text: /***************************************************************************/
  /*                                                                         */
  /*  ftstroke.c                                                             */
  /*                                                                         */
  /*    FreeType path stroker (body).                                        */
  /*                                                                         */
  /*  Copyright 2002-2006, 2008-2011 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.                                        */
  /*                                                                         */
  /***************************************************************************/
  
  
  #include <ft2build.h>
  #include FT_STROKER_H
  #include FT_TRIGONOMETRY_H
  #include FT_OUTLINE_H
  #include FT_INTERNAL_MEMORY_H
  #include FT_INTERNAL_DEBUG_H
  #include FT_INTERNAL_OBJECTS_H
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_StrokerBorder )
  
  FT_Outline_GetInsideBorder( FT_Outline
  
  *  outline )
  {
  FT_Orientation o = FT_Outline_Get_Orientation( outline );
  
  
  return o == FT_ORIENTATION_TRUETYPE ? FT_STROKER_BORDER_RIGHT
  :
  
  FT_STROKER_BORDER_LEFT;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_StrokerBorder )
  
  FT_Outline_GetOutsideBorder( FT_Outline
  
  *  outline )
  {
  FT_Orientation o = FT_Outline_Get_Orientation( outline );
  
  
  return o == FT_ORIENTATION_TRUETYPE ? FT_STROKER_BORDER_LEFT
  :
  
  FT_STROKER_BORDER_RIGHT;
  
  }
  
  
  /*************************************************************************/
  /*************************************************************************/
  /*****                                                               *****/
  /*****                      BEZIER COMPUTATIONS                      *****/
  /*****                                                               *****/
  /*************************************************************************/
  /*************************************************************************/
  
  #define FT_SMALL_CONIC_THRESHOLD  ( FT_ANGLE_PI / 6 )
  #define FT_SMALL_CUBIC_THRESHOLD  ( FT_ANGLE_PI / 8 )
  
  #define FT_EPSILON  2
  
  #define FT_IS_SMALL( x )  ( (x) > -FT_EPSILON && (x) < FT_EPSILON )
  
  
  static FT_Pos
  ft_pos_abs( FT_Pos x )
  {
      return x >= 0 ? x : -x;
  }
  
  
  static void
  ft_conic_split( FT_Vector *base )
  {
      FT_Pos 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 FT_Bool
  ft_conic_is_small_enough( FT_Vector *base,
                            FT_Angle *angle_in,
                            FT_Angle *angle_out )
  {
      FT_Vector d1, d2;
      FT_Angle theta;
      FT_Int close1, close2;
  
  
      d1.x = base[ 1 ].x - base[ 2 ].x;
      d1.y = base[ 1 ].y - base[ 2 ].y;
      d2.x = base[ 0 ].x - base[ 1 ].x;
      d2.y = base[ 0 ].y - base[ 1 ].y;
  
      close1 = FT_IS_SMALL( d1.x ) && FT_IS_SMALL( d1.y );
      close2 = FT_IS_SMALL( d2.x ) && FT_IS_SMALL( d2.y );
  
      if ( close1 )
      {
          if ( close2 )
          {
              /* basically a point;                      */
              /* do nothing to retain original direction */
          }
          else
          {
              *angle_in =
              *angle_out = FT_Atan2( d2.x, d2.y );
          }
      }
      else /* !close1 */
      {
          if ( close2 )
          {
              *angle_in =
              *angle_out = FT_Atan2( d1.x, d1.y );
          }
          else
          {
              *angle_in = FT_Atan2( d1.x, d1.y );
              *angle_out = FT_Atan2( d2.x, d2.y );
          }
      }
  
      theta = ft_pos_abs( FT_Angle_Diff( *angle_in, *angle_out ));
  
      return FT_BOOL( theta < FT_SMALL_CONIC_THRESHOLD);
  }
  
  
  static void
  ft_cubic_split( FT_Vector *base )
  {
      FT_Pos 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;
  }
  
  
  /* Return the average of `angle1' and `angle2'.            */
  /* This gives correct result even if `angle1' and `angle2' */
  /* have opposite signs.                                    */
  static FT_Angle
  ft_angle_mean( FT_Angle angle1,
                 FT_Angle angle2 )
  {
      return angle1 + FT_Angle_Diff( angle1, angle2 ) / 2;
  }
  
  
  static FT_Bool
  ft_cubic_is_small_enough( FT_Vector *base,
                            FT_Angle *angle_in,
                            FT_Angle *angle_mid,
                            FT_Angle *angle_out )
  {
      FT_Vector d1, d2, d3;
      FT_Angle theta1, theta2;
      FT_Int close1, close2, close3;
  
  
      d1.x = base[ 2 ].x - base[ 3 ].x;
      d1.y = base[ 2 ].y - base[ 3 ].y;
      d2.x = base[ 1 ].x - base[ 2 ].x;
      d2.y = base[ 1 ].y - base[ 2 ].y;
      d3.x = base[ 0 ].x - base[ 1 ].x;
      d3.y = base[ 0 ].y - base[ 1 ].y;
  
      close1 = FT_IS_SMALL( d1.x ) && FT_IS_SMALL( d1.y );
      close2 = FT_IS_SMALL( d2.x ) && FT_IS_SMALL( d2.y );
      close3 = FT_IS_SMALL( d3.x ) && FT_IS_SMALL( d3.y );
  
      if ( close1 )
      {
          if ( close2 )
          {
              if ( close3 )
              {
                  /* basically a point;                      */
                  /* do nothing to retain original direction */
              }
              else /* !close3 */
              {
                  *angle_in =
                  *angle_mid =
                  *angle_out = FT_Atan2( d3.x, d3.y );
              }
          }
          else /* !close2 */
          {
              if ( close3 )
              {
                  *angle_in =
                  *angle_mid =
                  *angle_out = FT_Atan2( d2.x, d2.y );
              }
              else /* !close3 */
              {
                  *angle_in =
                  *angle_mid = FT_Atan2( d2.x, d2.y );
                  *angle_out = FT_Atan2( d3.x, d3.y );
              }
          }
      }
      else /* !close1 */
      {
          if ( close2 )
          {
              if ( close3 )
              {
                  *angle_in =
                  *angle_mid =
                  *angle_out = FT_Atan2( d1.x, d1.y );
              }
              else /* !close3 */
              {
                  *angle_in = FT_Atan2( d1.x, d1.y );
                  *angle_out = FT_Atan2( d3.x, d3.y );
                  *angle_mid = ft_angle_mean( *angle_in, *angle_out );
              }
          }
          else /* !close2 */
          {
              if ( close3 )
              {
                  *angle_in = FT_Atan2( d1.x, d1.y );
                  *angle_mid =
                  *angle_out = FT_Atan2( d2.x, d2.y );
              }
              else /* !close3 */
              {
                  *angle_in = FT_Atan2( d1.x, d1.y );
                  *angle_mid = FT_Atan2( d2.x, d2.y );
                  *angle_out = FT_Atan2( d3.x, d3.y );
              }
          }
      }
  
      theta1 = ft_pos_abs( FT_Angle_Diff( *angle_in, *angle_mid ));
      theta2 = ft_pos_abs( FT_Angle_Diff( *angle_mid, *angle_out ));
  
      return FT_BOOL( theta1 < FT_SMALL_CUBIC_THRESHOLD &&
                      theta2 < FT_SMALL_CUBIC_THRESHOLD);
  }
  
