// Copyright (c) 2004-2008 Max-Planck-Institute Saarbruecken (Germany).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; either version 3 of the License,
// or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
// SPDX-License-Identifier: LGPL-3.0+
//
// Author(s)     : Pavel Emeliyanenko <asm@mpi-sb.mpg.de>
//
// ============================================================================

/*!\file CGAL/Curved_kernel_via_analysis_2/gfx/Curve_renderer_2.h
 * \brief definition of Curve_renderer_2<> 
 * rasterization of algebraic curves
 */

#ifndef CGAL_CKVA_CURVE_RENDERER_2_H
#define CGAL_CKVA_CURVE_RENDERER_2_H

#ifndef CGAL_AK_ENABLE_DEPRECATED_INTERFACE
#define CGAL_AK_ENABLE_DEPRECATED_INTERFACE 1
#endif

#include <vector>
#include <stack>
#include <cmath>
// #include <boost/multi_index_container.hpp>
// #include <boost/multi_index/member.hpp>
// #include <boost/multi_index/ordered_index.hpp>
// #include <boost/multi_index/sequenced_index.hpp>

#include <CGAL/Polynomial.h>
#include <CGAL/Interval_nt.h>

#include <CGAL/Arithmetic_kernel.h>
#include <CGAL/Algebraic_kernel_d/Bitstream_descartes.h>
#include <CGAL/Algebraic_kernel_d/Bitstream_descartes_rndl_tree_traits.h>
#include <CGAL/Algebraic_kernel_d/Bitstream_coefficient_kernel.h>

#include <CGAL/Curved_kernel_via_analysis_2/gfx/Curve_renderer_internals.h>
#include <CGAL/Curved_kernel_via_analysis_2/gfx/Curve_renderer_traits.h>

// using boost::multi_index::multi_index_container;
// using boost::multi_index::get;
// using boost::multi_index::project;

#ifdef CGAL_CKVA_CR_TIMING
extern CGAL::Timer refine_timer;
#endif

namespace CGAL {

#ifndef CGAL_CURVE_RENDERER_DEFS
#define CGAL_CURVE_RENDERER_DEFS

// subdivision level beyond which visibly coincide branches are not further
// discriminated
#define CGAL_COINCIDE_LEVEL 4 

// maximal recursion depth for derivative check
#define CGAL_DER_CHECK_DEPTH 5 

// # of pixels to enlarge the drawing window in y-direction: required to skip
// closely located clip-points
#define CGAL_WINDOW_ENLARGE  15 

// refine factor for intervals in x-direction (in pixel size)
#define CGAL_REFINE_X        100   // was 1000 previously 

// refine factor for intervals in y-direction (in pixel size) 
#define CGAL_REFINE_Y        100000  

// refine factor for clip-points 
#define CGAL_REFINE_CLIP_POINTS  1000 

// refine threshold for double point approximation
#define CGAL_REFINE_DOUBLE_APPROX 1e-10

#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
//!@note: points are drawn in any case even if they are outside the window
//#warning approximation hangs for CORE::BigFloat !!
     #define CGAL_CKVA_STORE_COORDS(container, pixel) \
        if(!std::isnan(pixel.xv)) \
           container.push_back(Coord_2(pixel.xv, pixel.yv)); 
#else
    #define CGAL_CKVA_STORE_COORDS(container, pixel) \
        container.push_back(Coord_2(pixel.x, pixel.y));
#endif

#endif // CGAL_CURVE_RENDERER_DEFS   

namespace {
// map from box sides to subpixel numbers (for h/v directions)
// these are old versions
static const int HV_SUBPIX_MAP[][4] = {
    {1,3,0,2},  // right(0)
    {3,2,1,0},  // top(1)
    {2,0,3,1},  // left(2)
    {0,1,2,3}   // bottom(3)
}; 

static const int D_SUBPIX_MAP[][4] = {
    {3,1,2,0},  // dir = 0
    {2,3,0,1},  // dir = 1
    {0,2,1,3},  // dir = 2
    {1,0,3,2}   // dir = 3
}; 
} // anonymous namespace                                  

/*!
 * \brief The class template \c Curve_renderer_2 and its associate functions.
 * 
 * The class implements rendering of distinct curve arcs and points as 
 * defined by \c CurvedKernelViaAnalysis_2::Arc_2 and 
 * \c CurvedKernelViaAnalysis_2::Point_2. \c Coeff_ template parameter
 * defines an underlying number type to be used in polynomial and range 
 * evaluations. Valid instantiations are \c double, multi-precision float or 
 * exact rational number type. The main float-point number type is defined by 
 * \c Curve_renderer_traits<Coeff_>::Float.
 */
//!@{
template <class CurvedKernelViaAnalysis_2, class Coeff_>
class Curve_renderer_2
{
public: 
    //! \name public typedefs 
    //!@{ 
    
    //! this instance's first template argument
    typedef CurvedKernelViaAnalysis_2 Curved_kernel_via_analysis_2;
    
    //! this instance's second template argument
    typedef Coeff_ Coeff;
    
    //! type of embedded curve kernel
    typedef typename Curved_kernel_via_analysis_2::Curve_kernel_2 
        Curve_kernel_2;
    
    //! type of x-monotone arc
    typedef typename Curved_kernel_via_analysis_2::X_monotone_curve_2 Arc_2;
    
    //! type of a point on curve
    typedef typename Curved_kernel_via_analysis_2::Point_2 Point_2;
    
    //! type of 1-curve analysis
    typedef typename Curve_kernel_2::Curve_analysis_2 Curve_analysis_2;
    
    //!@}
private:    
    //! \name private typedefs 
    //!@{

    //! type of X_coordinate
    typedef typename Curve_kernel_2::Coordinate_1 Coordinate_1;

    //! type of xy-coordinate ? ;)
    typedef typename Curve_kernel_2::Coordinate_2 Coordinate_2;

    //! rendering low-level routine
    typedef internal::Curve_renderer_internals<Curve_kernel_2, Coeff>
        Renderer_internals;

    //! exact rational number type
    typedef typename Renderer_internals::Rational Rational;

    /// polynomial traits should be used whenever possible
    typedef typename Renderer_internals::Polynomial_traits_2
            Polynomial_traits_2;

    //! curve renderer type conversion traits
    typedef typename Renderer_internals::Renderer_traits Renderer_traits;
    
    //! coercion between rational and polynomial coefficients number type
    typedef typename Renderer_internals::Rat_coercion_type Rat_coercion_type;

    //! approximate y coordinates
    typedef typename Curve_kernel_2::Approximate_absolute_y_2
        Approximate_absolute_y_2;
    //! bound number type for approximate
    typedef typename Approximate_absolute_y_2::result_type Bounds;

    //! specialized integer number type
    typedef typename Renderer_traits::Integer Integer;
    
    //! event line instance type
    typedef typename Curve_analysis_2::Status_line_1 Status_line_1;
    
    //! underlying bivariate polynomial type
    typedef typename Renderer_internals::Polynomial_2 Polynomial_2;
    
    //! underlying univariate polynomial type
    typedef typename Renderer_internals::Poly_dst_1 Poly_dst_1;
    
    //! basic number type used in all computations
    typedef typename Renderer_internals::NT NT;
    
    //! instance of a univariate polynomial
    typedef typename Renderer_internals::Poly_1 Poly_1;
    //! instance of a bivariate polynomial
    typedef typename Renderer_internals::Poly_2 Poly_2;
    
    //! conversion from the basic number type to doubles
    typename CGAL::Real_embeddable_traits<NT>::To_double to_double;
    
    //! conversion from the basic number type to integers
    typename Renderer_traits::Rat_to_integer rat2integer;
    //! conversion from \c Integer type to built-in integer
    typename Renderer_traits::Float_to_int float2int;
    
    //! conversion from \c Rational type to used number type
    typename Renderer_traits::Rat_to_float rat2float;
    //! makes the result exact after inexact operation (applicable only for
    //! exact number types
    typename Renderer_traits::Make_exact make_exact;
    
    //! returns \c true when the precision limit for a specified number type is
    //! reached
    typename Renderer_traits::Precision_limit limit;
    //! maximum level of subdivision dependending on speficied number type
    static const unsigned MAX_SUBDIVISION_LEVEL = 
            Renderer_traits::MAX_SUBDIVISION_LEVEL;
            
    //! pixel instance type
    typedef internal::Pixel_2_<Integer> Pixel_2;
    //! seed point instance type
    typedef internal::Seed_point_<Integer> Seed_point;
    //! support for multiple seed points
    typedef std::stack<Seed_point> Seed_stack;
    
    //! a range of x-coordinates to define bottom/top clip points
    typedef internal::Clip_point_<Rational, Coordinate_1>
            Clip_point_entry;
    //! a container of bottom/top clip points
    typedef std::vector<Clip_point_entry> Clip_points;
    //! an integer index container
    typedef std::vector<int> index_vector;
        
    //! map container element's type for maintaining a list of cache instances
    typedef std::pair<int, int> LRU_entry;
    //! LRU list used for cache switching
    typedef std::vector< LRU_entry > Cache_list;
    
    //! internal use only: a stripe defined by bounding polynomials
    //! and coordinates on sides, index 0 - lower boundary, 1 - upper boundary
    struct Stripe {
        Poly_1 poly[2];
        NT key[2];
        unsigned level[2];
    } ;
    
    //!@}
public: 
    //! \name public methods
    //!@{ 
    
    //! default constructor: a curve segment is undefined
    Curve_renderer_2() : cache_id(-1), IA_method(0), initialized(false),
       one(1) {
        arcno = -1;
        setup(Bbox_2(-1.0, -1.0, 1.0, 1.0), 640, 480);
    }
        
    //! sets up drawing window and pixel resolution
    void setup(const Bbox_2& bbox_, int res_w_, int res_h_) {
         
        initialized = engine.setup(bbox_, res_w_, res_h_);
        clip_pts_computed = false; // need to recompute clip points each time
                                   // the window dimensions are changed
    }
     
    //! \brief returns currently used supporting algebraic curve
    Curve_analysis_2 curve() const
    {
        return *support;
    }
    
    //! \brief returns the drawing window and resolution 
    void get_setup_parameters(CGAL::Bbox_2 *pbox, int& res_w_, 
                int& res_h_) const {
        if(pbox != NULL)
            *pbox = engine.window;
        res_w_ = engine.res_w; 
        res_h_ = engine.res_h;
    }

    //! \brief sets up preferred IA method: 0 - QF, 1 - MAA
    //! returns the old one
    int set_IA_method(int idx) {
        int tmp = IA_method;
        IA_method = idx;
        return tmp;
    }

    //! destructor
    ~Curve_renderer_2() {
        cache_list.clear();
    }
      
    //!@}    
private:
    //! \name Private methods
    //!@{ 
    
    //! \brief advances pixel's coordinates by given increments
    void advance_pixel(Pixel_2& pix, int new_dir)
    {
        int x_inc = internal::directions[new_dir].x,
            y_inc = internal::directions[new_dir].y;
        if(pix.level == 0) {
            pix.x += x_inc;
            pix.y += y_inc;
        } else {
            Integer x = pix.sub_x + x_inc,
                    y = pix.sub_y + y_inc, pow = (one << pix.level) - 1;
            (x < 0 ? pix.x-- : x > pow ? pix.x++ : x);
            (y < 0 ? pix.y-- : y > pow ? pix.y++ : y);
            pix.sub_x = x&pow;
            pix.sub_y = y&pow;
        }   
    }
    
    //! computes pixel coordinates from rational point
    void get_pixel_coords(const Rational& x, const Rational& y, 
            Pixel_2& pix, Rational *ppix_x=NULL, Rational *ppix_y=NULL)
    {
        Rational p_x = (x - engine.x_min_r) / engine.pixel_w_r,
                 p_y = (y - engine.y_min_r) / engine.pixel_h_r;

#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
        pix.xv = CGAL::to_double(x);
        pix.yv = CGAL::to_double(y);
#endif  
        pix.x = static_cast<int>(std::floor(CGAL::to_double(p_x)));
        pix.y = static_cast<int>(std::floor(CGAL::to_double(p_y)));   

        if(ppix_x != NULL && ppix_y != NULL) {
            *ppix_x = p_x;
            *ppix_y = p_y;
        }
    }
    
    //! refines y-coordinate of \c Coordinate_2 to a certain bound
    void refine_xy(const Coordinate_2& xy, const Rational& bound,
            Rational& low, Rational& up) {

        Approximate_absolute_y_2 ay =
	  xy.kernel()->approximate_absolute_y_2_object();

        Bounds bnd = ay(xy, -2); // do not approximate
        int prec = 1;
        while(bnd.second - bnd.first > bound) {
            bnd = ay(xy, prec);
            prec++;
        }
        low = bnd.first, up = bnd.second;
    }
               
    //!@}
private:
    //! \name Private properties
    //!@{ 
    
    unsigned max_level;            //! maximum subdivision level
    unsigned current_level;

    int arcno;  //! arcno ?
    
    bool clip_pts_computed;     //! indicates whether clip points are computed
    
    Pixel_2 isolated_l, isolated_h; //! isolating rectangles for lower  and
                                    //! upper end-points
                                    
    //! an instance of rendering engine
    internal::Curve_renderer_internals<Curve_kernel_2, Coeff> engine;
                                            
