ftgrays.c 52.4 KB
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/***************************************************************************/
/*                                                                         */
/*  ftgrays.c                                                              */
/*                                                                         */
/*    A new `perfect' anti-aliasing renderer (body).                       */
/*                                                                         */
/*  Copyright 2000 by                                                      */
/*  David Turner, Robert Wilhelm, and Werner Lemberg.                      */
/*                                                                         */
/*  This file is part of the FreeType project, and may only be used,       */
/*  modified, and distributed under the terms of the FreeType project      */
/*  license, LICENSE.TXT.  By continuing to use, modify, or distribute     */
/*  this file you indicate that you have read the license and              */
/*  understand and accept it fully.                                        */
/*                                                                         */
/***************************************************************************/

  /*************************************************************************/
  /*                                                                       */
  /*  This file can be compiled without the rest of the FreeType engine,   */
  /*  by defining the _STANDALONE_ macro when compiling it.  You also need */
  /*  to put the files `ftgrays.h' and `ftimage.h' into the current        */
  /*  compilation directory.  Typically, you could do something like       */
  /*                                                                       */
  /*  - copy `src/base/ftgrays.c' to your current directory                */
  /*                                                                       */
  /*  - copy `include/freetype/ftimage.h' and                              */
  /*    `include/freetype/ftgrays.h' to the same directory                 */
  /*                                                                       */
  /*  - compile `ftgrays' with the _STANDALONE_ macro defined, as in       */
  /*                                                                       */
  /*      cc -c -D_STANDALONE_ ftgrays.c                                   */
  /*                                                                       */
  /*  The renderer can be initialized with a call to                       */
  /*  `ft_grays_raster.grays_raster_new'; an anti-aliased bitmap can be    */
  /*  generated with a call to `ft_grays_raster.grays_raster_render'.      */
  /*                                                                       */
  /*  See the comments and documentation in the file `ftimage.h' for       */
  /*  more details on how the raster works.                                */
  /*                                                                       */
  /*************************************************************************/

  /*************************************************************************/
  /*                                                                       */
  /*  This is a new anti-aliasing scan-converter for FreeType 2.  The      */
  /*  algorithm used here is _very_ different from the one in the standard */
  /*  `ftraster' module.  Actually, `ftgrays' computes the _exact_         */
  /*  coverage of the outline on each pixel cell.                          */
  /*                                                                       */
  /*  It is based on ideas that I initially found in Raph Levien's         */
  /*  excellent LibArt graphics library (see http://www.levien.com/libart  */
  /*  for more information, though the web pages do not tell anything      */
  /*  about the renderer; you'll have to dive into the source code to      */
  /*  understand how it works).                                            */
  /*                                                                       */
  /*  Note, however, that this is a _very_ different implementation        */
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  /*  compared to Raph's.  Coverage information is stored in a very        */
  /*  different way, and I don't use sorted vector paths.  Also, it        */
  /*  doesn't use floating point values.                                   */
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  /*                                                                       */
  /*  This renderer has the following advantages:                          */
  /*                                                                       */
  /*  - It doesn't need an intermediate bitmap.  Instead, one can supply   */
  /*    a callback function that will be called by the renderer to draw    */
  /*    gray spans on any target surface.  You can thus do direct          */
  /*    composition on any kind of bitmap, provided that you give the      */
  /*    renderer the right callback.                                       */
  /*                                                                       */
  /*  - A perfect anti-aliaser, i.e., it computes the _exact_ coverage on  */
  /*    each pixel cell                                                    */
  /*                                                                       */
  /*  - It performs a single pass on the outline (the `standard' FT2       */
  /*    renderer makes two passes).                                        */
  /*                                                                       */
  /*  - It can easily be modified to render to _any_ number of gray levels */
  /*    cheaply.                                                           */
  /*                                                                       */
  /*  - For small (< 20) pixel sizes, it is faster than the standard       */
  /*    renderer.                                                          */
  /*                                                                       */
  /*************************************************************************/


#include <string.h>             /* for memcpy() */


  /*************************************************************************/
  /*                                                                       */
  /* The macro FT_COMPONENT is used in trace mode.  It is an implicit      */
  /* parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log  */
  /* messages during execution.                                            */
  /*                                                                       */
#undef  FT_COMPONENT
#define FT_COMPONENT  trace_aaraster


