0 )
        return QPolygonF();
 
    const int n = points.size();
    if ( n <= 1 )
        return points;
 
    if ( n == 2 )
    {
        // TODO
        return points;
    }
 
    QPolygonF path;
 
    const QVector< QLineF > controlLines = bezierControlLines( points );
 
    if ( controlLines.size() < n - 1 )
        return path;
 
    path += points.first();
    double t = distance;
 
    const QPointF* p = points.constData();
    const QLineF* cl = controlLines.constData();
 
    const QwtSplineParametrization* param = parametrization();
 
    for ( int i = 0; i < n - 1; i++ )
    {
        const double l = param->valueIncrement( p[i], p[i + 1] );
 
        while ( t < l )
        {
            path += QwtBezier::pointAt( p[i], cl[i].p1(),
                cl[i].p2(), p[i + 1], t / l );
 
            t += distance;
        }
 
        if ( withNodes )
        {
            if ( qFuzzyCompare( path.last().x(), p[i + 1].x() ) )
                path.last() = p[i + 1];
            else
                path += p[i + 1];
 
            t = distance;
        }
        else
        {
            t -= l;
        }
    }
 
    if ( ( boundaryType() == QwtSpline::ClosedPolygon )
        && ( controlLines.size() >= n ) )
    {
        const double l = param->valueIncrement( p[n - 1], p[0] );
 
        while ( t < l )
        {
            path += QwtBezier::pointAt( p[n - 1], cl[n - 1].p1(),
                cl[n - 1].p2(), p[0], t / l );
 
            t += distance;
        }
 
        if ( qFuzzyCompare( path.last().x(), p[0].x() ) )
            path.last() = p[0];
        else
            path += p[0];
    }
 
    return path;
}
 
QwtSplineG1::QwtSplineG1()
{
}
 
QwtSplineG1::~QwtSplineG1()
{
}
 
QwtSplineC1::QwtSplineC1()
{
    setParametrization( QwtSplineParametrization::ParameterX );
}
 
QwtSplineC1::~QwtSplineC1()
{
}
 
double QwtSplineC1::slopeAtBeginning( const QPolygonF& points, double slopeNext ) const
{
    if ( points.size() < 2 )
        return 0.0;
 
    return QwtSplineC1P::slopeBoundary(
        boundaryCondition( QwtSpline::AtBeginning ),
        boundaryValue( QwtSpline::AtBeginning ),
        points[0], points[1], slopeNext );
}
 
double QwtSplineC1::slopeAtEnd( const QPolygonF& points, double slopeBefore ) const
{
    const int n = points.size();
 
    const QPointF p1( points[n - 1].x(), -points[n - 1].y() );
    const QPointF p2( points[n - 2].x(), -points[n - 2].y() );
 
    const int condition = boundaryCondition( QwtSpline::AtEnd );
 
    double value = boundaryValue( QwtSpline::AtEnd );
    if ( condition != QwtSpline::LinearRunout )
    {
        // beside LinearRunout the boundaryValue is a slope or curvature
        // and needs to be inverted too
        value = -value;
    }
 
    const double slope = QwtSplineC1P::slopeBoundary( condition, value, p1, p2, -slopeBefore );
    return -slope;
}
 
QPainterPath QwtSplineC1::painterPath( const QPolygonF& points ) const
{
    const int n = points.size();
    if ( n <= 2 )
        return QwtSplineInterpolating::painterPath( points );
 
    using namespace QwtSplineC1P;
 
    PathStore store;
    switch( parametrization()->type() )
    {
        case QwtSplineParametrization::ParameterX:
        {
            store = qwtSplineC1PathParamX< PathStore >( this, points );
            break;
        }
        case QwtSplineParametrization::ParameterY:
        {
            store = qwtSplineC1PathParamY< PathStore >( this, points );
            break;
        }
        case QwtSplineParametrization::ParameterUniform:
        {
            store = qwtSplineC1PathParametric< PathStore >(
                this, points, paramUniform() );
            break;
        }
        case QwtSplineParametrization::ParameterCentripetal:
        {
            store = qwtSplineC1PathParametric< PathStore >(
                this, points, paramCentripetal() );
            break;
        }
        case QwtSplineParametrization::ParameterChordal:
        {
            store = qwtSplineC1PathParametric< PathStore >(
                this, points, paramChordal() );
            break;
        }
        default:
        {
            store = qwtSplineC1PathParametric< PathStore >(
                this, points, param( parametrization() ) );
        }
    }
 
    return store.path;
}
 
QVector< QLineF > QwtSplineC1::bezierControlLines( const QPolygonF& points ) const
{
    using namespace QwtSplineC1P;
 
    const int n = points.size();
    if ( n <= 2 )
        return QVector< QLineF >();
 