  
  /*************************************************************************/
  /*************************************************************************/
  /*****                                                               *****/
  /*****                       STROKE BORDERS                          *****/
  /*****                                                               *****/
  /*************************************************************************/
  /*************************************************************************/
  
  typedef enum FT_StrokeTags_
  {
      FT_STROKE_TAG_ON = 1,   /* on-curve point  */
      FT_STROKE_TAG_CUBIC = 2,   /* cubic off-point */
      FT_STROKE_TAG_BEGIN = 4,   /* sub-path start  */
      FT_STROKE_TAG_END = 8    /* sub-path end    */
  
  } FT_StrokeTags;
  
  #define  FT_STROKE_TAG_BEGIN_END  ( FT_STROKE_TAG_BEGIN | FT_STROKE_TAG_END )
  
  typedef struct FT_StrokeBorderRec_
  {
      FT_UInt num_points;
      FT_UInt max_points;
      FT_Vector *points;
      FT_Byte *tags;
      FT_Bool movable;  /* TRUE for ends of lineto borders */
      FT_Int start;    /* index of current sub-path start point */
      FT_Memory memory;
      FT_Bool valid;
  
  } FT_StrokeBorderRec, *FT_StrokeBorder;
  
  
  static FT_Error
  ft_stroke_border_grow( FT_StrokeBorder border,
                         FT_UInt new_points )
  {
      FT_UInt old_max = border->max_points;
      FT_UInt new_max = border->num_points + new_points;
      FT_Error error = FT_Err_Ok;
  
  
      if ( new_max > old_max )
      {
          FT_UInt cur_max = old_max;
          FT_Memory memory = border->memory;
  
  
          while ( cur_max < new_max )
          {
              cur_max += ( cur_max >> 1 ) + 16;
          }
  
          if ( FT_RENEW_ARRAY( border->points, old_max, cur_max ) ||
               FT_RENEW_ARRAY( border->tags, old_max, cur_max ))
          {
              goto Exit;
          }
  
          border->max_points = cur_max;
      }
  
      Exit:
      return error;
  }
  
  
  static void
  ft_stroke_border_close( FT_StrokeBorder border,
                          FT_Bool reverse )
  {
      FT_UInt start = border->start;
      FT_UInt count = border->num_points;
  
  
      FT_ASSERT( border->start >= 0 );
  
      /* don't record empty paths! */
      if ( count <= start + 1U )
      {
          border->num_points = start;
      }
      else
      {
          /* copy the last point to the start of this sub-path, since */
          /* it contains the `adjusted' starting coordinates          */
          border->num_points = --count;
          border->points[ start ] = border->points[ count ];
  
          if ( reverse )
          {
              /* reverse the points */
              {
                  FT_Vector *vec1 = border->points + start + 1;
                  FT_Vector *vec2 = border->points + count - 1;
  
  
                  for ( ; vec1 < vec2; vec1++, vec2-- )
                  {
                      FT_Vector tmp;
  
  
                      tmp = *vec1;
                      *vec1 = *vec2;
                      *vec2 = tmp;
                  }
              }
  
              /* then the tags */
              {
                  FT_Byte *tag1 = border->tags + start + 1;
                  FT_Byte *tag2 = border->tags + count - 1;
  
  
                  for ( ; tag1 < tag2; tag1++, tag2-- )
                  {
                      FT_Byte tmp;
  
  
                      tmp = *tag1;
                      *tag1 = *tag2;
                      *tag2 = tmp;
                  }
              }
          }
  
          border->tags[ start ] |= FT_STROKE_TAG_BEGIN;
          border->tags[ count - 1 ] |= FT_STROKE_TAG_END;
      }
  
      border->start = -1;
      border->movable = FALSE;
  }
  
  
  static FT_Error
  ft_stroke_border_lineto( FT_StrokeBorder border,
                           FT_Vector *to,
                           FT_Bool movable )
  {
      FT_Error error = FT_Err_Ok;
  
  
      FT_ASSERT( border->start >= 0 );
  
      if ( border->movable )
      {
          /* move last point */
          border->points[ border->num_points - 1 ] = *to;
      }
      else
      {
          /* don't add zero-length lineto */
          if ( border->num_points > 0 &&
               FT_IS_SMALL( border->points[ border->num_points - 1 ].x - to->x ) &&
               FT_IS_SMALL( border->points[ border->num_points - 1 ].y - to->y ))
          {
              return error;
          }
  
          /* add one point */
          error = ft_stroke_border_grow( border, 1 );
          if ( !error )
          {
              FT_Vector *vec = border->points + border->num_points;
              FT_Byte *tag = border->tags + border->num_points;
  
  
              vec[ 0 ] = *to;
              tag[ 0 ] = FT_STROKE_TAG_ON;
  
              border->num_points += 1;
          }
      }
      border->movable = movable;
      return error;
  }
  
  
  static FT_Error
  ft_stroke_border_conicto( FT_StrokeBorder border,
                            FT_Vector *control,
                            FT_Vector *to )
  {
      FT_Error error;
  
  
      FT_ASSERT( border->start >= 0 );
  
      error = ft_stroke_border_grow( border, 2 );
      if ( !error )
      {
          FT_Vector *vec = border->points + border->num_points;
          FT_Byte *tag = border->tags + border->num_points;
  
  
          vec[ 0 ] = *control;
          vec[ 1 ] = *to;
  
          tag[ 0 ] = 0;
          tag[ 1 ] = FT_STROKE_TAG_ON;
  
          border->num_points += 2;
      }
  
      border->movable = FALSE;
  
      return error;
  }
  
  
  static FT_Error
  ft_stroke_border_cubicto( FT_StrokeBorder border,
                            FT_Vector *control1,
                            FT_Vector *control2,
                            FT_Vector *to )
  {
      FT_Error error;
  
  
      FT_ASSERT( border->start >= 0 );
  
      error = ft_stroke_border_grow( border, 3 );
      if ( !error )
      {
          FT_Vector *vec = border->points + border->num_points;
          FT_Byte *tag = border->tags + border->num_points;
  
  
          vec[ 0 ] = *control1;
          vec[ 1 ] = *control2;
          vec[ 2 ] = *to;
  
          tag[ 0 ] = FT_STROKE_TAG_CUBIC;
          tag[ 1 ] = FT_STROKE_TAG_CUBIC;
          tag[ 2 ] = FT_STROKE_TAG_ON;
  
          border->num_points += 3;
      }
  
      border->movable = FALSE;
  
      return error;
  }
  
  
  #define FT_ARC_CUBIC_ANGLE  ( FT_ANGLE_PI / 2 )
  
  
  static FT_Error
  ft_stroke_border_arcto( FT_StrokeBorder border,
                          FT_Vector *center,
                          FT_Fixed radius,
                          FT_Angle angle_start,
                          FT_Angle angle_diff )
  {
      FT_Angle total, angle, step, rotate, next, theta;
      FT_Vector a, b, a2, b2;
      FT_Fixed length;
      FT_Error error = FT_Err_Ok;
  