#ifdef _STANDALONE_

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#define ErrRaster_Invalid_Mode     -2
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#define ErrRaster_Invalid_Outline  -1
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#include "ftimage.h"
#include "ftgrays.h"

  /* This macro is used to indicate that a function parameter is unused. */
  /* Its purpose is simply to reduce compiler warnings.  Note also that  */
  /* simply defining it as `(void)x' doesn't avoid warnings with certain */
  /* ANSI compilers (e.g. LCC).                                          */
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#define FT_UNUSED( x )  (x) = (x)
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  /* Disable the tracing mechanism for simplicity -- developers can      */
  /* activate it easily by redefining these two macros.                  */
#ifndef FT_ERROR
#define FT_ERROR( x )  do ; while ( 0 )     /* nothing */
#endif

#ifndef FT_TRACE
#define FT_TRACE( x )  do ; while ( 0 )     /* nothing */
#endif


#else /* _STANDALONE_ */

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#ifdef FT_FLAT_COMPILE
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#include "ftgrays.h"
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#else
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#include <smooth/ftgrays.h>
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#endif


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#include <freetype/internal/ftobjs.h>  /* for FT_UNUSED()               */
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#include <freetype/internal/ftdebug.h> /* for FT_TRACE() and FT_ERROR() */
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#include <freetype/ftoutln.h>          /* for FT_Outline_Decompose()    */
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#define ErrRaster_Invalid_Mode     FT_Err_Cannot_Render_Glyph
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#define ErrRaster_Invalid_Outline  FT_Err_Invalid_Outline
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#endif /* _STANDALONE_ */


  /* define this to dump debugging information */
#define xxxDEBUG_GRAYS

  /* as usual, for the speed hungry :-) */

#ifndef FT_STATIC_RASTER


#define RAS_ARG   PRaster  raster
#define RAS_ARG_  PRaster  raster,

#define RAS_VAR   raster
#define RAS_VAR_  raster,

#define ras       (*raster)


#else /* FT_STATIC_RASTER */


#define RAS_ARG   /* empty */
#define RAS_ARG_  /* empty */
#define RAS_VAR   /* empty */
#define RAS_VAR_  /* empty */

  static TRaster  ras;


#endif /* FT_STATIC_RASTER */


  /* must be at least 6 bits! */
#define PIXEL_BITS  8

#define ONE_PIXEL       ( 1L << PIXEL_BITS )
#define PIXEL_MASK      ( -1L << PIXEL_BITS )
#define TRUNC( x )      ( (x) >> PIXEL_BITS )
#define SUBPIXELS( x )  ( (x) << PIXEL_BITS )
#define FLOOR( x )      ( (x) & -ONE_PIXEL )
#define CEILING( x )    ( ( (x) + ONE_PIXEL - 1 ) & -ONE_PIXEL )
#define ROUND( x )      ( ( (x) + ONE_PIXEL / 2 ) & -ONE_PIXEL )

#if PIXEL_BITS >= 6
#define UPSCALE( x )    ( (x) << ( PIXEL_BITS - 6 ) )
#define DOWNSCALE( x )  ( (x) >> ( PIXEL_BITS - 6 ) )
#else
#define UPSCALE( x )    ( (x) >> ( 6 - PIXEL_BITS ) )
#define DOWNSCALE( x )  ( (x) << ( 6 - PIXEL_BITS ) )
#endif

  /* Define this if you want to use a more compact storage scheme.  This   */
  /* increases the number of cells available in the render pool but slows  */
  /* down the rendering a bit.  It is useful if you have a really tiny     */
  /* render pool.                                                          */
#define xxxGRAYS_COMPACT


  /*************************************************************************/
  /*                                                                       */
  /*   TYPE DEFINITIONS                                                    */
  /*                                                                       */
  typedef int   TScan;   /* integer scanline/pixel coordinate */
  typedef long  TPos;    /* sub-pixel coordinate              */

  /* maximal number of gray spans in a call to the span callback */
#define FT_MAX_GRAY_SPANS  32


#ifdef GRAYS_COMPACT

  typedef struct  TCell_
  {
    short  x     : 14;
    short  y     : 14;
    int    cover : PIXEL_BITS + 2;
    int    area  : PIXEL_BITS * 2 + 2;