    ControlPointsStore store;
    switch( parametrization()->type() )
    {
        case QwtSplineParametrization::ParameterX:
        {
            store = qwtSplineC1PathParamX< ControlPointsStore >( this, points );
            break;
        }
        case QwtSplineParametrization::ParameterY:
        {
            store = qwtSplineC1PathParamY< ControlPointsStore >( this, points );
            break;
        }
        case QwtSplineParametrization::ParameterUniform:
        {
            store = qwtSplineC1PathParametric< ControlPointsStore >(
                this, points, paramUniform() );
            break;
        }
        case QwtSplineParametrization::ParameterCentripetal:
        {
            store = qwtSplineC1PathParametric< ControlPointsStore >(
                this, points, paramCentripetal() );
            break;
        }
        case QwtSplineParametrization::ParameterChordal:
        {
            store = qwtSplineC1PathParametric< ControlPointsStore >(
                this, points, paramChordal() );
            break;
        }
        default:
        {
            store = qwtSplineC1PathParametric< ControlPointsStore >(
                this, points, param( parametrization() ) );
        }
    }
 
    return store.controlPoints;
}
 
QPolygonF QwtSplineC1::equidistantPolygon( const QPolygonF& points,
    double distance, bool withNodes ) const
{
    if ( parametrization()->type() == QwtSplineParametrization::ParameterX )
    {
        if ( points.size() > 2 )
        {
            const QVector< double > m = slopes( points );
            if ( m.size() != points.size() )
                return QPolygonF();
 
            return qwtPolygonParametric< QwtSplinePolynomial::fromSlopes >(
                distance, points, m, withNodes );
        }
    }
 
    return QwtSplineInterpolating::equidistantPolygon( points, distance, withNodes );
}
 
QVector< QwtSplinePolynomial > QwtSplineC1::polynomials(
    const QPolygonF& points ) const
{
    QVector< QwtSplinePolynomial > polynomials;
 
    const QVector< double > m = slopes( points );
    if ( m.size() < 2 )
        return polynomials;
 
    polynomials.reserve( m.size() - 1 );
    for ( int i = 1; i < m.size(); i++ )
    {
        polynomials += QwtSplinePolynomial::fromSlopes(
            points[i - 1], m[i - 1], points[i], m[i] );
    }
 
    return polynomials;
}
 
QwtSplineC2::QwtSplineC2()
{
}
 
QwtSplineC2::~QwtSplineC2()
{
}
 
QPainterPath QwtSplineC2::painterPath( const QPolygonF& points ) const
{
    // could be implemented from curvatures without the extra
    // loop for calculating the slopes vector. TODO ...
 
    return QwtSplineC1::painterPath( points );
}
 
QVector< QLineF > QwtSplineC2::bezierControlLines( const QPolygonF& points ) const
{
    // could be implemented from curvatures without the extra
    // loop for calculating the slopes vector. TODO ...
 
    return QwtSplineC1::bezierControlLines( points );
}
 
QPolygonF QwtSplineC2::equidistantPolygon( const QPolygonF& points,
    double distance, bool withNodes ) const
{
    if ( parametrization()->type() == QwtSplineParametrization::ParameterX )
    {
        if ( points.size() > 2 )
        {
            const QVector< double > cv = curvatures( points );
            if ( cv.size() != points.size() )
                return QPolygonF();
 
            return qwtPolygonParametric< QwtSplinePolynomial::fromCurvatures >(
                distance, points, cv, withNodes );
        }
    }
 
    return QwtSplineInterpolating::equidistantPolygon( points, distance, withNodes );
}
 
QVector< double > QwtSplineC2::slopes( const QPolygonF& points ) const
{
    const QVector< double > curvatures = this->curvatures( points );
    if ( curvatures.size() < 2 )
        return QVector< double >();
 
    QVector< double > slopes( curvatures.size() );
 
    const double* cv = curvatures.constData();
    double* m = slopes.data();
 
    const int n = points.size();
    const QPointF* p = points.constData();
 
    QwtSplinePolynomial polynomial;
 
    for ( int i = 0; i < n - 1; i++ )
    {
        polynomial = QwtSplinePolynomial::fromCurvatures( p[i], cv[i], p[i + 1], cv[i + 1] );
        m[i] = polynomial.c1;
    }
 
    m[n - 1] = polynomial.slopeAt( p[n - 1].x() - p[n - 2].x() );
 
    return slopes;
}
 
QVector< QwtSplinePolynomial > QwtSplineC2::polynomials( const QPolygonF& points ) const
{
    QVector< QwtSplinePolynomial > polynomials;
 
    const QVector< double > curvatures = this->curvatures( points );
    if ( curvatures.size() < 2 )
        return polynomials;
 
    const QPointF* p = points.constData();
    const double* cv = curvatures.constData();
    const int n = curvatures.size();
    polynomials.reserve( n - 1 );
 
    for ( int i = 1; i < n; i++ )
    {
        polynomials += QwtSplinePolynomial::fromCurvatures(
            p[i - 1], cv[i - 1], p[i], cv[i] );
    }
 
    return polynomials;
}