  
      /* compute start point */
      FT_Vector_From_Polar( &a, radius, angle_start );
      a.x += center->x;
      a.y += center->y;
  
      total = angle_diff;
      angle = angle_start;
      rotate = ( angle_diff >= 0 ) ? FT_ANGLE_PI2 : -FT_ANGLE_PI2;
  
      while ( total != 0 )
      {
          step = total;
          if ( step > FT_ARC_CUBIC_ANGLE)
          {
              step = FT_ARC_CUBIC_ANGLE;
          }
  
          else if ( step < -FT_ARC_CUBIC_ANGLE)
          {
              step = -FT_ARC_CUBIC_ANGLE;
          }
  
          next = angle + step;
          theta = step;
          if ( theta < 0 )
          {
              theta = -theta;
          }
  
          theta >>= 1;
  
          /* compute end point */
          FT_Vector_From_Polar( &b, radius, next );
          b.x += center->x;
          b.y += center->y;
  
          /* compute first and second control points */
          length = FT_MulDiv( radius, FT_Sin( theta ) * 4,
                              ( 0x10000L + FT_Cos( theta )) * 3 );
  
          FT_Vector_From_Polar( &a2, length, angle + rotate );
          a2.x += a.x;
          a2.y += a.y;
  
          FT_Vector_From_Polar( &b2, length, next - rotate );
          b2.x += b.x;
          b2.y += b.y;
  
          /* add cubic arc */
          error = ft_stroke_border_cubicto( border, &a2, &b2, &b );
          if ( error )
          {
              break;
          }
  
          /* process the rest of the arc ?? */
          a = b;
          total -= step;
          angle = next;
      }
  
      return error;
  }
  
  
  static FT_Error
  ft_stroke_border_moveto( FT_StrokeBorder border,
                           FT_Vector *to )
  {
      /* close current open path if any ? */
      if ( border->start >= 0 )
      {
          ft_stroke_border_close( border, FALSE );
      }
  
      border->start = border->num_points;
      border->movable = FALSE;
  
      return ft_stroke_border_lineto( border, to, FALSE );
  }
  
  
  static void
  ft_stroke_border_init( FT_StrokeBorder border,
                         FT_Memory memory )
  {
      border->memory = memory;
      border->points = NULL;
      border->tags = NULL;
  
      border->num_points = 0;
      border->max_points = 0;
      border->start = -1;
      border->valid = FALSE;
  }
  
  
  static void
  ft_stroke_border_reset( FT_StrokeBorder border )
  {
      border->num_points = 0;
      border->start = -1;
      border->valid = FALSE;
  }
  
  
  static void
  ft_stroke_border_done( FT_StrokeBorder border )
  {
      FT_Memory memory = border->memory;
  
  
      FT_FREE( border->points );
      FT_FREE( border->tags );
  
      border->num_points = 0;
      border->max_points = 0;
      border->start = -1;
      border->valid = FALSE;
  }
  
  
  static FT_Error
  ft_stroke_border_get_counts( FT_StrokeBorder border,
                               FT_UInt *anum_points,
                               FT_UInt *anum_contours )
  {
      FT_Error error = FT_Err_Ok;
      FT_UInt num_points = 0;
      FT_UInt num_contours = 0;
  
      FT_UInt count = border->num_points;
      FT_Vector *point = border->points;
      FT_Byte *tags = border->tags;
      FT_Int in_contour = 0;
  
  
      for ( ; count > 0; count--, num_points++, point++, tags++ )
      {
          if ( tags[ 0 ] & FT_STROKE_TAG_BEGIN )
          {
              if ( in_contour != 0 )
              {
                  goto Fail;
              }
  
              in_contour = 1;
          }
          else if ( in_contour == 0 )
          {
              goto Fail;
          }
  
          if ( tags[ 0 ] & FT_STROKE_TAG_END )
          {
              in_contour = 0;
              num_contours++;
          }
      }
  
      if ( in_contour != 0 )
      {
          goto Fail;
      }
  
      border->valid = TRUE;
  
      Exit:
      *anum_points = num_points;
      *anum_contours = num_contours;
      return error;
  
      Fail:
      num_points = 0;
      num_contours = 0;
      goto Exit;
  }
  
  
  static void
  ft_stroke_border_export( FT_StrokeBorder border,
                           FT_Outline *outline )
  {
      /* copy point locations */
      FT_ARRAY_COPY( outline->points + outline->n_points,
                     border->points,
                     border->num_points );
  
      /* copy tags */
      {
          FT_UInt count = border->num_points;
          FT_Byte *read = border->tags;
          FT_Byte *write = ( FT_Byte * ) outline->tags + outline->n_points;
  
  
          for ( ; count > 0; count--, read++, write++ )
          {
              if ( *read & FT_STROKE_TAG_ON )
              {
                  *write = FT_CURVE_TAG_ON;
              }
              else if ( *read & FT_STROKE_TAG_CUBIC )
              {
                  *write = FT_CURVE_TAG_CUBIC;
              }
              else
              {
                  *write = FT_CURVE_TAG_CONIC;
              }
          }
      }
  
      /* copy contours */
      {
          FT_UInt count = border->num_points;
          FT_Byte *tags = border->tags;
          FT_Short *write = outline->contours + outline->n_contours;
          FT_Short idx = ( FT_Short ) outline->n_points;
  
  
          for ( ; count > 0; count--, tags++, idx++ )
          {
              if ( *tags & FT_STROKE_TAG_END )
              {
                  *write++ = idx;
                  outline->n_contours++;
              }
          }
      }
  
      outline->n_points = ( short ) ( outline->n_points + border->num_points );
  
      FT_ASSERT( FT_Outline_Check( outline ) == 0 );
  }
  
  
  /*************************************************************************/
  /*************************************************************************/
  /*****                                                               *****/
  /*****                           STROKER                             *****/
  /*****                                                               *****/
  /*************************************************************************/
  /*************************************************************************/
  
  #define FT_SIDE_TO_ROTATE( s )   ( FT_ANGLE_PI2 - (s) * FT_ANGLE_PI )
  
  typedef struct FT_StrokerRec_
  {
      FT_Angle angle_in;             /* direction into curr join */
      FT_Angle angle_out;            /* direction out of join  */
      FT_Vector center;               /* current position */
      FT_Fixed line_length;          /* length of last lineto */
      FT_Bool first_point;          /* is this the start? */
      FT_Bool subpath_open;         /* is the subpath open? */
      FT_Angle subpath_angle;        /* subpath start direction */
      FT_Vector subpath_start;        /* subpath start position */
      FT_Fixed subpath_line_length;  /* subpath start lineto len */
      FT_Bool handle_wide_strokes;  /* use wide strokes logic? */
  
      FT_Stroker_LineCap line_cap;
      FT_Stroker_LineJoin line_join;
      FT_Stroker_LineJoin line_join_saved;
      FT_Fixed miter_limit;
      FT_Fixed radius;
  
      FT_StrokeBorderRec borders[2];
      FT_Library library;
  
  } FT_StrokerRec;
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_New( FT_Library
  
  library,
  
  FT_Stroker *astroker
  
  )
  {
  FT_Error error;
  
  FT_Memory memory;
  
  FT_Stroker stroker = NULL;
  
  
  if ( !library )
  return
  
  FT_Err_Invalid_Argument;
  
  memory = library->memory;
  
  if ( !
  