  } TCell, *PCell;

#else /* GRAYS_COMPACT */

  typedef struct  TCell_
  {
    TScan  x;
    TScan  y;
    int    cover;
    int    area;

  } TCell, *PCell;

#endif /* GRAYS_COMPACT */


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  typedef struct  TRaster_
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  {
    PCell  cells;
    int    max_cells;
    int    num_cells;

    TScan  min_ex, max_ex;
    TScan  min_ey, max_ey;

    int    area;
    int    cover;
    int    invalid;

    TScan  ex, ey;
    TScan  cx, cy;
    TPos   x,  y;

    TScan  last_ey;

    FT_Vector   bez_stack[32 * 3];
    int         lev_stack[32];

    FT_Outline  outline;
    FT_Bitmap   target;

    FT_Span     gray_spans[FT_MAX_GRAY_SPANS];
    int         num_gray_spans;

    FT_Raster_Span_Func  render_span;
    void*                render_span_data;
    int                  span_y;

    int    band_size;
    int    band_shoot;
    int    conic_level;
    int    cubic_level;

    void*  memory;

  } TRaster, *PRaster;


  /*************************************************************************/
  /*                                                                       */
  /* Initialize the cells table.                                           */
  /*                                                                       */
  static
  void  init_cells( RAS_ARG_ void*  buffer,
                    long            byte_size )
  {
    ras.cells     = (PCell)buffer;
    ras.max_cells = byte_size / sizeof ( TCell );
    ras.num_cells = 0;
    ras.area      = 0;
    ras.cover     = 0;
    ras.invalid   = 1;
  }


  /*************************************************************************/
  /*                                                                       */
  /* Compute the outline bounding box.                                     */
  /*                                                                       */
  static
  void  compute_cbox( RAS_ARG_ FT_Outline*  outline )
  {
    FT_Vector*  vec   = outline->points;
    FT_Vector*  limit = vec + outline->n_points;


    if ( outline->n_points <= 0 )
    {
      ras.min_ex = ras.max_ex = 0;
      ras.min_ey = ras.max_ey = 0;
      return;
    }

    ras.min_ex = ras.max_ex = vec->x;
    ras.min_ey = ras.max_ey = vec->y;

    vec++;

    for ( ; vec < limit; vec++ )
    {
      TPos  x = vec->x;
      TPos  y = vec->y;


      if ( x < ras.min_ex ) ras.min_ex = x;
      if ( x > ras.max_ex ) ras.max_ex = x;
      if ( y < ras.min_ey ) ras.min_ey = y;
      if ( y > ras.max_ey ) ras.max_ey = y;
    }

    /* truncate the bounding box to integer pixels */
    ras.min_ex = ras.min_ex >> 6;
    ras.min_ey = ras.min_ey >> 6;
    ras.max_ex = ( ras.max_ex + 63 ) >> 6;
    ras.max_ey = ( ras.max_ey + 63 ) >> 6;
  }


  /*************************************************************************/
  /*                                                                       */
  /* Record the current cell in the table.                                 */
  /*                                                                       */
  static
  int  record_cell( RAS_ARG )
  {
    PCell  cell;


    if ( !ras.invalid && ( ras.area | ras.cover ) )
    {
      if ( ras.num_cells >= ras.max_cells )
        return 1;

      cell        = ras.cells + ras.num_cells++;
      cell->x     = ras.ex - ras.min_ex;
      cell->y     = ras.ey - ras.min_ey;
      cell->area  = ras.area;
      cell->cover = ras.cover;
    }

    return 0;
  }


  /*************************************************************************/
  /*                                                                       */
  /* Set the current cell to a new position.                               */
  /*                                                                       */
  static
  int  set_cell( RAS_ARG_ TScan  ex,
                          TScan  ey )
  {
    int  invalid, record, clean;