    Curve_analysis_2 *support; //! supporting 1-curve analysis
    Curve_analysis_2 support_[CGAL_N_CACHES];
                                    
    Clip_points btm_clip, top_clip;  //! a set of bottom and top clip points
    Poly_dst_1 btm_poly, top_poly;   //! bottom and top polynomials 
    
    int cache_id;        //! index of currently used cache instance
    Cache_list cache_list; //! list of indices of cache instances
    
    Seed_stack s_stack;      //! a stack of seed points
    Seed_point current_seed; //! current seed point
    
    int IA_method; //! which IA method to use (0 - QF, 1 - MAA)
    
    bool initialized;  //! indicates whether the renderer has been initialized
                       //! with correct parameters
    const Integer one; //! just "one"
    bool branches_coincide; //! indicates that there are several branches
                           //! passing through one neighbourhood pixel
    int direction_taken;  //! stores a direction taken from the seed point 
                          //! during tracking, if it's possible to determine
                          //! 0 - towards lower point, 1 - towards upper
//!@}
//!\name public methods
//!@{

public: 

/*!\brief 
 * main rendering procedure for curve arcs
 * returns a list of sequences of coordinates as objects of type \c Coord_2
 * which must be constructible from a pair of ints / doubles
 *
 * An exception \c Insufficient_rasterize_precision_exception is 
 * thrown whenever the precision of currently used NT is not enough
 * to correctly render a curve arc. The exception has to be caught
 * outside the renderer to switch to a higher-precision arithmetic
 *
 * \c end_pt1/2 computes end-points (optionally)
 */
template < class Coord_2, template < class, class > class Container,
        class Allocator >
void draw(const Arc_2& arc, 
          Container< std::vector< Coord_2 >, Allocator >& points,
          boost::optional< Coord_2 > *end_pt1 = NULL, 
          boost::optional< Coord_2 > *end_pt2 = NULL) {

#ifdef CGAL_CKVA_CR_TIMING
    refine_timer.start();
#endif

    if(!initialized)
        return;
    select_cache_entry(arc); // lookup for appropriate cache instance 

    Gfx_OUT("\n////////////////////\n resolution: " << engine.res_w << " x "
            << engine.res_h << std::endl);
    Gfx_OUT("box: [" << engine.x_min << "; " << engine.y_min << "]x[" <<
        engine.x_max << "; " << engine.y_max << "]" << std::endl);
     
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT
    Rational lower = engine.x_min_r, upper = engine.x_max_r, y_lower = 0,
             y_upper = 0;
    bool clip_src = true, clip_tgt = true, x_outside_window = false;

    CGAL::Arr_parameter_space loc_p1 = arc.location(CGAL::ARR_MIN_END),
        loc_p2 = arc.location(CGAL::ARR_MAX_END);

    Rational ref_bound = engine.pixel_w_r / CGAL_REFINE_X;
    if(loc_p1 != CGAL::ARR_LEFT_BOUNDARY) {

        Gfx_OUT("refining left point\n");
        const Coordinate_1& x0 = arc.curve_end_x(CGAL::ARR_MIN_END);
        typename Curve_kernel_2::Algebraic_kernel_d_1::Approximate_relative_1
	  approximate_x;

        Bounds bnd = approximate_x(x0, 0); 
        while(1) {
            if(bnd.first >= engine.x_max_r) {
                x_outside_window = true;
                lower = bnd.second;
                break;
            }
            if(bnd.second < engine.x_min_r)
                break;
            else if(bnd.second - bnd.first <= ref_bound) {
                lower = bnd.second, clip_src = false;
                break;
            }
            bnd = approximate_x(x0, 1);
        }
    }            

    // since end-points are sorted lexicographically, no need to check for
    // lower boundary        
    if(loc_p2 == CGAL::ARR_TOP_BOUNDARY && arc.is_vertical()) 
        y_upper = engine.y_max_r;

    if(loc_p2 != CGAL::ARR_RIGHT_BOUNDARY) {

        Gfx_OUT("refining right point\n");
        const Coordinate_1& x0 = arc.curve_end_x(CGAL::ARR_MAX_END);

        typename Curve_kernel_2::Algebraic_kernel_d_1::Approximate_relative_1
	  approximate_x;

        Bounds bnd = approximate_x(x0, 0); // do not approximate
        while(1) {
            if(bnd.second <= engine.x_min_r) {
                x_outside_window = true;
                upper = bnd.first;
                break;
            }
            if(bnd.first > engine.x_max_r)
                break;
            else if(bnd.second - bnd.first <= ref_bound) {
                upper = bnd.first, clip_tgt = false;
                break;
            }
            bnd = approximate_x(x0, 1);
        }
    }

    if(x_outside_window) {
        return;
    }
#else

    Rational lower, upper, y_lower = 0, y_upper = 0;
    bool clip_src = false, clip_tgt = false;

    CGAL::Arr_parameter_space loc_p1 = arc.location(CGAL::ARR_MIN_END),
        loc_p2 = arc.location(CGAL::ARR_MAX_END);
    
    Rational ref_bound = engine.pixel_w_r/CGAL_REFINE_X;
    if(loc_p1 != CGAL::ARR_LEFT_BOUNDARY) {
        const Coordinate_1& x0 = arc.curve_end_x(CGAL::ARR_MIN_END);
        typename Curve_kernel_2::Algebraic_kernel_d_1::Approximate_relative_1
	  approximate_x;

        Bounds bnd = approximate_x(x0, 0); // do not approximate

        while(bnd.second - bnd.first > ref_bound) {
            bnd = approximate_x(x0, 1);
        }
        lower = bnd.second;
    } else {
        lower = engine.x_min_r; 
        clip_src = true;
    }            

    // since end-points are sorted lexicographically, no need to check for
    // lower boundary        
    if(loc_p2 == CGAL::ARR_TOP_BOUNDARY && arc.is_vertical()) 
        y_upper = engine.y_max_r;

    if(loc_p2 != CGAL::ARR_RIGHT_BOUNDARY) {
        const Coordinate_1& x0 = arc.curve_end_x(CGAL::ARR_MAX_END);
        typename Curve_kernel_2::Algebraic_kernel_d_1::Approximate_relative_1
	  approximate_x;
        Bounds bnd = approximate_x(x0, 0); // do not approximate

        while(bnd.second - bnd.first > ref_bound) {
            bnd = approximate_x(x0, 1);
        }
        upper = bnd.first;
//         while(ubound_x(x0) - lbound_x(x0) > ref_bound)
//             refine_x(x0);
//         upper = lbound_x(x0);
    } else {
        upper = engine.x_max_r;
        clip_tgt = true;
    }

    bool x_outside_window = false;
    Rational lower0 = lower, upper0 = upper, y_lower0(-1), y_upper0(-1);

    if(upper <= engine.x_min_r||lower >= engine.x_max_r) {
        x_outside_window = true;
        clip_src = clip_tgt = true;
    }
    
    if(lower < engine.x_min_r) {
        lower = engine.x_min_r;
        clip_src = true;
    }
    if(upper > engine.x_max_r) {
        upper = engine.x_max_r;
        clip_tgt = true;
    }
#endif

    if(!x_outside_window) {
        Rational height_r = (engine.y_max_r-engine.y_min_r)*2;
        ref_bound = engine.pixel_h_r/CGAL_REFINE_X;

#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
        ref_bound = std::min(ref_bound, Rational(CGAL_REFINE_DOUBLE_APPROX));
#endif
        Gfx_OUT("computing y-coordinates\n");

        if(loc_p1 == CGAL::ARR_INTERIOR || clip_src) 
            y_lower = get_endpoint_y(arc, lower, CGAL::ARR_MIN_END, clip_src,
                ref_bound);
    
        else if(loc_p1 == CGAL::ARR_BOTTOM_BOUNDARY) 
            y_lower = (arc.is_vertical() ? engine.y_min_r :
                engine.y_min_r - height_r);
    
        else { // endpoints must be sorted lexicographical
            CGAL_precondition(!arc.is_vertical());
            y_lower = engine.y_max_r + height_r;
        }

        if(loc_p2 == CGAL::ARR_INTERIOR || clip_tgt) 
            y_upper = get_endpoint_y(arc, upper, CGAL::ARR_MAX_END, clip_tgt,
                ref_bound);
        
        else if(loc_p2 == CGAL::ARR_BOTTOM_BOUNDARY) {
            CGAL_precondition(!arc.is_vertical());
            y_upper = engine.y_min_r - height_r;
    
        } else
            y_upper = (arc.is_vertical() ? engine.y_max_r :
                    engine.y_max_r + height_r);
    }       

#ifdef CGAL_CKVA_CR_TIMING
    refine_timer.stop();
#endif

#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
    if(end_pt1 != NULL && loc_p1 == CGAL::ARR_INTERIOR && 
         (clip_src || y_lower < engine.y_min_r || y_lower > engine.y_max_r)) {
        y_lower0 = (clip_src ? 
            get_endpoint_y(arc, lower0, CGAL::ARR_MIN_END, false,
                CGAL_REFINE_DOUBLE_APPROX) : y_lower);
        Gfx_OUT("lower end-point: [" << CGAL::to_double(lower0) << "; " <<
                CGAL::to_double(y_lower0) << "]\n");
        *end_pt1 = Coord_2(CGAL::to_double(lower0), 
                CGAL::to_double(y_lower0));  
    }
    if(end_pt2 != NULL && loc_p2 == CGAL::ARR_INTERIOR && 
         (clip_tgt || y_upper < engine.y_min_r || y_upper > engine.y_max_r)) { 
        y_upper0 = (clip_tgt ? 
            get_endpoint_y(arc, upper0, CGAL::ARR_MAX_END, false,
                CGAL_REFINE_DOUBLE_APPROX) : y_upper);
        Gfx_OUT("upper end-point: [" << CGAL::to_double(upper0) << "; " <<
                CGAL::to_double(y_upper0) << "]\n");
        *end_pt2 = Coord_2(CGAL::to_double(upper0), 
                CGAL::to_double(y_upper0));
    }
    if(x_outside_window)
        return;
#endif // CGAL_CKVA_RENDER_WITH_REFINEMENT

    Pixel_2 pix_1, pix_2;
    
    Gfx_OUT("lower: " << CGAL::to_double(lower) << "; upper: " <<
         CGAL::to_double(upper) << "; y_lower: " << CGAL::to_double(y_lower) <<
         "; y_upper: " << CGAL::to_double(y_upper) << "\n");

    get_pixel_coords(lower, y_lower, pix_1);  
    get_pixel_coords(upper, y_upper, pix_2);  
    
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT
    if(end_pt1 != NULL)
        *end_pt1 = Coord_2(pix_1.x, pix_1.y);
    if(end_pt2 != NULL)
        *end_pt2 = Coord_2(pix_2.x, pix_2.y);
#endif
    
     Gfx_OUT("lower pix: (" << pix_1.x << "; " << pix_1.y <<
          ") upper pix: (" << pix_2.x << "; " << pix_2.y <<
          ") pixel_w: " << engine.pixel_w << " pixel_h: " << engine.pixel_h <<
             std::endl);
     
    std::vector< Coord_2 > rev_points;
    // reserve at least enough space for arc's x-length
    rev_points.reserve(CGAL_ABS(pix_2.x - pix_1.x));
    
    if(arc.is_vertical()) {
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT
        CGAL_CKVA_STORE_COORDS(rev_points, pix_1);
        CGAL_CKVA_STORE_COORDS(rev_points, pix_2);
#else
        int inc = (pix_2.y > pix_1.y ? 1 : -1);
        for(; pix_1.y != pix_2.y; pix_1.y += inc) {
            CGAL_CKVA_STORE_COORDS(rev_points, pix_1);
            pix_1.yv += CGAL::to_double(engine.pixel_h * NT(inc));
        }

#endif
        points.push_back(rev_points);
        return;
    }

    if(!clip_pts_computed) {
        Gfx_OUT("computing clip points\n");
        horizontal_clip();
        clip_pts_computed = true;
    }
    
    typedef std::pair<int, int> Int_pair;
    typedef std::vector<Int_pair> index_pair_vector;
    index_vector btm_idx, top_idx;
    index_pair_vector seg_pts;
    
    Gfx_OUT("checking bottom clip: y = " << engine.y_min << "\n");
    segment_clip_points(lower, upper, y_lower, y_upper, engine.y_min_r,
        btm_poly, arc.arcno(), btm_clip, btm_idx);
        
    Gfx_OUT("checking top clip: y = " << engine.y_max << "\n");
    segment_clip_points(lower, upper, y_lower, y_upper, engine.y_max_r,
        top_poly, arc.arcno(), top_clip, top_idx);
        
    index_vector::iterator it1 = btm_idx.begin(), it2 = top_idx.begin();
    while(1) { 
        bool push_1 = true;
        if(it1 != btm_idx.end()) {
            if(it2 != top_idx.end())
                push_1 = ((btm_clip)[*it1].left < (top_clip)[*it2].left);
        } else if(it2 != top_idx.end()) 
            push_1 = false;
        else 
            break;
        if(push_1) {  // 0 - bottom point, 1 - top point
            seg_pts.push_back(Int_pair(*it1,0));
            it1++; 
        } else {
            seg_pts.push_back(Int_pair(*it2,1));
            it2++; 
        }
    }
    
    max_level = 0;
    Pixel_2 start;
    int dir[2], b_taken[2], last_x = -engine.res_w;
    bool b_coincide;
    Rational pt, mid;
    
    branches_coincide = false;
    // make a small tolerance
    bool pt1_inside = (pix_1.y >= -1 && pix_1.y <= engine.res_h),
         pt2_inside = (pix_2.y >= -1 && pix_2.y <= engine.res_h);
         