  FT_NEW( stroker )
  
  )
  {
  stroker->
  
  library = library;
  
  ft_stroke_border_init( &stroker->borders[0], memory );
  ft_stroke_border_init( &stroker->borders[1], memory );
  }
  
  *
  
  astroker = stroker;
  
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( void )
  
  FT_Stroker_Set( FT_Stroker
  
  stroker,
  
  FT_Fixed radius,
          FT_Stroker_LineCap
  
  line_cap,
  
  FT_Stroker_LineJoin line_join,
          FT_Fixed
  
  miter_limit )
  {
  stroker->
  
  radius = radius;
  
  stroker->
  
  line_cap = line_cap;
  
  stroker->
  
  line_join = line_join;
  
  stroker->
  
  miter_limit = miter_limit;
  
  /* ensure miter limit has sensible value */
  if ( stroker->miter_limit < 0x10000 )
  stroker->
  
  miter_limit = 0x10000;
  
  /* save line join style:                                           */
  /* line join style can be temporarily changed when stroking curves */
  stroker->
  
  line_join_saved = line_join;
  
  FT_Stroker_Rewind( stroker );
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( void )
  
  FT_Stroker_Rewind( FT_Stroker
  
  stroker )
  {
  if ( stroker )
  {
  ft_stroke_border_reset( &stroker->borders[0] );
  ft_stroke_border_reset( &stroker->borders[1] );
  }
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( void )
  
  FT_Stroker_Done( FT_Stroker
  
  stroker )
  {
  if ( stroker )
  {
  FT_Memory memory = stroker->library->memory;
  
  
  ft_stroke_border_done( &stroker->borders[0] );
  ft_stroke_border_done( &stroker->borders[1] );
  
  stroker->
  
  library = NULL;
  
  FT_FREE( stroker );
  
  }
  }
  
  
  /* create a circular arc at a corner or cap */
  static FT_Error
  ft_stroker_arcto( FT_Stroker stroker,
                    FT_Int side )
  {
      FT_Angle total, rotate;
      FT_Fixed radius = stroker->radius;
      FT_Error error = FT_Err_Ok;
      FT_StrokeBorder border = stroker->borders + side;
  
  
      rotate = FT_SIDE_TO_ROTATE( side );
  
      total = FT_Angle_Diff( stroker->angle_in, stroker->angle_out );
      if ( total == FT_ANGLE_PI )
      {
          total = -rotate * 2;
      }
  
      error = ft_stroke_border_arcto( border,
                                      &stroker->center,
                                      radius,
                                      stroker->angle_in + rotate,
                                      total );
      border->movable = FALSE;
      return error;
  }
  
  
  /* add a cap at the end of an opened path */
  static FT_Error
  ft_stroker_cap( FT_Stroker stroker,
                  FT_Angle angle,
                  FT_Int side )
  {
      FT_Error error = FT_Err_Ok;
  
  
      if ( stroker->line_cap == FT_STROKER_LINECAP_ROUND )
      {
          /* add a round cap */
          stroker->angle_in = angle;
          stroker->angle_out = angle + FT_ANGLE_PI;
  
          error = ft_stroker_arcto( stroker, side );
      }
      else if ( stroker->line_cap == FT_STROKER_LINECAP_SQUARE )
      {
          /* add a square cap */
          FT_Vector delta, delta2;
          FT_Angle rotate = FT_SIDE_TO_ROTATE( side );
          FT_Fixed radius = stroker->radius;
          FT_StrokeBorder border = stroker->borders + side;
  
  
          FT_Vector_From_Polar( &delta2, radius, angle + rotate );
          FT_Vector_From_Polar( &delta, radius, angle );
  
          delta.x += stroker->center.x + delta2.x;
          delta.y += stroker->center.y + delta2.y;
  
          error = ft_stroke_border_lineto( border, &delta, FALSE );
          if ( error )
          {
              goto Exit;
          }
  
          FT_Vector_From_Polar( &delta2, radius, angle - rotate );
          FT_Vector_From_Polar( &delta, radius, angle );
  
          delta.x += delta2.x + stroker->center.x;
          delta.y += delta2.y + stroker->center.y;
  
          error = ft_stroke_border_lineto( border, &delta, FALSE );
      }
      else if ( stroker->line_cap == FT_STROKER_LINECAP_BUTT )
      {
          /* add a butt ending */
          FT_Vector delta;
          FT_Angle rotate = FT_SIDE_TO_ROTATE( side );
          FT_Fixed radius = stroker->radius;
          FT_StrokeBorder border = stroker->borders + side;
  
  
          FT_Vector_From_Polar( &delta, radius, angle + rotate );
  
          delta.x += stroker->center.x;
          delta.y += stroker->center.y;
  
          error = ft_stroke_border_lineto( border, &delta, FALSE );
          if ( error )
          {
              goto Exit;
          }
  
          FT_Vector_From_Polar( &delta, radius, angle - rotate );
  
          delta.x += stroker->center.x;
          delta.y += stroker->center.y;
  
          error = ft_stroke_border_lineto( border, &delta, FALSE );
      }
  
      Exit:
      return error;
  }
  
  
  /* process an inside corner, i.e. compute intersection */
  static FT_Error
  ft_stroker_inside( FT_Stroker stroker,
                     FT_Int side,
                     FT_Fixed line_length )
  {
      FT_StrokeBorder border = stroker->borders + side;
      FT_Angle phi, theta, rotate;
      FT_Fixed length, thcos;
      FT_Vector delta;
      FT_Error error = FT_Err_Ok;
      FT_Bool intersect;          /* use intersection of lines? */
  
  
      rotate = FT_SIDE_TO_ROTATE( side );
  
      theta = FT_Angle_Diff( stroker->angle_in, stroker->angle_out ) / 2;
  
      /* Only intersect borders if between two lineto's and both */
      /* lines are long enough (line_length is zero for curves). */
      if ( !border->movable || line_length == 0 )
      {
          intersect = FALSE;
      }
      else
      {
          /* compute minimum required length of lines */
          FT_Fixed min_length = ft_pos_abs( FT_MulFix( stroker->radius,
                                                       FT_Tan( theta )));
  
  
          intersect = FT_BOOL( stroker->line_length >= min_length &&
                               line_length >= min_length );
      }
  
      if ( !intersect )
      {
          FT_Vector_From_Polar( &delta, stroker->radius,
                                stroker->angle_out + rotate );
          delta.x += stroker->center.x;
          delta.y += stroker->center.y;
  
          border->movable = FALSE;
      }
      else
      {
          /* compute median angle */
          phi = stroker->angle_in + theta;
  
          thcos = FT_Cos( theta );
  
          length = FT_DivFix( stroker->radius, thcos );
  