    /* Move the cell pointer to a new position.  We set the `invalid'      */
    /* flag to indicate that the cell isn't part of those we're interested */
    /* in during the render phase.  This means that:                       */
    /*                                                                     */
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    /* . the new vertical position must be within min_ey..max_ey-1.        */
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    /* . the new horizontal position must be strictly less than max_ex     */
    /*                                                                     */
    /* Note that if a cell is to the left of the clipping region, it is    */
    /* actually set to the (min_ex-1) horizontal position.                 */

    record  = 0;
    clean   = 1;

    invalid = ( ey < ras.min_ey || ey >= ras.max_ey || ex >= ras.max_ex );
    if ( !invalid )
    {
      /* All cells that are on the left of the clipping region go to the */
      /* min_ex - 1 horizontal position.                                 */
      if ( ex < ras.min_ex )
        ex = ras.min_ex - 1;

      /* if our position is new, then record the previous cell */
      if ( ex != ras.ex || ey != ras.ey )
        record = 1;
      else
        clean = ras.invalid;  /* do not clean if we didn't move from */
                              /* a valid cell                        */
    }

    /* record the previous cell if needed (i.e., if we changed the cell */
    /* position, of changed the `invalid' flag)                         */
    if ( ( ras.invalid != invalid || record ) && record_cell( RAS_VAR ) )
      return 1;

    if ( clean )
    {
      ras.area  = 0;
      ras.cover = 0;
    }

    ras.invalid = invalid;
    ras.ex      = ex;
    ras.ey      = ey;
    return 0;
  }


  /*************************************************************************/
  /*                                                                       */
  /* Start a new contour at a given cell.                                  */
  /*                                                                       */
  static
  void  start_cell( RAS_ARG_  TScan  ex,
                              TScan  ey )
  {
    if ( ex < ras.min_ex )
      ex = ras.min_ex - 1;

    ras.area    = 0;
    ras.cover   = 0;
    ras.ex      = ex;
    ras.ey      = ey;
    ras.last_ey = SUBPIXELS( ey );
    ras.invalid = 0;

    (void)set_cell( RAS_VAR_ ex, ey );
  }


  /*************************************************************************/
  /*                                                                       */
  /* Render a scanline as one or more cells.                               */
  /*                                                                       */
  static
  int  render_scanline( RAS_ARG_  TScan  ey,
                                  TPos   x1,
                                  TScan  y1,
                                  TPos   x2,
                                  TScan  y2 )
  {
    TScan  ex1, ex2, fx1, fx2, delta;
    long   p, first, dx;
    int    incr, lift, mod, rem;


    dx = x2 - x1;

    ex1 = TRUNC( x1 ); /* if (ex1 >= ras.max_ex) ex1 = ras.max_ex-1; */
    ex2 = TRUNC( x2 ); /* if (ex2 >= ras.max_ex) ex2 = ras.max_ex-1; */
    fx1 = x1 - SUBPIXELS( ex1 );
    fx2 = x2 - SUBPIXELS( ex2 );

    /* trivial case.  Happens often */
    if ( y1 == y2 )
      return set_cell( RAS_VAR_ ex2, ey );

    /* everything is located in a single cell.  That is easy! */
    /*                                                        */
    if ( ex1 == ex2 )
    {
      delta      = y2 - y1;
      ras.area  += ( fx1 + fx2 ) * delta;
      ras.cover += delta;
      return 0;
    }

    /* ok, we'll have to render a run of adjacent cells on the same */
    /* scanline...                                                  */
    /*                                                              */
    p     = ( ONE_PIXEL - fx1 ) * ( y2 - y1 );
    first = ONE_PIXEL;
    incr  = 1;

    if ( dx < 0 )
    {
      p     = fx1 * ( y2 - y1 );
      first = 0;
      incr  = -1;
      dx    = -dx;
    }

    delta = p / dx;
    mod   = p % dx;
    if ( mod < 0 )
    {
      delta--;
      mod += dx;
    }

    ras.area  += ( fx1 + first ) * delta;
    ras.cover += delta;

    ex1 += incr;
    if ( set_cell( RAS_VAR_ ex1, ey ) )
      goto Error;
    y1  += delta;

    if ( ex1 != ex2 )
    {
      p     = ONE_PIXEL * ( y2 - y1 );
      lift  = p / dx;
      rem   = p % dx;
      if ( rem < 0 )
      {
        lift--;
        rem += dx;
      }

      mod -= dx;

      while ( ex1 != ex2 )
      {
        delta = lift;
        mod  += rem;
        if ( mod >= 0 )
        {
          mod -= dx;
          delta++;
        }

        ras.area  += ONE_PIXEL * delta;
        ras.cover += delta;
        y1        += delta;
        ex1       += incr;
        if ( set_cell( RAS_VAR_ ex1, ey ) )
          goto Error;
      }
    }

    delta      = y2 - y1;
    ras.area  += ( fx2 + ONE_PIXEL - first ) * delta;
    ras.cover += delta;

    return 0;