//TODO: use Event1_info::multiplicity(i) - to gather information about roots ?
    if(seg_pts.size() == 1 && pt1_inside && pt2_inside)
        seg_pts.clear();
           
    arcno = arc.arcno();
    // easy case: no clip-points found    
    if(seg_pts.size() == 0) {
        if(!pt1_inside && !pt2_inside) {
            Gfx_OUT("segment is outside\n");
            return;
        }
    /// WARNING: if x-interval is small while y coordinates are far away from
    /// the window, we can get into the troubles..
        if(pix_2.x - pix_1.x <= 1) {// it goes away right here
            CGAL_CKVA_STORE_COORDS(rev_points, pix_1);  
            CGAL_CKVA_STORE_COORDS(rev_points, pix_2);
            points.push_back(rev_points);
            return;
        } 

        Gfx_OUT("NO clip points\n");
        mid = (lower + upper)/2;
        pt = engine.x_min_r + (pix_1.x*2+5)*engine.pixel_w_r/2;
        // segment is almost vertical - start from a middle point
        //if(pt > mid) 
            pt = mid; 
        if(!get_seed_point(pt, start, dir, b_taken, b_coincide)) {
            Gfx_OUT("generic error occurred..\n");   
            return;
        }
         Gfx_OUT("starting pixel found: " << start << " directions: " << dir[0]
             << " and " << dir[1] << std::endl);
        // only one point list
        draw_lump(rev_points, last_x, pix_1, pix_2);
        points.push_back(rev_points);
        
        Gfx_OUT("exit normal, max_level: " << max_level << std::endl);
        return;
    } 
    
    // clip-points presented
    Rational l, r, y_clip;
    Coordinate_1 alpha;
    Poly_dst_1 *ppoly;
    Clip_point_entry *pclip, *ptmp;
        
    Gfx_OUT("segment is not completely inside the window..\n");
    index_pair_vector::iterator it = seg_pts.begin(), 
            eend = seg_pts.end();
    
    Gfx_OUT("\n\nSEGMENT clip points: \n");
    Pixel_2 pix_beg, pix_end;
    bool straight_segment;
    
    while(it != eend) {
        if(it != seg_pts.begin() && it == eend-1 && !pt2_inside)
            break;
        straight_segment = false;
        
        int idx = it->first;
        if(it->second == 0) {// bottom points
            pclip = &btm_clip[idx];
            ppoly = &btm_poly;
            y_clip = engine.y_min_r;
            Gfx_OUT("bottom clip-point: [");
        } else { // top points
            pclip = &top_clip[idx];
            ppoly = &top_poly;
            y_clip = engine.y_max_r;
            Gfx_OUT("up clip-point: [");
        }
        l = pclip->left;
        r = pclip->right;
        Gfx_OUT(CGAL::to_double(l) << "; " << CGAL::to_double(r) << "\n");   

        if(last_x != -engine.res_w) {  // compute screen coordinates of a pixel
            Rational last_pt = engine.x_min_r + static_cast<Rational>(last_x)*
                engine.pixel_w_r; 
            if(r <= last_pt) { // skip a clip-point if already drawn
                Gfx_OUT("clip point skipped..\n");
                it++;
                continue;
            }
        }
        
        rev_points.clear();
        if(it == seg_pts.begin() && pt1_inside) { 

            Gfx_OUT("starting point is near the left end-point\n");
            //pt = engine.x_min_r + (pix_1.x*2+5)*engine.pixel_w_r/2;
            pt = (lower + l)/2;
            if(pt <= lower) {
                Gfx_OUT("refining the left end-point\n");

                refine_alg_point(l, r, *ppoly, lower, 1);
                pt = l; 
                if(l - lower <= engine.pixel_w_r*2) 
                    straight_segment = true;
            }
            pix_beg = pix_1;
            get_pixel_coords(l, y_clip, pix_end);  

        } else if(it == eend - 1 && pt2_inside) {

            Gfx_OUT("starting point is near the right end-point\n");
            //pt = engine.x_min_r + (pix_2.x*2-3)*engine.pixel_w_r/2;
            pt = (r + upper)/2;
            Gfx_OUT("pt is: " << CGAL::to_double(pt) << "; r is: " <<
                CGAL::to_double(r) << ": pixel_w_r: " <<
                    CGAL::to_double(engine.pixel_w_r) << "\n");
            
            if(pt >= upper) {
                // ensure that clip-point interval is sufficiently small
                refine_alg_point(l, r, *ppoly, upper, 1);
                pt = r; 
                if(upper - r <= engine.pixel_w_r*2) 
                    straight_segment = true;
            }
            pix_end = pix_2;
            get_pixel_coords(r, y_clip, pix_beg);  

        } else {
            Gfx_OUT("starting point is between clip-points\n");
            it++;
            
            if(it == seg_pts.end()) {
                Gfx_OUT("ERROR: clip point missed ?\n");
                break;
            }
            
            idx = it->first;
            ptmp = (it->second ? &top_clip[idx] : &btm_clip[idx]);
            pt = pclip->alpha.rational_between(ptmp->alpha);
            
            get_pixel_coords(l, y_clip, pix_beg);  
            get_pixel_coords(ptmp->left, it->second ? engine.y_max_r :
                engine.y_min_r, pix_end); 
            if(CGAL_ABS(ptmp->left - l) <= engine.pixel_w_r*2) {
                
                Coordinate_2 xy(Coordinate_1(pt), *support, arc.arcno());
                Rational _;
                refine_xy(xy, engine.pixel_h_r/CGAL_REFINE_Y, _, mid);
//                 mid = ubound_y(xy);

                CGAL_CKVA_STORE_COORDS(rev_points, pix_beg);
                get_pixel_coords(pt, mid, pix_beg); 
                straight_segment = true;
            } 
        }
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT            
        if(straight_segment) {
            Gfx_OUT("straight subsegment found\n");
            CGAL_CKVA_STORE_COORDS(rev_points, pix_beg);
            CGAL_CKVA_STORE_COORDS(rev_points, pix_end);

            points.push_back(rev_points);
            it++;
            continue;
        }
#endif // !CGAL_CKVA_RENDER_WITH_REFINEMENT
            
        if(!get_seed_point(pt, start, dir, b_taken, b_coincide)) {
            Gfx_OUT("get_seed_point: a problem occurred..\n");
            it++; 
            continue;
        }
            
        Gfx_OUT("starting pixel found: " << start << " directions: " <<
              dir[0] << " and " << dir[1] << std::endl);
        draw_lump(rev_points, last_x, pix_beg, pix_end);
        points.push_back(rev_points);
        if(it == eend)
            break;
        it++;
    }
}

/*!\brief
 * overloaded version to rasterize distinct points on curves
 *
 * \return \c false indicating that the point lies outside the drawing window
 * \c true in case the point coordinates were successfully computed
 */
template < class Coord_2 >
bool draw(const Point_2& pt, Coord_2& coord) {

    Gfx_OUT("rasterizing point: " << CGAL::to_double(pt.x()) << std::endl);

    const Coordinate_1& x0 = pt.x();
    Rational ref_bound = engine.pixel_w_r / 2;
#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
        ref_bound = std::min(ref_bound, Rational(CGAL_REFINE_DOUBLE_APPROX));
#endif

    typename Curve_kernel_2::Algebraic_kernel_d_1::Approximate_relative_1
      approximate_x;
    Bounds bnd = approximate_x(x0, 0); // do not approximate
    while(bnd.second - bnd.first > ref_bound) {
        bnd = approximate_x(x0, 1);
    }

//     while(ubound_x(x0) - lbound_x(x0) > ref_bound)
//         refine_x(x0);
    Rational x_s = bnd.first, y_s;
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT 
    if(x_s < engine.x_min_r || x_s > engine.x_max_r)
        return false;
#endif
    ref_bound = engine.pixel_h_r / CGAL_REFINE_X;

#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
    ref_bound = std::min(ref_bound, Rational(CGAL_REFINE_DOUBLE_APPROX));
#endif

    Coordinate_2 xy(x0, pt.curve(), pt.arcno());
    Rational _;
    refine_xy(xy, ref_bound, _, y_s);

    Pixel_2 pix;
    get_pixel_coords(x_s, y_s, pix);
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT 
    if(pix.x < 0 || pix.x >= engine.res_w || pix.y < 0 || 
            pix.y >= engine.res_h)
        return false;

    coord = Coord_2(pix.x, pix.y);
#else
    coord = Coord_2(pix.xv, pix.yv);
#endif
    return true;
}

//!@}
private:
//!\name private methods
//!@{

//! draws a segment's piece from pix_1 to pix_2, the starting pixel is taken
//! from the seed point stack
template < class Coord_2 >
void draw_lump(std::vector< Coord_2 >& rev_points, int& last_x,
    const Pixel_2& pix_1, const Pixel_2& pix_2) {

    Pixel_2 start, pix, witness, prev_pix, stored_pix, stored_prev;
    int back_dir, new_dir, stored_dir, dir[2], b_taken[2], ux, uy;
    bool b_coincide, failed;
    start = s_stack.top().start;
    
    bool pix_1_close = (CGAL_ABS(start.x - pix_1.x) <= 1&&
            CGAL_ABS(start.y - pix_1.y) <= 1),
         pix_2_close = (CGAL_ABS(start.x - pix_2.x) <= 1&&
            CGAL_ABS(start.y - pix_2.y) <= 1);
    
    // stores an x-coordinate of the last traced pixel in --> direction:
    // required to "jump" over clip-points
    last_x = -engine.res_w; 
    
    if(pix_1_close && pix_2_close) { // suppress drawing of trivial segments
        CGAL_CKVA_STORE_COORDS(rev_points, pix_1);
        CGAL_CKVA_STORE_COORDS(rev_points, pix_2);
        return;
    }

    Gfx_OUT("draw_lump: pix_1 = " << pix_1 << "; pix_2 = " << pix_2 << "\n");
    std::vector< Coord_2 > points;
    // reserve a list of point coordinates
    points.reserve(CGAL_ABS(pix_2.x - pix_1.x));

    direction_taken = -1;
    bool overstep = (pix_1_close||pix_2_close);
    bool ready = false;

    while(!s_stack.empty())
    {

    current_seed = s_stack.top();
    s_stack.pop();
    branches_coincide = current_seed.branches_coincide;
    if(direction_taken == -1) 
        direction_taken = current_seed.direction_taken;

    witness = current_seed.start;
    pix = witness;
    current_level = witness.level;
    pix.sub_x = 0;
    pix.sub_y = 0;
    pix.level = 0;
    back_dir = current_seed.back_dir;
    stored_dir = back_dir;
    stored_pix = pix;
    stored_prev = pix;

    // store result in a list or reversed list depending on the orientation
    std::vector< Coord_2 > & ppoints = (direction_taken == 0 ?
        rev_points : points);
    
    Gfx_OUT("continuing from a seed point: " << witness << "; back_dir: " <<
        back_dir << " direction_taken: " << direction_taken << std::endl);
        
    bool is_exit = false;
    while(1) {
        if(pix.x < 0 || pix.x > engine.res_w || pix.y < -CGAL_WINDOW_ENLARGE||
            pix.y > engine.res_h + CGAL_WINDOW_ENLARGE) {
            branches_coincide = false;
            break;
        } 