          FT_Vector_From_Polar( &delta, length, phi + rotate );
          delta.x += stroker->center.x;
          delta.y += stroker->center.y;
      }
  
      error = ft_stroke_border_lineto( border, &delta, FALSE );
  
      return error;
  }
  
  
  /* process an outside corner, i.e. compute bevel/miter/round */
  static FT_Error
  ft_stroker_outside( FT_Stroker stroker,
                      FT_Int side,
                      FT_Fixed line_length )
  {
      FT_StrokeBorder border = stroker->borders + side;
      FT_Error error;
      FT_Angle rotate;
  
  
      if ( stroker->line_join == FT_STROKER_LINEJOIN_ROUND )
      {
          error = ft_stroker_arcto( stroker, side );
      }
      else
      {
          /* this is a mitered (pointed) or beveled (truncated) corner */
          FT_Fixed sigma = 0, radius = stroker->radius;
          FT_Angle theta = 0, phi = 0;
          FT_Fixed thcos = 0;
          FT_Bool bevel, fixed_bevel;
  
  
          rotate = FT_SIDE_TO_ROTATE( side );
  
          bevel =
                  FT_BOOL( stroker->line_join == FT_STROKER_LINEJOIN_BEVEL );
  
          fixed_bevel =
                  FT_BOOL( stroker->line_join != FT_STROKER_LINEJOIN_MITER_VARIABLE );
  
          if ( !bevel )
          {
              theta = FT_Angle_Diff( stroker->angle_in, stroker->angle_out );
  
              if ( theta == FT_ANGLE_PI )
              {
                  theta = rotate;
                  phi = stroker->angle_in;
              }
              else
              {
                  theta /= 2;
                  phi = stroker->angle_in + theta + rotate;
              }
  
              thcos = FT_Cos( theta );
              sigma = FT_MulFix( stroker->miter_limit, thcos );
  
              /* is miter limit exceeded? */
              if ( sigma < 0x10000L )
              {
                  /* don't create variable bevels for very small deviations; */
                  /* FT_Sin(x) = 0 for x <= 57                               */
                  if ( fixed_bevel || ft_pos_abs( theta ) > 57 )
                  {
                      bevel = TRUE;
                  }
              }
          }
  
          if ( bevel )  /* this is a bevel (broken angle) */
          {
              if ( fixed_bevel )
              {
                  /* the outer corners are simply joined together */
                  FT_Vector delta;
  
  
                  /* add bevel */
                  FT_Vector_From_Polar( &delta,
                                        radius,
                                        stroker->angle_out + rotate );
                  delta.x += stroker->center.x;
                  delta.y += stroker->center.y;
  
                  border->movable = FALSE;
                  error = ft_stroke_border_lineto( border, &delta, FALSE );
              }
              else /* variable bevel */
              {
                  /* the miter is truncated */
                  FT_Vector middle, delta;
                  FT_Fixed length;
  
  
                  /* compute middle point */
                  FT_Vector_From_Polar( &middle,
                                        FT_MulFix( radius, stroker->miter_limit ),
                                        phi );
                  middle.x += stroker->center.x;
                  middle.y += stroker->center.y;
  
                  /* compute first angle point */
                  length = FT_MulFix( radius,
                                      FT_DivFix( 0x10000L - sigma,
                                                 ft_pos_abs( FT_Sin( theta ))));
  
                  FT_Vector_From_Polar( &delta, length, phi + rotate );
                  delta.x += middle.x;
                  delta.y += middle.y;
  
                  error = ft_stroke_border_lineto( border, &delta, FALSE );
                  if ( error )
                  {
                      goto Exit;
                  }
  
                  /* compute second angle point */
                  FT_Vector_From_Polar( &delta, length, phi - rotate );
                  delta.x += middle.x;
                  delta.y += middle.y;
  
                  error = ft_stroke_border_lineto( border, &delta, FALSE );
                  if ( error )
                  {
                      goto Exit;
                  }
  
                  /* finally, add an end point; only needed if not lineto */
                  /* (line_length is zero for curves)                     */
                  if ( line_length == 0 )
                  {
                      FT_Vector_From_Polar( &delta,
                                            radius,
                                            stroker->angle_out + rotate );
  
                      delta.x += stroker->center.x;
                      delta.y += stroker->center.y;
  
                      error = ft_stroke_border_lineto( border, &delta, FALSE );
                  }
              }
          }
          else /* this is a miter (intersection) */
          {
              FT_Fixed length;
              FT_Vector delta;
  
  
              length = FT_DivFix( stroker->radius, thcos );
  
              FT_Vector_From_Polar( &delta, length, phi );
              delta.x += stroker->center.x;
              delta.y += stroker->center.y;
  
              error = ft_stroke_border_lineto( border, &delta, FALSE );
              if ( error )
              {
                  goto Exit;
              }
  
              /* now add an end point; only needed if not lineto */
              /* (line_length is zero for curves)                */
              if ( line_length == 0 )
              {
                  FT_Vector_From_Polar( &delta,
                                        stroker->radius,
                                        stroker->angle_out + rotate );
                  delta.x += stroker->center.x;
                  delta.y += stroker->center.y;
  
                  error = ft_stroke_border_lineto( border, &delta, FALSE );
              }
          }
      }
  
      Exit:
      return error;
  }
  
  
  static FT_Error
  ft_stroker_process_corner( FT_Stroker stroker,
                             FT_Fixed line_length )
  {
      FT_Error error = FT_Err_Ok;
      FT_Angle turn;
      FT_Int inside_side;
  
  
      turn = FT_Angle_Diff( stroker->angle_in, stroker->angle_out );
  
      /* no specific corner processing is required if the turn is 0 */
      if ( turn == 0 )
      {
          goto Exit;
      }
  
      /* when we turn to the right, the inside side is 0 */
      inside_side = 0;
  
      /* otherwise, the inside side is 1 */
      if ( turn < 0 )
      {
          inside_side = 1;
      }
  
      /* process the inside side */
      error = ft_stroker_inside( stroker, inside_side, line_length );
      if ( error )
      {
          goto Exit;
      }
  
      /* process the outside side */
      error = ft_stroker_outside( stroker, 1 - inside_side, line_length );
  
      Exit:
      return error;
  }
  
  
  /* add two points to the left and right borders corresponding to the */
  /* start of the subpath                                              */
  static FT_Error
  ft_stroker_subpath_start( FT_Stroker stroker,
                            FT_Angle start_angle,
                            FT_Fixed line_length )
  {
      FT_Vector delta;
      FT_Vector point;
      FT_Error error;
      FT_StrokeBorder border;
  
  
      FT_Vector_From_Polar( &delta, stroker->radius,
                            start_angle + FT_ANGLE_PI2 );
  
      point.x = stroker->center.x + delta.x;
      point.y = stroker->center.y + delta.y;
  
      border = stroker->borders;
      error = ft_stroke_border_moveto( border, &point );
      if ( error )
      {
          goto Exit;
      }
  
      point.x = stroker->center.x - delta.x;
      point.y = stroker->center.y - delta.y;
  
      border++;
      error = ft_stroke_border_moveto( border, &point );
  
      /* save angle, position, and line length for last join */
      /* (line_length is zero for curves)                    */
      stroker->subpath_angle = start_angle;
      stroker->first_point = FALSE;
      stroker->subpath_line_length = line_length;
  
      Exit:
      return error;
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_LineTo( FT_Stroker
  
  stroker,
  
  FT_Vector *to
  
  )
  {
  FT_Error error = FT_Err_Ok;
  
  FT_StrokeBorder border;
  
  FT_Vector delta;
  
  FT_Angle angle;
  