  Error:
    return 1;
  }


  /*************************************************************************/
  /*                                                                       */
  /* Render a given line as a series of scanlines.                         */
  /*                                                                       */
  static
  int  render_line( RAS_ARG_ TPos  to_x,
                             TPos  to_y )
  {
    TScan  ey1, ey2, fy1, fy2;
    TPos   dx, dy, x, x2;
    int    p, rem, mod, lift, delta, first, incr;


    ey1 = TRUNC( ras.last_ey );
    ey2 = TRUNC( to_y ); /* if (ey2 >= ras.max_ey) ey2 = ras.max_ey-1; */
    fy1 = ras.y - ras.last_ey;
    fy2 = to_y - SUBPIXELS( ey2 );

    dx = to_x - ras.x;
    dy = to_y - ras.y;

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    /* XXX: we should do something about the trivial case where dx == 0, */
    /*      as it happens very often!                                    */
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    /* perform vertical clipping */
    {
      TScan  min, max;


      min = ey1;
      max = ey2;
      if ( ey1 > ey2 )
      {
        min = ey2;
        max = ey1;
      }
      if ( min >= ras.max_ey || max < ras.min_ey )
        goto End;
    }

    /* everything is on a single scanline */
    if ( ey1 == ey2 )
    {
      if ( render_scanline( RAS_VAR_ ey1, ras.x, fy1, to_x, fy2 ) )
        goto Error;
      goto End;
    }

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    /* ok, we have to render several scanlines */
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    p     = ( ONE_PIXEL - fy1 ) * dx;
    first = ONE_PIXEL;
    incr  = 1;

    if ( dy < 0 )
    {
      p     = fy1 * dx;
      first = 0;
      incr  = -1;
      dy    = -dy;
    }

    delta = p / dy;
    mod   = p % dy;
    if ( mod < 0 )
    {
      delta--;
      mod += dy;
    }

    x = ras.x + delta;
    if ( render_scanline( RAS_VAR_ ey1, ras.x, fy1, x, first ) )
      goto Error;

    ey1 += incr;
    if ( set_cell( RAS_VAR_ TRUNC( x ), ey1 ) )
      goto Error;

    if ( ey1 != ey2 )
    {
      p     = ONE_PIXEL * dx;
      lift  = p / dy;
      rem   = p % dy;
      if ( rem < 0 )
      {
        lift--;
        rem += dy;
      }
      mod -= dy;

      while ( ey1 != ey2 )
      {
        delta = lift;
        mod  += rem;
        if ( mod >= 0 )
        {
          mod -= dy;
          delta++;
        }

        x2 = x + delta;
        if ( render_scanline( RAS_VAR_ ey1,
                              x, ONE_PIXEL - first, x2, first ) )
          goto Error;
        x = x2;
        ey1 += incr;
        if ( set_cell( RAS_VAR_ TRUNC( x ), ey1 ) )
          goto Error;
      }
    }

    if ( render_scanline( RAS_VAR_ ey1,
                          x, ONE_PIXEL - first, to_x, fy2 ) )
      goto Error;

  End:
    ras.x       = to_x;
    ras.y       = to_y;
    ras.last_ey = SUBPIXELS( ey2 );

    return 0;

  Error:
    return 1;
  }


  static
  void  split_conic( FT_Vector*  base )
  {
    TPos  a, b;


    base[4].x = base[2].x;
    b = base[1].x;
    a = base[3].x = ( base[2].x + b ) / 2;
    b = base[1].x = ( base[0].x + b ) / 2;
    base[2].x = ( a + b ) / 2;

    base[4].y = base[2].y;
    b = base[1].y;
    a = base[3].y = ( base[2].y + b ) / 2;
    b = base[1].y = ( base[0].y + b ) / 2;
    base[2].y = ( a + b ) / 2;
  }


  static
  int  render_conic( RAS_ARG_ FT_Vector*  control,
                              FT_Vector*  to )
  {
    TPos        dx, dy;
    int         top, level;
    int*        levels;
    FT_Vector*  arc;


    dx = DOWNSCALE( ras.x ) + to->x - ( control->x << 1 );
    if ( dx < 0 )
      dx = -dx;
    dy = DOWNSCALE( ras.y ) + to->y - ( control->y << 1 );
    if ( dy < 0 )
      dy = -dy;
    if ( dx < dy )
      dx = dy;

    level = 1;
    dx = dx / ras.conic_level;
    while ( dx > 0 )
    {
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      dx >>= 2;
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      level++;
    }