        CGAL_CKVA_STORE_COORDS(ppoints, pix);
#ifndef CGAL_CKVA_RENDER_WITH_REFINEMENT 
        bool bb1 = (direction_taken == 0 && pix.x <= pix_1.x),
             bb2 = (direction_taken == 1 && pix.x >= pix_2.x);
        if((bb1 || bb2)) {
            //Gfx_OUT("STOP: reached end-point x-coordinate\n");
            branches_coincide = false;
            is_exit = true;
            break;
        }
#endif

        bool set_ready = false;
        if(CGAL_ABS(pix.x - pix_1.x) <= 1 && CGAL_ABS(pix.y - pix_1.y) <= 1) {
            if(!overstep || ready) {
                pix = pix_1; 
                branches_coincide = false;
                break;
            }
            set_ready = true;
        } 

        if(CGAL_ABS(pix.x - pix_2.x) <= 1 && CGAL_ABS(pix.y - pix_2.y) <= 1) {
            if(!overstep || ready) {
                pix = pix_2; 
                branches_coincide = false;
                break;
            }
            ready = true;
        }
        if(set_ready)
            ready = true;

        if(!test_neighbourhood(pix, back_dir, new_dir)) {
            ux = pix.x;
            uy = pix.y;
            if(witness == pix) { // witness subpixel is a pixel itself
                if(!subdivide(pix,back_dir,new_dir)) {
                    is_exit = true;
                    break;
                }
                prev_pix = pix;
                advance_pixel(pix,new_dir); 
                back_dir = (new_dir+4)&7;
            } else {
                back_dir = stored_dir;
                pix = witness;
                prev_pix = witness;
            }

            stored_dir = -1;
            failed = false;
            while(ux == pix.x && uy == pix.y) {
                if(!failed && ((prev_pix.sub_x & 2) != (pix.sub_x & 2)||
                    (prev_pix.sub_y & 2) != (pix.sub_y & 2))) {
                    stored_pix = pix;
                    pix.sub_x >>= 1;
                    pix.sub_y >>= 1;
                    pix.level--;
                    stored_dir = back_dir;
                    stored_prev = prev_pix;
                }

                if(!test_neighbourhood(pix, back_dir, new_dir)) {
                    if(stored_dir != -1) {
                        pix = stored_pix;
                        prev_pix = stored_prev;
                        back_dir = stored_dir;
                        stored_dir = -1;
                        failed = true;
                        continue;
                    }
                    if(!subdivide(pix,back_dir,new_dir)) {
                        is_exit = true;
                        break;
                    }
                }
                //Gfx_OUT(pix << " dir = " << new_dir << std::endl);
                prev_pix = pix;
                advance_pixel(pix, new_dir);
                if(is_isolated_pixel(pix)) {
                    branches_coincide = false;
                    is_exit = true;
                    break;
                }
                back_dir = (new_dir+4)&7;
            }
            if(is_exit)
                break;

            stored_dir = back_dir;
            witness = pix; 
            pix.level = 0;
            pix.sub_x = 0;
            pix.sub_y = 0;
            //Gfx_OUT(witness << " " << prev_pix << std::endl);
        } else {
            ux = pix.x;
            uy = pix.y;
            advance_pixel(pix, new_dir);
            witness = pix;
        }
        back_dir = (new_dir+4)&7;
    }
    
    if(branches_coincide) { // oops, need another seed point
        if(direction_taken == -1) {
            Gfx_OUT("\n\nFATAL: unknown direction in coincide mode!\n\n");
            return;
        }
        
        (direction_taken == 0) ? pix.x -= 1 : pix.x += 1;
        if(pix.x >= pix_2.x-1||pix.x <= pix_1.x+1)
            goto Lexit;
            
        //std::cout << "New seed point required at: " << pix << std::endl;
        if(!get_seed_point(engine.x_min_r+pix.x*engine.pixel_w_r, start, dir,
                 b_taken, b_coincide)) { 
            std::cerr << " wrong seed point found " << std::endl;
            throw internal::Insufficient_rasterize_precision_exception();
        }
        
        Gfx_OUT("new seed point found: " << start << " directions: " << dir[0]
             << " and " << dir[1] << "; taken = " << direction_taken <<
                 std::endl);
                 
        b_taken[0] = internal::DIR_TAKEN_MAP[dir[0]];
        b_taken[1] = internal::DIR_TAKEN_MAP[dir[1]];

        if(b_taken[0] != -1) 
            new_dir = dir[b_taken[0] != direction_taken ? 0: 1];
        else if(b_taken[1] != -1) 
            new_dir = dir[b_taken[1] != direction_taken ? 1: 0];
        else {
            std::cerr << "ERROR: wrong backward dir after seed point: " << 
                dir[0] << " " << dir[1] << std::endl;
            throw internal::Insufficient_rasterize_precision_exception();
        }

        s_stack.push(Seed_point(start, new_dir, direction_taken, b_coincide));
        continue;

    } else if(is_exit) {
Lexit:  
        pix = (direction_taken == 0 ? pix_1 : pix_2);
    }

    //!@note: this can cause missing end-points !!
    //! if they proceeded by pt1/2_inside (-1 or res_h+1)
    //if(pix.y >= 0 && pix.y < engine.res_h)
        CGAL_CKVA_STORE_COORDS(ppoints, pix);
            
    // we were tracing in --> direction: store the pixel where we stopped
    if(direction_taken == 1)
        last_x = pix.x;

    if(direction_taken != -1)
        direction_taken = 1 - direction_taken;
    ready = false;
    
    }  // while(!s_stack.empty())

    std::reverse(rev_points.begin(), rev_points.end());
    // resize rev_points to accomodate the size of points vector
    unsigned rsize = rev_points.size();
    rev_points.resize(rsize + points.size());
    std::copy(points.begin(), points.end(), rev_points.begin() + rsize);
}


//! recursively subdivides pixel into 4 subpixels, returns a new tracking
//! direction
bool subdivide(Pixel_2& pix, int back_dir, int& new_dir) {   

    //Gfx_OUT("\n\nSubdivision of a pixel" << pix << " with back_dir =  " << 
        //back_dir << std::endl);
    NT inv = NT(1) / NT(one << pix.level);
    make_exact(inv);
    int idx, pref_dir = back_dir>>1;
    
    if(pix.level >= MAX_SUBDIVISION_LEVEL) {
        std::cerr << "reached maximum subdivision level: " << pix << 
            std::endl;
        throw internal::Insufficient_rasterize_precision_exception();
    }
    
    if(limit(engine.pixel_w * inv) || limit(engine.pixel_h * inv)) {
        std::cerr << "too small subpixel size: " << pix << std::endl;
        throw internal::Insufficient_rasterize_precision_exception();
    }
    
    // if several branches coincide withing this pixel we cannot perform
    // a subdivision
    if(branches_coincide) 
        return false;
        
    if(back_dir & 1) { // diagonal direction
        idx = get_subpixel_diag(pix,pref_dir);
        if(idx == -1) {
            std::cerr << "wrong diag subpixel: " << pix << " direction: " <<
                pref_dir << std::endl;
            throw internal::Insufficient_rasterize_precision_exception();
        }
        idx = D_SUBPIX_MAP[pref_dir][idx];
        
    } else {
        idx = get_subpixel_hv(pix,pref_dir);
        if(idx == -1) {
            std::cerr << "wrong h/v subpixel" << pix << " direction: " <<
                pref_dir << std::endl;
            throw internal::Insufficient_rasterize_precision_exception();
        }
        idx = HV_SUBPIX_MAP[pref_dir][idx];
    }
    
    pix.level++;
    if(max_level < pix.level)
        max_level = pix.level;
    if(current_level < pix.level)
        current_level = pix.level;
    
    pix.sub_x = (pix.sub_x<<1) + (idx&1);
    pix.sub_y = (pix.sub_y<<1) + (idx>>1);
    //Gfx_DETAILED_OUT("subpixel index: " << idx << " (" << pix.sub_x << "; "
        // << pix.sub_y << ")" << std::endl);
    if(!test_neighbourhood(pix, back_dir, new_dir))
        return subdivide(pix,back_dir,new_dir);
    //Gfx_DETAILED_OUT("new direction found: " << new_dir << " at a pixel:" <<
        //pix << std::endl);
    return true;
}

//! returns a "seed" point of a segment and two possible directions from it
bool get_seed_point(const Rational& seed, Pixel_2& start, int *dir, 
        int *b_taken, bool& b_coincide) {

    Rational x_s = seed, y_s;

    Gfx_OUT("get seed point: " << rat2float(seed) << "\n");
    //NOTE dirty HACK: we have no kernel instance ..
    Coordinate_2 xy(Coordinate_1(seed), *support, arcno);

    Rational _;    
    refine_xy(xy, engine.pixel_h_r/CGAL_REFINE_Y, _, y_s);
//     y_s = ubound_y(xy);

    Integer lvl = one;
    Rational x_seed, y_seed;
    get_pixel_coords(x_s, y_s, start, &x_seed, &y_seed);
    
    Gfx_OUT("y_seed = " << rat2float(y_seed) << std::endl);
    
    // we allow a small tolerance for seed point coordinates due to
    // round off errors
    if(start.y < -CGAL_WINDOW_ENLARGE || 
        start.y > engine.res_h + CGAL_WINDOW_ENLARGE) {
        Gfx_OUT("get_seed_point: starting pixel does not fit the window " 
            "boundaries " << std::endl);
        return false;
    }
    
    start.level = 0;
    start.sub_x = 0;
    start.sub_y = 0;
    current_level = 0;
    
    //Gfx_OUT("refining starting pixel: " << start << std::endl);
//     Gfx_OUT("x/y seed: (" << CGAL::to_double(x_seed) << "; " <<
//         CGAL::to_double(y_seed) << ")" << std::endl);
    b_taken[0] = b_taken[1] = -1;
    b_coincide = false;
    
    int coincide_level = 0;
    while(!test_pixel(start, (int *)dir, (int *)b_taken, b_coincide)) {
        
        //Gfx_OUT("refining starting pixel: " << start << std::endl);
        if(start.level >= MAX_SUBDIVISION_LEVEL) {
            std::cerr << "get_seed_point: reached maximum subdivision level "
                << start.level << std::endl;
            throw internal::Insufficient_rasterize_precision_exception();
        }
        //dump_neighbourhood(start);
        
        if(limit(engine.pixel_w/NT(lvl))||limit(engine.pixel_h/NT(lvl))) {
            std::cerr << "get_seed_point: too small subpixel size: " <<
                 start.level << std::endl;
            throw internal::Insufficient_rasterize_precision_exception();
        }
        
        start.level++;
        if(start.level > max_level)
            max_level = start.level;
        if(start.level > current_level)
            current_level = start.level;
        lvl <<= 1;
        
        if(lvl > CGAL_REFINE_Y) {
            Rational _;
            refine_xy(xy, engine.pixel_h_r/(lvl*2), _, y_s);
            y_seed = (y_s - engine.y_min_r)/engine.pixel_h_r;
        }
        
        start.sub_x = rat2integer((x_seed - start.x)*lvl);
        start.sub_y = rat2integer((y_seed - start.y)*lvl);
                
        if(start.sub_x < 0)
            start.sub_x = 0;
        start.sub_x &= (lvl-1);
        if(start.sub_y < 0)
            start.sub_y = 0;
        start.sub_y &= (lvl-1);
        
        if(current_level >= CGAL_COINCIDE_LEVEL) {
        
            Pixel_2 test = start;
            test.sub_x = start.sub_x >> (start.level - coincide_level);
            test.sub_y = start.sub_y >> (start.level - coincide_level);
            test.level = coincide_level;

            if(test_pixel(test, (int *)dir, (int *)b_taken, b_coincide)) {
                start = test;
                break;
            }
            coincide_level++;
        }
        //Gfx_DETAILED_OUT("\nTesting pixel: " << start << std::endl);
    }

    //Gfx_OUT("directions found: " << dir[0] << "; " << dir[1] << "\n");
    int t0 = internal::DIR_TAKEN_MAP[dir[0]], t1 = internal::DIR_TAKEN_MAP[dir[1]];
    if(t0 != -1&&t0 == t1) {
        Gfx_OUT("get_seed_point: one-side directions found: " <<
                 dir[0] << " and " << dir[1] << std::endl);
    }

    if(!branches_coincide) {
        if(t0 == -1) {
            if(t1 == -1) {
                // vx = -df/dy; vy = df/dx
                typename Renderer_internals::Coercion::Cast rccast;
                Rat_coercion_type xcs = rccast(x_s), ycs = rccast(y_s);

                Rat_coercion_type
                  vx = -engine.substitute_xy(*(engine.rational_fy), x_s, y_s),
                  vy = engine.substitute_xy(*(engine.rational_fx), x_s, y_s);
                
                NT vvx = rat2float(vx), vvy = rat2float(vy);
                if(vvy < 0) {
                    vvx = -vvx;
                }                

                // vx > 0: 2 - right(1), 6 - left(0)
                // vx < 0: 2 - left(0), 6 - right(1)
                bool flag = (vvx > 0);
                if(dir[0] == 2||dir[1] == 6) {
                // taken 2 = 0 if vx>0, 1 if vx<0
                // taken 6 = 1 if vx>0, 0 if vx<0
                    b_taken[0] = !flag; // !(vx > 0)
                    b_taken[1] = flag; //  vx > 0
                } else if(dir[0] == 6||dir[1] == 2) {
                    b_taken[0] = flag;  // vx > 0 
                    b_taken[1] = !flag;  // !(vx > 0)
                } 
            } else  // t1 != -1
               b_taken[0] = 1 - b_taken[1];
        } else if(t1 == -1)
            b_taken[1] = 1 - b_taken[0];
            
        s_stack.push(Seed_point(start, dir[0], b_taken[0], b_coincide));
        s_stack.push(Seed_point(start, dir[1], b_taken[1], b_coincide));
    }
    if(b_coincide)
        Gfx_OUT("seed point with coincide branches found" << std::endl);
    return true;
}

//! checks whether only one curve branch crosses the pixel, required to
//! compute a starting witness pixel
bool test_pixel(const Pixel_2& pix, int *dir, int *b_taken, bool& b_coincide)
{
    Stripe box[2]; // 0 - left-right stripe, 1 - bottom-top stripe
    NT lvl = NT(one << pix.level), inv = NT(1) / lvl;
    make_exact(inv);
    get_boundaries(CGAL_X_RANGE, pix, box[1]);
    get_boundaries(CGAL_Y_RANGE, pix, box[0]);
    NT bottom = box[1].key[0], //top = box[1].key[1], 
        left = box[0].key[0], lower;//, right = box[0].key[1], lower;
    // bottom(2), top(3), left(0), right(1)
    int n_sign = 0, i, j, n_dir = 0, shift, n_local, new_dir;
    get_polynomials(CGAL_Y_RANGE, box[0]);