  FT_Int side;
  
  FT_Fixed line_length;
  
  
  delta.
  
  x = to->x - stroker->center.x;
  
  delta.
  
  y = to->y - stroker->center.y;
  
  /* a zero-length lineto is a no-op; avoid creating a spurious corner */
  if ( delta.x == 0 && delta.y == 0 )
  goto
  
  Exit;
  
  /* compute length of line */
  line_length = FT_Vector_Length( &delta );
  
  angle = FT_Atan2( delta.x, delta.y );
  
  FT_Vector_From_Polar( &delta, stroker->radius, angle + FT_ANGLE_PI2 );
  
  /* process corner if necessary */
  if ( stroker->first_point )
  {
  /* This is the first segment of a subpath.  We need to     */
  /* add a point to each border at their respective starting */
  /* point locations.                                        */
  error = ft_stroker_subpath_start( stroker, angle, line_length );
  
  if ( error )
  goto
  
  Exit;
  
  }
  else
  {
  /* process the current corner */
  stroker->
  
  angle_out = angle;
  
  error = ft_stroker_process_corner( stroker, line_length );
  
  if ( error )
  goto
  
  Exit;
  
  }
  
  /* now add a line segment to both the `inside' and `outside' paths */
  for (
  
  border = stroker->borders, side = 1;
  
  side >= 0; side--, border++ )
  {
  FT_Vector point;
  
  
  point.
  
  x = to->x + delta.x;
  
  point.
  
  y = to->y + delta.y;
  
  /* the ends of lineto borders are movable */
  error = ft_stroke_border_lineto( border, &point, TRUE );
  
  if ( error )
  goto
  
  Exit;
  
  delta.
  
  x = -delta.x;
  
  delta.
  
  y = -delta.y;
  
  }
  
  stroker->
  
  angle_in = angle;
  
  stroker->
  
  center = *to;
  
  stroker->
  
  line_length = line_length;
  
  Exit:
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_ConicTo( FT_Stroker
  
  stroker,
  
  FT_Vector *control,
          FT_Vector
  
  *  to )
  {
  FT_Error error = FT_Err_Ok;
  
  FT_Vector bez_stack[34];
  
  FT_Vector *arc;
  
  FT_Vector *limit = bez_stack + 30;
  
  FT_Bool first_arc = TRUE;
  
  
  /* if all control points are coincident, this is a no-op; */
  /* avoid creating a spurious corner                       */
  if ( FT_IS_SMALL( stroker->center.x - control->x ) &&
  FT_IS_SMALL( stroker->center.y - control->y ) &&
  FT_IS_SMALL( control->x - to->x ) &&
  FT_IS_SMALL( control->y - to->y ))
  {
  stroker->
  
  center = *to;
  
  goto
  
  Exit;
  
  }
  
  arc = bez_stack;
  
  arc[0] = *
  
  to;
  
  arc[1] = *
  
  control;
  
  arc[2] = stroker->
  
  center;
  
  while ( arc >= bez_stack )
  {
  FT_Angle angle_in, angle_out;
  
  
  /* initialize with current direction */
  angle_in = angle_out = stroker->angle_in;
  
  if ( arc < limit                                             &&
  !
  
  ft_conic_is_small_enough( arc,
  
  &angle_in, &angle_out ))
  {
  if ( stroker->first_point )
  stroker->
  
  angle_in = angle_in;
  
  ft_conic_split( arc );
  
  arc += 2;
  continue;
  }
  
  if ( first_arc )
  {
  first_arc = FALSE;
  
  /* process corner if necessary */
  if ( stroker->first_point )
  
  error = ft_stroker_subpath_start( stroker, angle_in, 0 );
  
  else
  {
  stroker->
  
  angle_out = angle_in;
  
  error = ft_stroker_process_corner( stroker, 0 );
  
  }
  }
  else if (
  
  ft_pos_abs ( FT_Angle_Diff( stroker
  
  ->angle_in, angle_in )) >
  FT_SMALL_CONIC_THRESHOLD / 4                             )
  {
  /* if the deviation from one arc to the next is too great, */
  /* add a round corner                                      */
  stroker->
  
  center = arc[ 2 ];
  
  stroker->
  
  angle_out = angle_in;
  
  stroker->
  
  line_join = FT_STROKER_LINEJOIN_ROUND;
  
  error = ft_stroker_process_corner( stroker, 0 );
  
  /* reinstate line join style */
  stroker->
  
  line_join = stroker->line_join_saved;
  
  }
  
  if ( error )
  goto
  
  Exit;
  
  /* the arc's angle is small enough; we can add it directly to each */
  /* border                                                          */
  {
  FT_Vector ctrl, end;
  
  FT_Angle theta, phi, rotate, alpha0 = 0;
  
  FT_Fixed length;
  
  FT_StrokeBorder border;
  
  FT_Int side;
  
  
  theta = FT_Angle_Diff( angle_in, angle_out ) / 2;
  
  phi = angle_in + theta;
  
  length = FT_DivFix( stroker->radius, FT_Cos( theta ));
  
  /* compute direction of original arc */
  if ( stroker->handle_wide_strokes )
  
  alpha0 = FT_Atan2( arc[ 0 ].x - arc[ 2 ].x, arc[ 0 ].y - arc[ 2 ].y );
  
  for (
  
  border = stroker->borders, side = 0;
  
  side <= 1;
  side++, border++ )
  {
  rotate = FT_SIDE_TO_ROTATE( side );
  
  /* compute control point */
  FT_Vector_From_Polar( &ctrl, length, phi + rotate );
  ctrl.x += arc[1].
  
  x;
  
  ctrl.y += arc[1].
  
  y;
  
  /* compute end point */
  FT_Vector_From_Polar( &end, stroker->radius, angle_out + rotate );
  end.x += arc[0].
  
  x;
  
  end.y += arc[0].
  
  y;
  
  if ( stroker->handle_wide_strokes )
  {
  FT_Vector start;
  
  FT_Angle alpha1;
  
  
  /* determine whether the border radius is greater than the */
  /* radius of curvature of the original arc                 */
  start = border->points[ border->num_points - 1 ];
  
  alpha1 = FT_Atan2( end.x - start.x, end.y - start.y );
  
  /* is the direction of the border arc opposite to */
  /* that of the original arc? */
  if (
  
  ft_pos_abs ( FT_Angle_Diff( alpha0, alpha1 ))
  
  >
  FT_ANGLE_PI / 2                             )
  {
  FT_Angle beta, gamma;
  
  FT_Vector bvec, delta;
  
  FT_Fixed blen, sinA, sinB, alen;
  
  
  /* use the sine rule to find the intersection point */
  beta = FT_Atan2( arc[ 2 ].x - start.x, arc[ 2 ].y - start.y );
  
  gamma = FT_Atan2( arc[ 0 ].x - end.x, arc[ 0 ].y - end.y );
  
  bvec.
  
  x = end.x - start.x;
  
  bvec.
  
  y = end.y - start.y;
  
  blen = FT_Vector_Length( &bvec );
  
  sinA = ft_pos_abs( FT_Sin( alpha1 - gamma ));
  
  sinB = ft_pos_abs( FT_Sin( beta - gamma ));
  
  alen = FT_DivFix( FT_MulFix( blen, sinA ), sinB );
  