    /* a shortcut to speed things up */
    if ( level <= 1 )
    {
      /* we compute the mid-point directly in order to avoid */
      /* calling split_conic()                               */
      TPos   to_x, to_y, mid_x, mid_y;


      to_x  = UPSCALE( to->x );
      to_y  = UPSCALE( to->y );
      mid_x = ( ras.x + to_x + 2 * UPSCALE( control->x ) ) / 4;
      mid_y = ( ras.y + to_y + 2 * UPSCALE( control->y ) ) / 4;

      return render_line( RAS_VAR_ mid_x, mid_y ) ||
             render_line( RAS_VAR_ to_x, to_y );
    }

    arc       = ras.bez_stack;
    levels    = ras.lev_stack;
    top       = 0;
    levels[0] = level;

    arc[0].x = UPSCALE( to->x );
    arc[0].y = UPSCALE( to->y );
    arc[1].x = UPSCALE( control->x );
    arc[1].y = UPSCALE( control->y );
    arc[2].x = ras.x;
    arc[2].y = ras.y;

    while ( top >= 0 )
    {
      level = levels[top];
      if ( level > 1 )
      {
        /* check that the arc crosses the current band */
        TPos  min, max, y;


        min = max = arc[0].y;

        y = arc[1].y;
        if ( y < min ) min = y;
        if ( y > max ) max = y;

        y = arc[2].y;
        if ( y < min ) min = y;
        if ( y > max ) max = y;

        if ( TRUNC( min ) >= ras.max_ey || TRUNC( max ) < 0 )
          goto Draw;

        split_conic( arc );
        arc += 2;
        top++;
        levels[top] = levels[top - 1] = level - 1;
        continue;
      }

    Draw:
      {
        TPos  to_x, to_y, mid_x, mid_y;


        to_x  = arc[0].x;
        to_y  = arc[0].y;
        mid_x = ( ras.x + to_x + 2 * arc[1].x ) / 4;
        mid_y = ( ras.y + to_y + 2 * arc[1].y ) / 4;

        if ( render_line( RAS_VAR_ mid_x, mid_y ) ||
             render_line( RAS_VAR_ to_x, to_y )   )
          return 1;

        top--;
        arc -= 2;
      }
    }
    return 0;
  }


  static
  void  split_cubic( FT_Vector*  base )
  {
    TPos  a, b, c, d;


    base[6].x = base[3].x;
    c = base[1].x;
    d = base[2].x;
    base[1].x = a = ( base[0].x + c ) / 2;
    base[5].x = b = ( base[3].x + d ) / 2;
    c = ( c + d ) / 2;
    base[2].x = a = ( a + c ) / 2;
    base[4].x = b = ( b + c ) / 2;
    base[3].x = ( a + b ) / 2;

    base[6].y = base[3].y;
    c = base[1].y;
    d = base[2].y;
    base[1].y = a = ( base[0].y + c ) / 2;
    base[5].y = b = ( base[3].y + d ) / 2;
    c = ( c + d ) / 2;
    base[2].y = a = ( a + c ) / 2;
    base[4].y = b = ( b + c ) / 2;
    base[3].y = ( a + b ) / 2;
  }


  static
  int  render_cubic( RAS_ARG_ FT_Vector*  control1,
                              FT_Vector*  control2,
                              FT_Vector*  to )
  {
    TPos        dx, dy, da, db;
    int         top, level;
    int*        levels;
    FT_Vector*  arc;


    dx = DOWNSCALE( ras.x ) + to->x - ( control1->x << 1 );
    if ( dx < 0 )
      dx = -dx;
    dy = DOWNSCALE( ras.y ) + to->y - ( control1->y << 1 );
    if ( dy < 0 )
      dy = -dy;
    if ( dx < dy )
      dx = dy;
    da = dx;