/*
    Gfx_OUT("test pixel: " << pix << "--------------------------------\n");
    dump_neighbourhood(pix);
    Gfx_OUT("----------------------------------------------\n\n");*/
    
    b_coincide = false;
    int n_corner_dir = 0, corner_dir[] = {-1, -1};
    // process subsegments: left/right
    for(i = 0; i < 2; i++) { 
        for(j = 0, n_local = 0, lower = bottom; j < 3; j++, lower += inv)  

            if(get_range_1(CGAL_Y_RANGE, lower, lower+inv, box[0].key[i],
                box[0].poly[i], 3) || engine.zero_bounds) {
                
                Gfx_DETAILED_OUT("intersection detected at subsegment: " << j
                  << " side: " << i << "; left(0), right(1), bottom(2), top(3)"
                  << std::endl);
                
                if(engine.zero_bounds) {
                    
                    bool is_corner = false;
                    int diff = float2int((lower - pix.y)*lvl - pix.sub_y);
                    int low_sign = engine.evaluate_generic(CGAL_Y_RANGE, lower,
                        box[0].key[i], box[0].poly[i]);

                    Gfx_DETAILED_OUT("zero bounds detected, low_sign: "
                        << low_sign << "\n");
                        
                    if(diff != 0) {
                        if(diff == 1) {
                            is_corner = (engine.evaluate_generic(CGAL_Y_RANGE,
                               lower+inv, box[0].key[i], box[0].poly[i]) == 0);
                            // in case there is intersection at lower point =>
                            // it is counted as normal intersection
                            if(!is_corner && low_sign != 0)
                                continue;
                        } else {
                            is_corner = (low_sign == 0);
                            if(!is_corner)
                                continue;
                        }
                            
                        if(is_corner) {
                            // compute corner direction
                            diff = internal::DIR_MAP[i][diff+1];
                            if(n_corner_dir >= 2) {
                                Gfx_DETAILED_OUT("too many corners\n");
                                return false;
                            }
                           // corner dirs on vertical segs cannot be identified
                            corner_dir[n_corner_dir++] = diff;
                            Gfx_DETAILED_OUT("corner direction saved: "
                                << diff << "\n");
                        }
                    } else if(low_sign != 0)
                        continue;
                }
                  
                n_local++;
                if(n_local > 1) {
             Gfx_DETAILED_OUT("more than 1 intersection along vertical side "
                     << i << std::endl);
                    return false;
                }
                // no need to check the derivative if already coincide mode
                if(!b_coincide)
                    if(engine.first_der && !recursive_check(CGAL_Y_RANGE,
                        lower, lower+inv, box[0].key[i],box[0].poly[i])) {
                Gfx_DETAILED_OUT("\nrecursive_check failed" << std::endl);
                        if(current_level < CGAL_COINCIDE_LEVEL)
                            return false;
                        b_coincide = true;
                    }
                shift = float2int((lower - pix.y)*lvl - pix.sub_y);
                new_dir = internal::DIR_MAP[i][shift+1];
            // left side (i = 0) - direction taken = 1
            // right side (i = 1) - direction taken = 0
                b_taken[n_dir] = 1 - i;
                Gfx_DETAILED_OUT("new dir found: " << new_dir << "\n");
                dir[n_dir++] = new_dir;
            }
        n_sign += n_local;
    }
    
    get_polynomials(CGAL_X_RANGE,box[1]);
    Gfx_DETAILED_OUT("computing bottom/top sides" << std::endl);
    for(i = 0; i < 2; i++) {
        for(j = 0, n_local = 0, lower = left; j < 3; j++, lower += inv)  {
        
            if(get_range_1(CGAL_X_RANGE,lower,lower+inv,box[1].key[i],
                box[1].poly[i], 3) || engine.zero_bounds) {
                
           Gfx_DETAILED_OUT("intersection detected at subsegment: " << j <<
            " side: " << i + 2 << "; left(0), right(1), bottom(2), top(3) "
                  << std::endl);

                if(engine.zero_bounds) {

                    bool is_corner = false;
                    int diff = float2int((lower - pix.x)*lvl - pix.sub_x);
                    int low_sign = engine.evaluate_generic(CGAL_X_RANGE, lower,
                        box[1].key[i], box[1].poly[i]);

                    Gfx_DETAILED_OUT("zero bounds detected, low_sign: "
                        << low_sign << "\n");
                    
                    if(diff != 0) {
                        if(diff == 1) {
                            is_corner = (engine.evaluate_generic(CGAL_X_RANGE,
                               lower+inv, box[1].key[i], box[1].poly[i]) == 0);
                            // in case there is intersection at lower point =>
                            // it is counted as normal intersection
                            if(!is_corner && low_sign != 0)
                                continue;
                        } else {
                            is_corner = (low_sign == 0);
                            if(!is_corner)
                                continue;
                        }
                            
                        if(is_corner) {
                            Gfx_DETAILED_OUT("corner direction detected\n");
                            // compute corner direction
                            diff = internal::DIR_MAP[i+2][diff+1];
                            if(n_corner_dir > 0 && (corner_dir[0] == diff ||
                                    corner_dir[1] == diff)) {
                           Gfx_DETAILED_OUT("corner_dir identified: "
                            << diff << "\n");
                                continue;
                            }
                        }
                    } else if(low_sign != 0)
                        continue;
                }  
    
                n_local++;
                n_sign++;
                if(n_local > 1||n_sign > 2) {
                Gfx_DETAILED_OUT("more than 1 intersection along horizontal "
                         "side " << i+2 << std::endl);
                    return false;
                }
                
                if(!b_coincide)
                    if(engine.first_der && !recursive_check(CGAL_X_RANGE,
                    lower, lower+inv, box[1].key[i],box[1].poly[i])) {
                Gfx_DETAILED_OUT("\nrecursive_check failed" << std::endl);
                        if(current_level < CGAL_COINCIDE_LEVEL)
                            return false;
                        b_coincide = true;
                    }
                shift = float2int((lower - pix.x)*lvl - pix.sub_x);
                new_dir = internal::DIR_MAP[i+2][shift+1];
                b_taken[n_dir] = 1 - internal::DIR_TAKEN_MAP[new_dir];

                Gfx_DETAILED_OUT(" new dir found: " << new_dir << "\n");
                dir[n_dir++] = new_dir;
            }
        }
    }
    if(n_sign < 2) {
        Gfx_DETAILED_OUT("ERROR: not enough intersections found" <<
             std::endl);
        return false;
    }
    return true;

    /*Stripe box[2]; // 0 - left-right stripe, 1 - bottom-top stripe
    NT lvl = NT(one << pix.level), inv = NT(1) / lvl, llow[2];
    make_exact(inv);
    get_boundaries(CGAL_X_RANGE,pix,box[1]);
    get_boundaries(CGAL_Y_RANGE,pix,box[0]);
    //NT bottom = box[1].key[0], //top = box[1].key[1], 
        //left = box[0].key[0], right = box[0].key[1];
    NT lower;
    // bottom(2), top(3), left(0), right(1)
    int f_der[2];
    int idx[2], ibox, ikey, var, n_sign = 0, i, j, n_dir = 0, shift, 
        n_local, new_dir;
    get_polynomials(CGAL_Y_RANGE,box[0]);
    b_coincide = false;
    for(i = 0; i < 4; i++) { 
        if(i == 2)
            get_polynomials(CGAL_X_RANGE,box[1]);
        var = (i < 2 ? CGAL_Y_RANGE : CGAL_X_RANGE);
        ibox = i >> 1;
        ikey = i & 1;
        lower = box[1-ibox].key[0]; 
        for(j = 0, n_local = 0; j < 3; j++, lower += inv)  
            if(get_range_1(var,lower,lower+inv,box[ibox].key[ikey],
                box[ibox].poly[ikey])||zero_bounds) {
                Gfx_DETAILED_OUT("intersection detected at subsegment: " << j
                  << " side: " << i << "; left(0), right(1), bottom(2), top(3)"
                    << std::endl);
                if(zero_bounds&&evaluate_modular(var, lower, 
                    box[ibox].key[ikey], true) != 0)
                    continue;
                n_local++;
                n_sign++;
                if(n_local > 1||n_sign > 2) {
                  Gfx_DETAILED_OUT("more than 1 intersection along side " << i 
                        << std::endl);
                    return false;
                }
                f_der[n_dir] = engine.first_der;
                llow[n_dir] = lower;
                idx[n_dir++] = i;
            }
    }
    if(n_sign < 2) {
        Gfx_DETAILED_OUT("not enough intersections found" << std::endl);
        return false;
    }
    for(i = 0; i < 2; i++) {
        var = (idx[i] < 2 ? CGAL_Y_RANGE : CGAL_X_RANGE);
        ibox = idx[i] >> 1;
        ikey = idx[i] & 1;
        lower = llow[i];
        if(f_der[i] == -1) { // the 1st derivative needs to be computed
            get_range_1(var,lower,lower+inv,box[ibox].key[ikey],
                box[ibox].poly[ikey],2);
            f_der[i] = engine.first_der;
        }
        // no need to check the derivative if already coincide mode
        if(!b_coincide)
        if(f_der[i]&&!recursive_check(var,lower,lower+inv,box[ibox].key[ikey],
            box[ibox].poly[ikey])) {
        Gfx_DETAILED_OUT("\nrecursive_check failed" << std::endl);
            if(current_level < CGAL_COINCIDE_LEVEL)
                return false;
            b_coincide = true;
        }
        if(var == CGAL_Y_RANGE)
            shift = float2int((lower - pix.y)*lvl - pix.sub_y);
        else
            shift = float2int((lower - pix.x)*lvl - pix.sub_x);
        new_dir = internal::DIR_MAP[idx[i]][shift+1];
        b_taken[i] = 1 - internal::DIR_TAKEN_MAP[new_dir];
        dir[i] = new_dir;
        Gfx_DETAILED_OUT(" direction " << i << " found: " << new_dir << 
            " taken: " << b_taken[i] << std::endl);
    }
    return true;*/
}

void horizontal_clip()
{
    typedef typename CGAL::Fraction_traits<Rational> F_traits;
    typename F_traits::Numerator_type num;
    typename F_traits::Denominator_type denom;
    typename F_traits::Decompose decompose;

    top_clip.clear();
    btm_clip.clear();
    Rational l, r;
    
    typename CGAL::Polynomial_traits_d< Poly_dst_1 >::Make_square_free msf;

    decompose(engine.y_min_r, num, denom);
    btm_poly = msf(support->polynomial_2().evaluate_homogeneous(num, denom));
    decompose(engine.y_max_r, num, denom);
    top_poly = msf(support->polynomial_2().evaluate_homogeneous(num, denom));

    CGAL::internal::Bitstream_descartes<
        CGAL::internal::Bitstream_descartes_rndl_tree_traits
        < CGAL::internal::Bitstream_coefficient_kernel
            < typename Poly_dst_1::NT >
        > > isolator_btm(btm_poly), isolator_top(top_poly);

    int n_roots = isolator_btm.number_of_real_roots(), i;
    Rational criteria = engine.pixel_w_r/CGAL_REFINE_CLIP_POINTS;
    
    for(i = 0; i < n_roots; i++) {
        l = isolator_btm.left_bound(i);
        r = isolator_btm.right_bound(i);
        refine_alg_point(l, r, btm_poly, criteria);
        if(l > engine.x_max_r || r < engine.x_min_r)
            continue;
        btm_clip.push_back(Clip_point_entry(l,r));
    }

    n_roots = isolator_top.number_of_real_roots();
    for(i = 0; i < n_roots; i++) {
        l = isolator_top.left_bound(i),
        r = isolator_top.right_bound(i);
        refine_alg_point(l, r, top_poly, criteria);
        if(l > engine.x_max_r || r < engine.x_min_r)
            continue;
        top_clip.push_back(Clip_point_entry(l,r));
    }
}

// low_pix, up_pix and y_clip define segment end-points in pixel space
void segment_clip_points(const Rational& x_lower, const Rational& x_upper,
        const Rational& y_lower, const Rational& y_upper,
            const Rational& y_clip, const Poly_dst_1& poly, int arcno,
             Clip_points& clip_points, index_vector& clip_indices) {

    int n_arcs, i, j = 0;
    typename Clip_points::iterator it = clip_points.begin();
    Rational low, high, d, l, r;
    Gfx_DETAILED_OUT("segment: (" << rat2float(x_lower) << "; " << 
        rat2float(x_upper) << "); arcno: " << arcno << "\n");
    
    while(it != clip_points.end()) {
        l = it->left;
        r = it->right;
        if(r > x_lower && l < x_upper) {
            Coordinate_1 alpha = (l == r ? Coordinate_1(r) :
                 Coordinate_1(poly, l, r));
               
            // need precise comparisons in case of tight boundaries
            if(l < x_lower && alpha.compare(x_lower) != ::CGAL::LARGER) {
                it++; j++;
                continue;
            }
            if(r > x_upper && alpha.compare(x_upper) != ::CGAL::SMALLER) {
                it++; j++;
                continue;
            }
            if(it->arcno == -1) {
                Status_line_1 sline = support->status_line_for_x(alpha);
                n_arcs = sline.number_of_events();
                for(i = 0; i < n_arcs; i++) {
        