  FT_Vector_From_Polar( &delta, alen, beta );
  delta.x += start.
  
  x;
  
  delta.y += start.
  
  y;
  
  /* circumnavigate the negative sector backwards */
  border->
  
  movable = FALSE;
  
  error = ft_stroke_border_lineto( border, &delta, FALSE );
  
  if ( error )
  goto
  
  Exit;
  
  error = ft_stroke_border_lineto( border, &end, FALSE );
  
  if ( error )
  goto
  
  Exit;
  
  error = ft_stroke_border_conicto( border, &ctrl, &start );
  
  if ( error )
  goto
  
  Exit;
  
  /* and then move to the endpoint */
  error = ft_stroke_border_lineto( border, &end, FALSE );
  
  if ( error )
  goto
  
  Exit;
  
  continue;
  }
  
  /* else fall through */
  }
  
  /* simply add an arc */
  error = ft_stroke_border_conicto( border, &ctrl, &end );
  
  if ( error )
  goto
  
  Exit;
  
  }
  }
  
  arc -= 2;
  
  stroker->
  
  angle_in = angle_out;
  
  }
  
  stroker->
  
  center = *to;
  
  Exit:
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_CubicTo( FT_Stroker
  
  stroker,
  
  FT_Vector *control1,
          FT_Vector
  
  *  control2,
  
  FT_Vector *to
  
  )
  {
  FT_Error error = FT_Err_Ok;
  
  FT_Vector bez_stack[37];
  
  FT_Vector *arc;
  
  FT_Vector *limit = bez_stack + 32;
  
  FT_Bool first_arc = TRUE;
  
  
  /* if all control points are coincident, this is a no-op; */
  /* avoid creating a spurious corner */
  if ( FT_IS_SMALL( stroker->center.x - control1->x ) &&
  FT_IS_SMALL( stroker->center.y - control1->y ) &&
  FT_IS_SMALL( control1->x - control2->x ) &&
  FT_IS_SMALL( control1->y - control2->y ) &&
  FT_IS_SMALL( control2->x - to->x ) &&
  FT_IS_SMALL( control2->y - to->y ))
  {
  stroker->
  
  center = *to;
  
  goto
  
  Exit;
  
  }
  
  arc = bez_stack;
  
  arc[0] = *
  
  to;
  
  arc[1] = *
  
  control2;
  
  arc[2] = *
  
  control1;
  
  arc[3] = stroker->
  
  center;
  
  while ( arc >= bez_stack )
  {
  FT_Angle angle_in, angle_mid, angle_out;
  
  
  /* initialize with current direction */
  angle_in = angle_out = angle_mid = stroker->angle_in;
  
  if ( arc < limit                                         &&
  !
  
  ft_cubic_is_small_enough( arc,
  
  &angle_in,
  &angle_mid, &angle_out ))
  {
  if ( stroker->first_point )
  stroker->
  
  angle_in = angle_in;
  
  ft_cubic_split( arc );
  
  arc += 3;
  continue;
  }
  
  if ( first_arc )
  {
  first_arc = FALSE;
  
  /* process corner if necessary */
  if ( stroker->first_point )
  
  error = ft_stroker_subpath_start( stroker, angle_in, 0 );
  
  else
  {
  stroker->
  
  angle_out = angle_in;
  
  error = ft_stroker_process_corner( stroker, 0 );
  
  }
  }
  else if (
  
  ft_pos_abs ( FT_Angle_Diff( stroker
  
  ->angle_in, angle_in )) >
  FT_SMALL_CUBIC_THRESHOLD / 4                             )
  {
  /* if the deviation from one arc to the next is too great, */
  /* add a round corner                                      */
  stroker->
  
  center = arc[ 3 ];
  
  stroker->
  
  angle_out = angle_in;
  
  stroker->
  
  line_join = FT_STROKER_LINEJOIN_ROUND;
  
  error = ft_stroker_process_corner( stroker, 0 );
  
  /* reinstate line join style */
  stroker->
  
  line_join = stroker->line_join_saved;
  
  }
  
  if ( error )
  goto
  
  Exit;
  
  /* the arc's angle is small enough; we can add it directly to each */
  /* border                                                          */
  {
  FT_Vector ctrl1, ctrl2, end;
  
  FT_Angle theta1, phi1, theta2, phi2, rotate, alpha0 = 0;
  
  FT_Fixed length1, length2;
  
  FT_StrokeBorder border;
  
  FT_Int side;
  
  
  theta1 = FT_Angle_Diff( angle_in, angle_mid ) / 2;
  
  theta2 = FT_Angle_Diff( angle_mid, angle_out ) / 2;
  
  phi1 = ft_angle_mean( angle_in, angle_mid );
  
  phi2 = ft_angle_mean( angle_mid, angle_out );
  
  length1 = FT_DivFix( stroker->radius, FT_Cos( theta1 ));
  
  length2 = FT_DivFix( stroker->radius, FT_Cos( theta2 ));
  
  /* compute direction of original arc */
  if ( stroker->handle_wide_strokes )
  
  alpha0 = FT_Atan2( arc[ 0 ].x - arc[ 3 ].x, arc[ 0 ].y - arc[ 3 ].y );
  
  for (
  
  border = stroker->borders, side = 0;
  
  side <= 1;
  side++, border++ )
  {
  rotate = FT_SIDE_TO_ROTATE( side );
  
  /* compute control points */
  FT_Vector_From_Polar( &ctrl1, length1, phi1 + rotate );
  ctrl1.x += arc[2].
  
  x;
  
  ctrl1.y += arc[2].
  
  y;
  
  FT_Vector_From_Polar( &ctrl2, length2, phi2 + rotate );
  ctrl2.x += arc[1].
  
  x;
  
  ctrl2.y += arc[1].
  
  y;
  
  /* compute end point */
  FT_Vector_From_Polar( &end, stroker->radius, angle_out + rotate );
  end.x += arc[0].
  
  x;
  
  end.y += arc[0].
  
  y;
  
  if ( stroker->handle_wide_strokes )
  {
  FT_Vector start;
  
  FT_Angle alpha1;
  
  
  /* determine whether the border radius is greater than the */
  /* radius of curvature of the original arc                 */
  start = border->points[ border->num_points - 1 ];
  
  alpha1 = FT_Atan2( end.x - start.x, end.y - start.y );
  
  /* is the direction of the border arc opposite to */
  /* that of the original arc? */
  if (
  
  ft_pos_abs ( FT_Angle_Diff( alpha0, alpha1 ))
  
  >
  FT_ANGLE_PI / 2                             )
  {
  FT_Angle beta, gamma;
  
  FT_Vector bvec, delta;
  
  FT_Fixed blen, sinA, sinB, alen;
  
  
  /* use the sine rule to find the intersection point */
  beta = FT_Atan2( arc[ 3 ].x - start.x, arc[ 3 ].y - start.y );
  
  gamma = FT_Atan2( arc[ 0 ].x - end.x, arc[ 0 ].y - end.y );
  
  bvec.
  
  x = end.x - start.x;
  
  bvec.
  