    dx = DOWNSCALE( ras.x ) + to->x - 3 * ( control1->x + control2->x );
    if ( dx < 0 )
      dx = -dx;
    dy = DOWNSCALE( ras.y ) + to->y - 3 * ( control1->x + control2->y );
    if ( dy < 0 )
      dy = -dy;
    if ( dx < dy )
      dx = dy;
    db = dx;

    level = 1;
    da    = da / ras.cubic_level;
    db    = db / ras.conic_level;
    while ( da > 0 || db > 0 )
    {
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      da >>= 2;
      db >>= 3;
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      level++;
    }

    if ( level <= 1 )
    {
      TPos   to_x, to_y, mid_x, mid_y;


      to_x  = UPSCALE( to->x );
      to_y  = UPSCALE( to->y );
      mid_x = ( ras.x + to_x +
                3 * UPSCALE( control1->x + control2->x ) ) / 8;
      mid_y = ( ras.y + to_y +
                3 * UPSCALE( control1->y + control2->y ) ) / 8;

      return render_line( RAS_VAR_ mid_x, mid_y ) ||
             render_line( RAS_VAR_ to_x, to_y );
    }

    arc      = ras.bez_stack;
    arc[0].x = UPSCALE( to->x );
    arc[0].y = UPSCALE( to->y );
    arc[1].x = UPSCALE( control2->x );
    arc[1].y = UPSCALE( control2->y );
    arc[2].x = UPSCALE( control1->x );
    arc[2].y = UPSCALE( control1->y );
    arc[3].x = ras.x;
    arc[3].y = ras.y;

    levels    = ras.lev_stack;
    top       = 0;
    levels[0] = level;

    while ( top >= 0 )
    {
      level = levels[top];
      if ( level > 1 )
      {
        /* check that the arc crosses the current band */
        TPos  min, max, y;


        min = max = arc[0].y;
        y = arc[1].y;
        if ( y < min ) min = y;
        if ( y > max ) max = y;
        y = arc[2].y;
        if ( y < min ) min = y;
        if ( y > max ) max = y;
        y = arc[3].y;
        if ( y < min ) min = y;
        if ( y > max ) max = y;
        if ( TRUNC( min ) >= ras.max_ey || TRUNC( max ) < 0 )
          goto Draw;
        split_cubic( arc );
        arc += 3;
        top ++;
        levels[top] = levels[top - 1] = level - 1;
        continue;
      }

    Draw:
      {
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        TPos  to_x, to_y, mid_x, mid_y;
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        to_x  = arc[0].x;
        to_y  = arc[0].y;
        mid_x = ( ras.x + to_x + 3 * ( arc[1].x + arc[2].x ) ) / 8;
        mid_y = ( ras.y + to_y + 3 * ( arc[1].y + arc[2].y ) ) / 8;

        if ( render_line( RAS_VAR_ mid_x, mid_y ) ||
             render_line( RAS_VAR_ to_x, to_y )   )
          return 1;
        top --;
        arc -= 3;
      }
    }
    return 0;
  }


  /* a macro comparing two cell pointers.  Returns true if a <= b. */
#if 1
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#define PACK( a )          ( ( (long)(a)->y << 16 ) + (a)->x )
#define LESS_THAN( a, b )  ( PACK( a ) < PACK( b ) )
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#else /* 1 */
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#define LESS_THAN( a, b )  ( (a)->y < (b)->y || \
                             ( (a)->y == (b)->y && (a)->x < (b)->x ) )
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#endif /* 1 */

#define SWAP_CELLS( a, b, temp )  do             \
                                  {              \
                                    temp = *(a); \
                                    *(a) = *(b); \
                                    *(b) = temp; \
                                  } while ( 0 )
#define DEBUG_SORT
#define QUICK_SORT

#ifdef SHELL_SORT

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  /* a simple shell sort algorithm that works directly on our */
  /* cells table                                              */
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  static
  void  shell_sort ( PCell  cells,
                     int    count )
  {
    PCell  i, j, limit = cells + count;
    TCell  temp;
    int    gap;


    /* compute initial gap */
    for ( gap = 0; ++gap < count; gap *= 3 )
      ;

    while ( gap /= 3 )