                    Coordinate_2 xy(alpha, *support, i);
                    Bounds _ = xy.kernel()->approximate_absolute_y_2_object()(xy, -2);
//                     low = lbound_y(xy);
//                     high = ubound_y(xy);
    //TODO: no need to refine y-intervals: you need only to check whether
    // polynomial vanishes at y_clip, since algebraic real you specify is exact
                    if((_.first < y_clip && _.second > y_clip)||
                            _.first == y_clip || _.second == y_clip) {
                        //event.refine_to(i, engine.pixel_h_r/CGAL_REFINE_Y);
                        Rational _1, _2;
                        refine_xy(xy, engine.pixel_h_r/CGAL_REFINE_Y, _1, _2);
                        if(_1 <= y_clip && _2 >= y_clip) {
                            it->arcno = i;
                            it->alpha = alpha;
                            Gfx_DETAILED_OUT("clip point assigned to arcno: "
                                << i << "\n");
                            break;
                        } 
                    }
                }
            }
            if(it->arcno == arcno) {
                bool set = true;
                Coordinate_2 xy(it->alpha, *support, arcno);
                Bounds _ = xy.kernel()->approximate_absolute_y_2_object()(xy, -2);

                if(r - x_lower <= engine.pixel_w_r) {
//                     d = ubound_y(xy) - y_lower;
                    d = _.second - y_lower;
                   set = (CGAL_ABS(d) > engine.pixel_h_r);
                } else if(x_upper - r <= engine.pixel_w_r) {
//                     d = ubound_y(xy) - y_upper;
                    d = _.second - y_upper;
                    set = (CGAL_ABS(d) > engine.pixel_h_r);
                }   
                if(set) {
                    Gfx_DETAILED_OUT("clip-point found\n");
                    clip_indices.push_back(j); // store point index
                } 
            }
        }
        it++; j++;
    }
}

//! mode = 0: criteria specifies the length of the refined interval
//! mode = 1: criteria specifies a point that must be lie outside the interval
void refine_alg_point(Rational& l, Rational& r, const Poly_dst_1& poly, 
            const Rational& criteria, int mode = 0) {

    CGAL::Sign eval_l, eval_m;
    Rational mid;
    eval_l = poly.sign_at(l);
    while(1) {
        if(mode == 0) {
            if(r - l < criteria)
                break;
        } else if(l > criteria || r < criteria)
            break;
        mid = (l+r)/2;
        eval_m = poly.sign_at(mid);
        if(eval_m == CGAL::EQUAL)
            l = r = mid;
        else if(eval_m == eval_l) 
            l = mid;
        else
            r = mid;
    } 
}

//! recursively checks whether only one curve branch crosses a line segment
bool recursive_check(int var, const NT& beg_, const NT& end_, 
    const NT& key, const Poly_1& poly, int depth = 0)
{
    // if polynomial is linear/quadratic and there is sign-change over interval
    // => only one curve branch is guaranteed
    //if(poly.degree() < 3) 
      //  return true;

    int val_1, val_2, val_3;
    NT mid = (beg_+end_)/2, key_1, key_2, beg = beg_, end = end_;
    make_exact(mid);
    
    Gfx_DETAILED_OUT("executing recursive check; poly = " << poly << std::endl);
    val_1 = engine.evaluate_generic(var, beg, key, poly);
    val_3 = engine.evaluate_generic(var, end, key, poly);
    
    Gfx_DETAILED_OUT("beg: " << val_1 << " end: " << val_3 << std::endl);
    // no sing change: at least two curve branches inside
    if((val_1^val_3)==0)
        return false;
    
    val_2 = engine.evaluate_generic(var, mid, key, poly);
    // select the half with even number of intersections
    if((val_1^val_2)==0)
        key_1 = beg;    
    else
        key_1 = end;        
    if(get_range_1(var, key_1, mid, key, poly)) 
        return false;   // at least three intersections over the interval: done
     key_2 = beg + end - key_1;
    
    get_range_1(var, key_2, mid, key, poly, 3);
 
    // the first derivative does not straddle zero then only one intersection
    if(!engine.first_der)     
        return true;  
    // if second derivative does not stranddle zero - only 1 first derivative
    //if(engine.second_der == 0) 
        //return true;
    if(depth > CGAL_DER_CHECK_DEPTH) 
        return true;
    return recursive_check(var, key_2, mid, key, poly, depth+1);
}

//! computes lower/upper boundaries for pixel's neighbourhood
void get_boundaries(int var, const Pixel_2& pix, Stripe& stripe) {
   
    int level = pix.level, val = pix.y;
    Integer sub = pix.sub_y, cur_sub;
    if(var == CGAL_Y_RANGE) {
        sub = pix.sub_x;
        val = pix.x;
    }
    cur_sub = sub;
    if(level > 0) {
        cur_sub += 2; // obtain local coordinates for upper boundary
        // for even boundaries raise the level up
        while(level > 0 && ((cur_sub & 1) == 0)) {
            level--; 
            cur_sub >>= 1;
        }
    }
    stripe.level[1] = level;
    if(level == 0) 
        cur_sub = 2 - cur_sub;
    stripe.key[1] = val + NT(cur_sub) / NT(one << level);
    make_exact(stripe.key[1]);
    level = pix.level; 
    cur_sub = sub - 1; 
    while(level > 0 && ((cur_sub & 1) == 0)) {
        level--; 
        cur_sub >>= 1;
    }
    stripe.level[0] = level;
    stripe.key[0] = val + NT(cur_sub) / NT(one << level);
    make_exact(stripe.key[0]);
}

//! shortcut to get precached polynomials
inline void get_polynomials(int var, Stripe& stripe) {
    engine.get_precached_poly(var, stripe.key[1], stripe.level[1],
        stripe.poly[1]);
    engine.get_precached_poly(var, stripe.key[0], stripe.level[0],
         stripe.poly[0]);
}

/*! 
 * checks 8-pixel neighbourhood of a pixel, returns \c true if 
 * only one curve branch intersects pixel's neighbourhood, \c dir
 * defines backward direction, \c new_dir is a new tracking direction
 *
 * if \c CGAL_CKVA_RENDER_WITH_REFINEMENT is set, in case of success \c pix
 * receives double approximations of intersection point 
 */
bool test_neighbourhood(Pixel_2& pix, int dir, int& new_dir)
{
    NT lvl = NT(one << pix.level);
    NT inv = NT(1.0) / lvl;
    make_exact(inv);
    int back_dir = dir, idx[5], n_pts = 5, n_sign;
    int shift, i, j, res, var, s_shift = 0, tmp;
    // box[0].key[0] - x-coordinate of left pixel boundary
    // box[0].key[1] - x-coordinate of right pixel boundary
    // box[1].key[0] - y-coordinate of bottom pixel boundary
    // box[1].key[1] - y-coordinate of top pixel boundary
    Stripe box[2]; // 0 - left-right stripe, 1 - bottom-top stripe
    NT p1, p2, lower; 
    get_boundaries(CGAL_X_RANGE,pix,box[1]);
    get_boundaries(CGAL_Y_RANGE,pix,box[0]);
    
    struct {         // local point descriptor 
        NT key;      // respective "key"
        int flag;    // 0th bit: 0: lower polynomial(left/bottom); 
                     //          1: upper polynomial(right/top)
                     // 1st bit: 0: x-range poly; 1: y-range poly
                     // 2nd bit: 0: c is in lower poly; 1: c is in upper poly 
                     // (for regular points only)
    } pts[6] = {{box[0].key[0], 0}, {box[0].key[0], 1}, {box[0].key[1], 1|4}, 
            {box[0].key[1], 0|4}};
            
    int ix = internal::directions[back_dir].x;
    int e_dir = back_dir, _3and5 = 0, _3and7 = 0;
    if(back_dir&1) {// 5 and 3: 4; 7 and 1: 0
        _3and5 = (e_dir == 5||e_dir == 3);
        _3and7 = e_dir&2;
        e_dir = _3and5 ? 4: 0;
    }
    int _4and6 = e_dir>>2, _2and6 = (e_dir&2)>>1;
    int _2and4 = (e_dir==2)||(e_dir==4);
    
    pts[4].key = box[_2and6].key[_4and6] + (1-(_4and6<<1))*inv;
    pts[4].flag = (_2and6<<1) + _2and4;
    pts[5].key = pts[4].key;
    pts[5].flag = pts[4].flag^1; //(_2and6<<1) + (!_2and4); // pts[4].flag^1 ?
    idx[0] = 4;
    idx[3] = 5;
    if(back_dir&1) 
        (_3and7) ? pts[5].key += ix*inv : pts[4].key += ix*inv;
    shift = (6 - e_dir) >> 1; 
    n_pts = 4;
    idx[1] = (1 + shift) & 3;
    idx[2] = (2 + shift) & 3;
/*  NT xx, yy;
        if(var == CGAL_X_RANGE) {
            xx = pts[idx[i]].key;
            yy = box[shift].key[fl&1];
        } else {
            yy = pts[idx[i]].key;
            xx = box[shift].key[fl&1];
        }
        xx = xx*20+ofsxx;
        yy = yy*20+100;
        if(i == 0)
            ppnt->moveTo(int(to_double(xx)),700-int(to_double(yy)));
        else
            ppnt->lineTo(int(to_double(xx)),700-int(to_double(yy)));
        prev = curr;
    }*/

    Gfx_DETAILED_OUT("test_neigh for " << pix << "; dir = " << dir << "\n");

    n_sign = 0;
    get_polynomials(CGAL_Y_RANGE,box[0]); 
    get_polynomials(CGAL_X_RANGE,box[1]); 
    int f_der = -1, corner_dir = -1;
    bool set_coincide = false;
    // bool s_change = false;
    new_dir = -1;
    for(i = 0; i < n_pts - 1; i++) {
        int f1 = pts[idx[i]].flag, f2 = pts[idx[i+1]].flag;
        var = CGAL_Y_RANGE;
        p1 = pts[idx[i]].key; 
        p2 = pts[idx[i+1]].key;
        if((f1&2) + (f2&2) == 0) {// both can use x-range polynomials
         // different polys in x-range - use y-range instead
            if((f1&1)!=(f2&1)) {
                shift = (f1&4)>>2;
                p1 = box[1].key[0]; 
                p2 = box[1].key[1]; 
            } else { // same polys in x-range
                shift = f1&1;
                var = CGAL_X_RANGE; 
                shift += 2;
            }
        } else { 
            if((f1&2) == 0) { // first point is in x-range (convert to y-range)
                shift = f2&1;
                p1 = box[1].key[f1&1]; 
            } else {   // second point is in x-range
                shift = f1&1;
                p2 = box[1].key[f2&1]; 
            }
        } 
        if(p1 > p2) {
            NT tmp1 = p1;
            p1 = p2;
            p2 = tmp1;
        }
        res = float2int((p2 - p1)*lvl); 
        int local_sign = 0;
        for(j = 0; j < res; j++, p1 += inv) {

            NT lkey = box[shift>>1].key[shift&1];
            const Poly_1& lpoly = box[shift>>1].poly[shift&1];
             
            if(get_range_1(var, p1, p1+inv, lkey, lpoly, 3) ||
                    engine.zero_bounds) {

                Gfx_DETAILED_OUT("range including 0 found at: " << i << 
                "; subseg: " << j <<  "; first der = " << engine.first_der <<
                     std::endl);

                if(engine.zero_bounds) {
                    Gfx_DETAILED_OUT("also with zero_bounds");
                
                    bool is_corner = false;
                    int diff = float2int(var == CGAL_X_RANGE ?
                        ((p1 - pix.x)*lvl - pix.sub_x) :
                        ((p1 - pix.y)*lvl - pix.sub_y));
                    int low_sign = engine.evaluate_generic(var, p1, lkey,
                            lpoly);
                    
                    if(diff != 0) {
                        // diff == -1: test lower corner: p1
                        // diff ==  1: test upper corver: p1+inv
                        if(diff == 1) {
                            is_corner = (engine.evaluate_generic(var, p1+inv,
                                lkey, lpoly) == 0);
                            // in case there is intersection at lower point =>
                            // it is counted as normal intersection
                            if(!is_corner && low_sign != 0)
                                continue;
                        } else {
                            is_corner = (low_sign == 0);
                            if(!is_corner)
                                continue;
                        }
                            
                        if(is_corner) {
                            // compute corner direction
                            tmp = internal::DIR_MAP[shift][diff+1];
                            Gfx_DETAILED_OUT("corner direction detected: "
                                << tmp << "\n");
                                
                            if(corner_dir == -1)
                                corner_dir = tmp;
                            else if(corner_dir != tmp) {
                                Gfx_DETAILED_OUT("different corner \
                                    directions found\n");
                                return false;
                            } else {
                                // decrement since sign change occurs in the
                                // corner
                                local_sign--;
                                Gfx_DETAILED_OUT("corner found: sign: "
                                    << n_sign << "; local: " << local_sign
                                        << "\n");
                            }
                        }
                    } else if(low_sign != 0)
                        continue;
                }
                local_sign++; 
              // Gfx_DETAILED_OUT("a curve branch detected at interval: (" << i
                 //   << "; " << i+1 << ") segment: " << j << std::endl);
                if(local_sign > 1) {
                    Gfx_DETAILED_OUT("more than 1 curve branch detected" <<
                        std::endl);
                    return false;
                }
                lower = p1; 
                s_shift = shift;
                f_der = engine.first_der;
                // s_change = engine.sign_change;
            }
        }
        n_sign += local_sign;
        if(local_sign == 1) {
            tmp = float2int(var == CGAL_X_RANGE ? // bottom/top
                ((lower - pix.x)*lvl - pix.sub_x) :
                ((lower - pix.y)*lvl - pix.sub_y));
            tmp = internal::DIR_MAP[s_shift][tmp+1];
//             if(new_dir != -1&&new_dir != tmp) // more than 1 direction found
//                 return false;
//             else {
//                 if(!branches_coincide&&
//                     (current_level < CGAL_COINCIDE_LEVEL))
//                     return false; // level is too small 
//                 set_coincide = true; // diagonal coincide mode
//                 //std::cerr << "diagonal coincide mode\n";
//             }
            new_dir = tmp;
            if(n_sign > 1) {
                Gfx_DETAILED_OUT("test_neigh: too many intersections found");
                return false;
            }
        }
    }
    
    //Gfx_DETAILED_OUT("processing segment: [" << lower << "; " << lower+inv <<
      //  "]" <<std::endl);
    if(n_sign == 0) {
        Gfx_DETAILED_OUT("ERROR: test_neigh: no sign changes detected..\n");
        return false;
    }
    var = ((s_shift & 2) ? CGAL_X_RANGE : CGAL_Y_RANGE);
    int ibox = s_shift>>1, ikey = s_shift&1;
    if(f_der == -1) { // need to compute the first derivative
        get_range_1(var, lower, lower+inv, box[ibox].key[ikey],
            box[ibox].poly[ikey], 3);
        f_der = engine.first_der;
    }
    
    // if coincide already set - no need to check the derivative
//     if(!set_coincide && f_der && 
//         (!s_change || !recursive_check(var,lower,lower+inv,
//         box[ibox].key[ikey], box[ibox].poly[ikey]))) {
    if(!set_coincide && f_der && !recursive_check(var,lower,lower+inv,
        box[ibox].key[ikey], box[ibox].poly[ikey])) {
        
        if(!branches_coincide &&    
              (current_level < CGAL_COINCIDE_LEVEL))//||direction_taken == -1))
            return false;
        if(!branches_coincide)
            Gfx_OUT("Branches coincide at pixel: " << pix <<
                std::endl);
        set_coincide = true;
    }

    int taken = internal::DIR_TAKEN_MAP[new_dir];
    if(taken != -1 && taken != direction_taken) {
    
       Gfx_DETAILED_OUT("\nERROR: wrong direction " << new_dir << " at pixel: "
            << pix << "; back_dir = " << back_dir << "; taken = " <<
                direction_taken << std::endl);
        if(back_dir == 2)
            new_dir = 6;
        else if(back_dir == 6)
            new_dir = 2;
        else 
            return false;
    }
    if(set_coincide) {
        branches_coincide = true;
#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT

        if(var == CGAL_X_RANGE) {
            Gfx_OUT("WARNING: unable to approximate point \
                 in vertical coincide mode!\n");
//             NT xx = lower+inv/2, yy = box[ibox].key[ikey];
//             xx = engine.x_min + xx*engine.pixel_w;
//             yy = engine.y_min + yy*engine.pixel_h;

            pix.xv = NAN; // mark this point as invalid (to be further skipped)
            pix.yv = NAN;
        } else {
            NT seed = box[ibox].key[ikey];
            seed = engine.x_min + seed*engine.pixel_w;

          Gfx_DETAILED_OUT("approximating in coincide mode: " << seed << "\n");
            Coordinate_2 xy(Coordinate_1(Rational(seed)),
                *support, arcno);
            Rational yseed, _;
            refine_xy(xy, CGAL_REFINE_DOUBLE_APPROX, _, yseed);
//             Rational yseed = ubound_y(xy);
            pix.xv = CGAL::to_double(seed); 
            pix.yv = CGAL::to_double(yseed);   
        }
#endif
    } 
#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
    else {
        Gfx_DETAILED_OUT("new dir = " << new_dir << 
        "; compute_double_approx for lower = " << lower << "; key = " <<
            box[ibox].key[ikey] << "\n");
        compute_double_approx(var, lower, lower+inv, box[ibox].key[ikey],
            box[ibox].poly[ikey], pix);
    }
#endif
    return true;
}

#ifdef CGAL_CKVA_RENDER_WITH_REFINEMENT
bool compute_double_approx(int var, const NT& l_, const NT& r_, 
    const NT& key, const Poly_1& poly, Pixel_2& pix) {

    NT l(l_), r(r_);
    if(l > r) {
        l = r_;
        r = l_;
    }
    
    bool ret = true;
    int eval1, eval2, mid_eval;
    NT threshold(CGAL_REFINE_DOUBLE_APPROX);
    eval1 = engine.evaluate_generic(var, l, key, poly);
    if(eval1 == 0)
        goto Lexit;
    
    eval2 = engine.evaluate_generic(var, r, key, poly);
    if(eval2 == 0) {
        l = r;
        goto Lexit;
    }

    if(eval1 == eval2) {
        std::cerr << "ERROR: no sign change in compute_double_approx: " <<
            eval1 << " and " << eval2 << "\n";
        l = (l+r)/NT(2);
        make_exact(l);
        ret = false;  
        goto Lexit;
    }
   
    while((r - l) > threshold) {
        NT mid = (l + r) / NT(2);
        make_exact(mid);
        //Gfx_DETAILED_OUT("current approx: " << (r - l) << "; threshold = " << threshold << "\n");
        //Gfx_DETAILED_OUT("l = " << l << "; r = " << r << "; mid = " << mid <<  "\n");
        mid_eval = engine.evaluate_generic(var, mid, key, poly);
        if(mid_eval == 0) {
            l = r = mid;    
            break;
        }   
        if(mid_eval == eval1)
            l = mid;
        else
            r = mid;
    }
Lexit:     
    NT x = l, y = key;
    if(var == CGAL_Y_RANGE) {
        x = key;
        y = l;
    }
    pix.xv = CGAL::to_double(engine.x_min + x*engine.pixel_w);
    pix.yv = CGAL::to_double(engine.y_min + y*engine.pixel_h);
    return ret;
}
#endif // CGAL_CKVA_RENDER_WITH_REFINEMENT

//! \brief returns whether a polynomial has zero over an interval,
//! we are not interested in concrete values
//!
//! a bit mask \c check indicates which boundaries are to be computed
//! 0th bit - of a polynomial itself (0th derivative, default)
//! 1st bit - of the first derivative (sets \c first_der flag)
//! 2nd bit - of the second derivative
inline bool get_range_1(int var, const NT& lower, const NT& upper, 
    const NT& key, const Poly_1& poly, int check = 1)
{
    if(IA_method == 0)
        return engine.get_range_QF_1(var, lower, upper, key, poly, check);
 
    return engine.get_range_MAA_1(var, lower, upper, key, poly, check);
}

//! \brief copmutes an isolating interval for y-coordinate of an end-point,
//! uses caching if possible
Rational get_endpoint_y(const Arc_2& arc, const Rational& x,
     CGAL::Arr_curve_end end, bool is_clipped, Rational refine_bound) {
        
    int arcno;
    Coordinate_2 xy;
    
    if(arc.location(end) != CGAL::ARR_LEFT_BOUNDARY &&
            arc.location(end) != CGAL::ARR_RIGHT_BOUNDARY && !is_clipped) {
        arcno = arc.curve_end(end).arcno();
        xy = Coordinate_2(arc.curve_end_x(end),
            arc.curve_end(end).curve(), arcno);
        
    } else {
        arcno = arc.arcno();
        xy = Coordinate_2(Coordinate_1(x), *support, arcno);
        //refine_bound = engine.pixel_h_r / 6; //! use other bound HACK 
    }

    // refine the y-interval until its size is smaller than the pixel size
    ////////////////////////// CHANGED REFINE_Y by REFINE_X
    Rational _1, _2;
    refine_xy(xy, refine_bound, _1, _2);
    ////////////////////////// CHANGED REFINE_Y by REFINE_X
    return _2;//event.upper_boundary(arcno);
}

//! \brief switches to a certain cache instance depending on currently used 
//! algebraic curve
void select_cache_entry(const Arc_2& arc) {
        
    int cid = arc.curve().id();
    typename Cache_list::iterator it = cache_list.begin();
    while(it != cache_list.end()) {
        if(it->first == cid)
            break;
        it++;
    }

    int new_id, new_entry = (it == cache_list.end());
    if(new_entry) {
        new_id = cache_list.size();
        if(new_id >= CGAL_N_CACHES) {
            new_id = cache_list.back().second;
            cache_list.pop_back();
        } 
        cache_list.insert(cache_list.begin(), LRU_entry(cid, new_id));

    } else { // mark this entry as MRU
        new_id = it->second;
        LRU_entry lru = *it;
        cache_list.erase(it);
        cache_list.insert(cache_list.begin(), lru);
    }
    
    if(cache_id != new_id) 
        clip_pts_computed = false;
    cache_id = new_id;
    
    support = support_ + cache_id;
    engine.select_cache_entry(cache_id);
    
    while(!s_stack.empty())
        s_stack.pop(); 
        
    if(new_entry) {
        *support = arc.curve();
        engine.precompute(support->polynomial_2());
    }
}

//! \brief returns one of 4 subpixels of a pixel which is crossed by the curve
//! branch
//!
//! subpixels are chosen with the priority \c dir which defines a preferred 
//! direction, i.e. right(0), top(1), left(2) or bottom(3) this is only for h/v
//! directions
int get_subpixel_hv(const Pixel_2& pix, int dir)
{
    Poly_1 box[4];
    NT inv = NT(1) / NT(one << pix.level), inv_2;
    make_exact(inv);
    inv_2 = inv/2;
    make_exact(inv_2);
    NT pix_x = pix.x + pix.sub_x * inv;
    NT pix_y = pix.y + pix.sub_y * inv;
    NT s_c, s_key, inc_c, inc_key;
    int var, inv_var;
    if(dir&1) { // 1 or 3 (vertical direction)
        var = CGAL_X_RANGE;
        s_c = pix_x;
        s_key = pix_y;
    } else {  // 0 or 2 (horizontal direction)
        var = CGAL_Y_RANGE;
        s_c = pix_y;
        s_key = pix_x;
    }
    inc_key = ((dir&2)-1)*inv_2;
    inc_c = (1-2*((dir>>1)^(dir&1)))*inv_2;
    inv_var = 1 - var;
    s_c = s_c + inv_2 - inc_c;
    s_key = s_key + inv_2 - inc_key;
    engine.get_precached_poly(var, s_key, pix.level, box[0]);
    
    //Gfx_DETAILED_OUT("hv subdpixel" << std::endl);
    int p0 = engine.evaluate_generic(var, s_c, s_key, box[0]); // 0
    int p1 = engine.evaluate_generic(var, s_c + inc_c, s_key, box[0]); // 1
    if(p0^p1)
        return 0; 
        
    int p2 = engine.evaluate_generic(var, s_c + inc_c*2, s_key, box[0]); // 2
    if(p2^p1)
        return 1;
        
    engine.get_precached_poly(inv_var, s_c, pix.level, box[1]);    
    int p3 = engine.evaluate_generic(inv_var, s_key+inc_key, s_c, box[1]); // 3
    if(p3^p0)
        return 0;
        
    engine.get_precached_poly(inv_var, s_c + inc_c*2, pix.level, box[2]);
    int p4 = engine.evaluate_generic(inv_var, s_key+inc_key, s_c + inc_c*2,
        box[2]);
     
    if(p4^p2)
        return 1;
        
    int p5 = engine.evaluate_generic(inv_var, s_key+inc_key*2, s_c, box[1]); 
    if(p3^p5)
        return 2;
        
    int p7 = engine.evaluate_generic(inv_var, s_key+inc_key*2, s_c + inc_c*2,
        box[2]); // 7
    if(p4^p7)
        return 3;
        
    engine.get_precached_poly(var, s_key+inc_key*2, pix.level, box[3]);
    int p6 = engine.evaluate_generic(var, s_c + inc_c, s_key+inc_key*2,
        box[3]); // 6
    if(p5^p6)
        return 2;
    if(p6^p7)
        return 3;
    return -1;  
}

//! \brief the same for diagonal directions
//! 
//! preferred direction: NE(0), NW(1), SW(2), SE(3)
int get_subpixel_diag(const Pixel_2& pix, int dir)
{
    Poly_1 box[4];
    NT inv = NT(1) / NT(one << pix.level);
    make_exact(inv);
    NT inv_2 = inv/2;
    make_exact(inv_2);
    NT pix_x = pix.x + pix.sub_x * inv;
    NT pix_y = pix.y + pix.sub_y * inv;
    NT key_1, key_2, inc_1, inc_2;
    int var, inv_var;
    if(dir&1) { // 1 or 3 (vertical direction)
        var = CGAL_X_RANGE;
        key_1 = pix_x;
        key_2 = pix_y;
    } else {  // 0 or 2 (horizontal direction)
        var = CGAL_Y_RANGE;
        key_1 = pix_y;
        key_2 = pix_x;
    }
    inc_1 = (2*((dir>>1)^(dir&1))-1)*inv_2;
    inc_2 = ((dir&2)-1)*inv_2;
    inv_var = 1 - var;
    key_1 = key_1 + inv_2 - inc_1;
    key_2 = key_2 + inv_2 - inc_2;
    
    engine.get_precached_poly(var, key_2, pix.level, box[0]);
    int p0 = engine.evaluate_generic(var, key_1, key_2, box[0]); // 0
    int p1 = engine.evaluate_generic(var, key_1 + inc_1, key_2, box[0]); // 1
    //Gfx_DETAILED_OUT("p0: " << p0 << "; p1: " << p1 << std::endl);
    
    if(p0^p1)
        return 0; 
    
    engine.get_precached_poly(inv_var, key_1, pix.level, box[1]);  
    int p2 = engine.evaluate_generic(inv_var, key_2+inc_2, key_1, box[1]); // 2
    //Gfx_DETAILED_OUT << "p2: " << p2 << std::endl;
    if(p2^p0)
        return 0;   
    
    int p3 = engine.evaluate_generic(var, key_1 + inc_1*2, key_2, box[0]); // 3
    //Gfx_DETAILED_OUT << "p3: " << p3 << std::endl;
    if(p3^p1)
        return 1;
    
    int p4 = engine.evaluate_generic(inv_var, key_2+inc_2*2, key_1, box[1]);
    //Gfx_DETAILED_OUT << "p4: " << p4 << std::endl;
    if(p4^p2)
        return 2;
        
    engine.get_precached_poly(inv_var, key_1 + inc_1*2, pix.level, box[2]);    
    int p5 = engine.evaluate_generic(inv_var, key_2+inc_2, key_1 + inc_1*2,
        box[2]);
    //Gfx_DETAILED_OUT << "p5: " << p5 << std::endl;
    if(p3^p5)
        return 1;
        
    engine.get_precached_poly(var, key_2+inc_2*2, pix.level, box[3]);
    int p6 = engine.evaluate_generic(var, key_1 + inc_1, key_2+inc_2*2,
        box[3]); // 6
    //Gfx_DETAILED_OUT << "p6: " << p6 << std::endl;
    if(p4^p6)
        return 2;   
        
    int p7 = engine.evaluate_generic(var, key_1+inc_1*2, key_2 + inc_2*2,
        box[3]); // 7
    //Gfx_DETAILED_OUT << "p7: " << p7 << std::endl;
    if(p5^p7)
        return 3;

    if(p6^p7)
        return 3;
    return -1;  
}

//! attempts to find an isolating box for an end-point whose upper/lower
//! boundaries do not intersect with a curve 
bool get_isolating_box(const Rational& x_s, const Rational& y_s, Pixel_2& res)
{
    Integer lvl = one;
    Rational x_seed, y_seed;
    get_pixel_coords(x_s, y_s, res, &x_seed, &y_seed);
    res.level = 0;
    res.sub_x = 0;
    res.sub_y = 0;
    
    NT inv = NT(1), bottom = NT(res.y), top = bottom + inv, left = NT(res.x); 
    
    Poly_1 poly_btm, poly_top;
    engine.get_precached_poly(CGAL_X_RANGE, bottom, 0, poly_btm);
    engine.get_precached_poly(CGAL_X_RANGE, top, 0, poly_top);
    
    while(1) {
        inv = NT(1) / NT(lvl);
        make_exact(inv);
        left = NT(res.x) + NT(res.sub_x) * inv;
        
        Gfx_DETAILED_OUT("processing pixel: " << res << std::endl);
        if(get_range_1(CGAL_X_RANGE,left,left+inv,bottom,poly_btm)&&
            !engine.zero_bounds) { 
            
            Gfx_DETAILED_OUT("bottom boundary intersection found at pixel: " <<
                res << std::endl);
        } else {
            if(!get_range_1(CGAL_X_RANGE,left,left+inv,top,poly_top)) {
            
                Gfx_DETAILED_OUT("no bottom/top intersections found at pixel:"
                    << res << std::endl);
                break;
            } else {    
                Gfx_DETAILED_OUT("top boundary intersection found at pixel: "
                    << res << std::endl);
            }
        }
        lvl <<= 1;
        /*if(lvl > CGAL_REFINE_Y) {
            event.refine_to(arcno, pixel_h_r/(lvl*2));
            y_seed = (event.lower_boundary(arcno) + 
                event.upper_boundary(arcno))/2;
        }*/
        if(res.level >= MAX_SUBDIVISION_LEVEL) {
        std::cerr("get_isolating_box: reached maximum subdivision level "
                << res.level << std::endl);
            return false;
        }
        res.level++;
        res.sub_x = rat2integer((x_seed - res.x)*lvl);
        res.sub_y = rat2integer((y_seed - res.y)*lvl);
        if(res.sub_x < 0)
            res.sub_x = 0;
        res.sub_x &= (lvl-1);
        if(res.sub_y < 0)
            res.sub_y = 0;
        res.sub_y &= (lvl-1);
    } 
    return true;
}

//! returns true if \c pix is encompassed into one of isolating rectangles
//! (stopping criteria)
inline bool is_isolated_pixel(const Pixel_2& /* pix */) {
    /*Integer sub_x;
    if(isolated_l.level != -1u&&isolated_l.y == pix.y&&isolated_l.x == pix.x&&
        pix.level >= isolated_l.level) {
        sub_x = pix.sub_x >> (pix.level - isolated_l.level);
        if(CGAL_ABS(sub_x - isolated_l.sub_x) <= 1)
            return true;
    }
    if(isolated_h.level != -1u&&isolated_h.y == pix.y&&isolated_h.x == pix.x&&
        pix.level >= isolated_h.level) {
        sub_x = pix.sub_x >> (pix.level - isolated_h.level);
        if(CGAL_ABS(sub_x - isolated_h.sub_x) <= 1)
            return true;
    }*/
    return false;
}

// DEBUG ONLY
void dump_neighbourhood(const Pixel_2& pix) {
#ifdef Gfx_USE_OUT
    CGAL::set_mode(std::cerr, CGAL::IO::PRETTY);
    CGAL::set_mode(std::cout, CGAL::IO::PRETTY);

    Stripe box[2]; // 0 - left-right stripe, 1 - bottom-top stripe
    //NT inv = NT(1) / NT(one << pix.level);
    get_boundaries(CGAL_X_RANGE,pix,box[1]);
    get_boundaries(CGAL_Y_RANGE,pix,box[0]);
    NT bottom = box[1].key[0], top = box[1].key[1],
        left = box[0].key[0], right = box[0].key[1];
    
    Gfx_OUT("\n\nComputing nighbourhood for pixel: " << pix << std::endl);
    get_polynomials(CGAL_X_RANGE, box[1]);
    get_polynomials(CGAL_Y_RANGE, box[0]);
    NT a,b,range,inc,val;
    ///////////////////////////////////////////////////////////////////////
    Gfx_OUT("\nevaluate RIGHT side:" << std::endl);
    range = top - bottom; 
    inc = range / 3;
    val = bottom;
    
    if(get_range_1(CGAL_Y_RANGE, bottom, top, right, box[0].poly[1]))
        Gfx_OUT("segment RIGHT registered" << std::endl);
    if(get_range_1(CGAL_Y_RANGE, val, val + inc, right, box[0].poly[1]))
        Gfx_OUT("segment 0 registered" << std::endl);
    
    a = engine.evaluate_generic(CGAL_Y_RANGE, val, right, box[0].poly[1]);
    Gfx_OUT("val = " << a << std::endl);
    val += inc;
    
    b = engine.evaluate_generic(CGAL_Y_RANGE, val, right, box[0].poly[1]);
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 0" << std::endl);
    a = b;
    if(get_range_1(CGAL_Y_RANGE, val, val + inc, right, box[0].poly[1]))
        Gfx_OUT("segment 1 registered" << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_Y_RANGE, val, right, box[0].poly[1]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 1" << std::endl);
    if(get_range_1(CGAL_Y_RANGE, val, val + inc, right, box[0].poly[1]))
        Gfx_OUT("segment 2 registered" << std::endl);
    a = b;
    val = top;

    Gfx_OUT("poly right: " << box[0].poly[1] << " at (" << val << "; " <<
        right << ")\n");
    
    b = engine.evaluate_generic(CGAL_Y_RANGE, val, right, box[0].poly[1]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 2" << std::endl);
    ///////////////////////////////////////////////////////////////////////    
    Gfx_OUT("\nevaluate BOTTOM side:" << std::endl);
    range = right - left; 
    inc = range / 3;
    val = left;
    if(get_range_1(CGAL_X_RANGE, left, right, bottom, box[1].poly[0]))
        Gfx_OUT("segment BOTTOM registered" << std::endl);
    if(get_range_1(CGAL_X_RANGE, val, val + inc, bottom, box[1].poly[0]))
        Gfx_OUT("segment 0 registered" << std::endl);

    Gfx_OUT("poly bottom: " << box[1].poly[0] << " at (" << val << "; " <<
        bottom << ")\n");
        
    a = engine.evaluate_generic(CGAL_X_RANGE, val, bottom, box[1].poly[0]);
    
    Gfx_OUT("val = " << a << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_X_RANGE, val, bottom, box[1].poly[0]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 0" << std::endl);
    a = b;
    if(get_range_1(CGAL_X_RANGE, val, val + inc, bottom, box[1].poly[0]))
        Gfx_OUT("segment 1 registered" << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_X_RANGE, val, bottom, box[1].poly[0]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 1" << std::endl);
    a = b;
    if(get_range_1(CGAL_X_RANGE, val, val + inc, bottom, box[1].poly[0]))
        Gfx_OUT("segment 2 registered" << std::endl);
    val = right;
    b = engine.evaluate_generic(CGAL_X_RANGE, val, bottom, box[1].poly[0]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 2" << std::endl);
    ///////////////////////////////////////////////////////////////////////    
    Gfx_OUT("\nevaluate LEFT side:" << std::endl);
    range = top - bottom; 
    inc = range / 3;
    val = bottom;
    if(get_range_1(CGAL_Y_RANGE, bottom, top, left, box[0].poly[0]))
        Gfx_OUT("segment LEFT registered" << std::endl);
    if(get_range_1(CGAL_Y_RANGE, val, val + inc, left, box[0].poly[0]))
        Gfx_OUT("segment 0 registered" << std::endl);

    Gfx_OUT("poly left: " << box[0].poly[0] << " at (" << val << "; " <<
        left << ")\n");
    a = engine.evaluate_generic(CGAL_Y_RANGE, val, left, box[0].poly[0]);
        
    Gfx_OUT("val = " << a << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_Y_RANGE, val, left, box[0].poly[0]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 0" << std::endl);
    a = b;
    if(get_range_1(CGAL_Y_RANGE, val, val + inc, left, box[0].poly[0]))
        Gfx_OUT("segment 1 registered" << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_Y_RANGE, val, left, box[0].poly[0]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 1" << std::endl);
    a = b;
    if(get_range_1(CGAL_Y_RANGE, val, val + inc, left, box[0].poly[0]))
        Gfx_OUT("segment 2 registered" << std::endl);
    val = top;
    b = engine.evaluate_generic(CGAL_Y_RANGE, val, left, box[0].poly[0]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 2" << std::endl);
    ///////////////////////////////////////////////////////////////////////    
    Gfx_OUT("\nevaluate TOP side:" << std::endl);
    range = right - left; 
    inc = range / 3;
    val = left;
    if(get_range_1(CGAL_X_RANGE, left, right, top, box[1].poly[1]))
        Gfx_OUT("segment TOP registered" << std::endl);
        
    if(get_range_1(CGAL_X_RANGE, val, val + inc, top, box[1].poly[1]))
        Gfx_OUT("segment 0 registered" << std::endl);
    a = engine.evaluate_generic(CGAL_X_RANGE, val, top, box[1].poly[1]);
    
    Gfx_OUT("val = " << a << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_X_RANGE, val, top, box[1].poly[1]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 0" << std::endl);
    a = b;
    if(get_range_1(CGAL_X_RANGE, val, val + inc, top, box[1].poly[1]))
        Gfx_OUT("segment 1 registered" << std::endl);
    val += inc;
    b = engine.evaluate_generic(CGAL_X_RANGE, val, top, box[1].poly[1]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 1" << std::endl);
    a = b;
    
    if(get_range_1(CGAL_X_RANGE, val, val + inc, top, box[1].poly[1]))
        Gfx_OUT("segment 2 registered" << std::endl);

    val = right;

    Gfx_OUT("poly top: " << box[1].poly[1] << " at (" << val << "; " <<
        top << ")\n");
    b = engine.evaluate_generic(CGAL_X_RANGE, val, top, box[1].poly[1]);
    
    Gfx_OUT("val = " << b << std::endl);
    if(a*b < 0) 
        Gfx_OUT("sign change at segment 2" << std::endl);
#endif // Gfx_USE_OUT
}

//!@} 
}; // class Curve_renderer_2<>

//!@}
} //namespace CGAL

#endif // CGAL_CKVA_CURVE_RENDERER_2_H