  y = end.y - start.y;
  
  blen = FT_Vector_Length( &bvec );
  
  sinA = ft_pos_abs( FT_Sin( alpha1 - gamma ));
  
  sinB = ft_pos_abs( FT_Sin( beta - gamma ));
  
  alen = FT_DivFix( FT_MulFix( blen, sinA ), sinB );
  
  FT_Vector_From_Polar( &delta, alen, beta );
  delta.x += start.
  
  x;
  
  delta.y += start.
  
  y;
  
  /* circumnavigate the negative sector backwards */
  border->
  
  movable = FALSE;
  
  error = ft_stroke_border_lineto( border, &delta, FALSE );
  
  if ( error )
  goto
  
  Exit;
  
  error = ft_stroke_border_lineto( border, &end, FALSE );
  
  if ( error )
  goto
  
  Exit;
  
  error = ft_stroke_border_cubicto( border,
                                    &ctrl2,
                                    &ctrl1,
                                    &start );
  
  if ( error )
  goto
  
  Exit;
  
  /* and then move to the endpoint */
  error = ft_stroke_border_lineto( border, &end, FALSE );
  
  if ( error )
  goto
  
  Exit;
  
  continue;
  }
  
  /* else fall through */
  }
  
  /* simply add an arc */
  error = ft_stroke_border_cubicto( border, &ctrl1, &ctrl2, &end );
  
  if ( error )
  goto
  
  Exit;
  
  }
  }
  
  arc -= 3;
  
  stroker->
  
  angle_in = angle_out;
  
  }
  
  stroker->
  
  center = *to;
  
  Exit:
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_BeginSubPath( FT_Stroker
  
  stroker,
  
  FT_Vector *to,
          FT_Bool
  
  open )
  {
  /* We cannot process the first point, because there is not enough      */
  /* information regarding its corner/cap.  The latter will be processed */
  /* in the `FT_Stroker_EndSubPath' routine.                             */
  /*                                                                     */
  stroker->
  
  first_point = TRUE;
  
  stroker->
  
  center = *to;
  
  stroker->
  
  subpath_open = open;
  
  /* Determine if we need to check whether the border radius is greater */
  /* than the radius of curvature of a curve, to handle this case       */
  /* specially.  This is only required if bevel joins or butt caps may  */
  /* be created, because round & miter joins and round & square caps    */
  /* cover the negative sector created with wide strokes.               */
  stroker->
  
  handle_wide_strokes =
  FT_BOOL( stroker->line_join != FT_STROKER_LINEJOIN_ROUND ||
           ( stroker->subpath_open &&
             stroker->line_cap == FT_STROKER_LINECAP_BUTT ));
  
  /* record the subpath start point for each border */
  stroker->
  
  subpath_start = *to;
  
  stroker->
  
  angle_in = 0;
  
  return
  
  FT_Err_Ok;
  
  }
  
  
  static FT_Error
  ft_stroker_add_reverse_left( FT_Stroker stroker,
                               FT_Bool open )
  {
      FT_StrokeBorder right = stroker->borders + 0;
      FT_StrokeBorder left = stroker->borders + 1;
      FT_Int new_points;
      FT_Error error = FT_Err_Ok;
  
  
      FT_ASSERT( left->start >= 0 );
  
      new_points = left->num_points - left->start;
      if ( new_points > 0 )
      {
          error = ft_stroke_border_grow( right, ( FT_UInt ) new_points );
          if ( error )
          {
              goto Exit;
          }
  
          {
              FT_Vector * dst_point = right->points + right->num_points;
              FT_Byte *dst_tag = right->tags + right->num_points;
              FT_Vector * src_point = left->points + left->num_points - 1;
              FT_Byte *src_tag = left->tags + left->num_points - 1;
  
  
              while ( src_point >= left->points + left->start )
              {
                  *dst_point = *src_point;
                  *dst_tag = *src_tag;
  
                  if ( open )
                  {
                      dst_tag[ 0 ] &= ~FT_STROKE_TAG_BEGIN_END;
                  }
                  else
                  {
                      FT_Byte ttag =
                              ( FT_Byte )( dst_tag[ 0 ] & FT_STROKE_TAG_BEGIN_END);
  
  
                      /* switch begin/end tags if necessary */
                      if ( ttag == FT_STROKE_TAG_BEGIN ||
                           ttag == FT_STROKE_TAG_END )
                      {
                          dst_tag[ 0 ] ^= FT_STROKE_TAG_BEGIN_END;
                      }
                  }
  
                  src_point--;
                  src_tag--;
                  dst_point++;
                  dst_tag++;
              }
          }
  
          left->num_points = left->start;
          right->num_points += new_points;
  
          right->movable = FALSE;
          left->movable = FALSE;
      }
  
      Exit:
      return error;
  }
  
  
  /* documentation is in ftstroke.h */
  
  /* there's a lot of magic in this function! */
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_EndSubPath( FT_Stroker
  
  stroker )
  {
  FT_Error error = FT_Err_Ok;
  
  
  if ( stroker->subpath_open )
  {
  FT_StrokeBorder right = stroker->borders;
  
  
  /* All right, this is an opened path, we need to add a cap between */
  /* right & left, add the reverse of left, then add a final cap     */
  /* between left & right.                                           */
  error = ft_stroker_cap( stroker, stroker->angle_in, 0 );
  
  if ( error )
  goto
  
  Exit;
  
  /* add reversed points from `left' to `right' */
  error = ft_stroker_add_reverse_left( stroker, TRUE );
  
  if ( error )
  goto
  
  Exit;
  
  /* now add the final cap */
  stroker->
  
  center = stroker->subpath_start;
  
  error = ft_stroker_cap( stroker,
                          stroker->subpath_angle + FT_ANGLE_PI, 0 );
  
  if ( error )
  goto
  
  Exit;
  
  /* Now end the right subpath accordingly.  The left one is */
  /* rewind and doesn't need further processing.             */
  ft_stroke_border_close( right, FALSE
  
  );
  }
  else
  {
  FT_Angle turn;
  
  FT_Int inside_side;
  
  
  /* close the path if needed */
  if ( stroker->center.x != stroker->subpath_start.x ||
  stroker->center.y != stroker->subpath_start.y )
  {
  error = FT_Stroker_LineTo( stroker, &stroker->subpath_start );
  
  if ( error )
  goto
  
  Exit;
  
  }
  
  /* process the corner */
  stroker->
  
  angle_out = stroker->subpath_angle;
  
  turn = FT_Angle_Diff( stroker->angle_in,
                        stroker->angle_out );
  
  /* no specific corner processing is required if the turn is 0 */
  if ( turn != 0 )
  {
  /* when we turn to the right, the inside side is 0 */
  inside_side = 0;
  
  /* otherwise, the inside side is 1 */
  if ( turn < 0 )
  
  inside_side = 1;
  
  error = ft_stroker_inside( stroker,
                             inside_side,
                             stroker->subpath_line_length );
  
  if ( error )
  goto
  
  Exit;
  
  /* process the outside side */
  error = ft_stroker_outside( stroker,
                              1 - inside_side,
                              stroker->subpath_line_length );
  
  if ( error )
  goto
  
  Exit;
  
  }
  
  /* then end our two subpaths */
  ft_stroke_border_close( stroker
  
  ->borders + 0, FALSE );
  
  ft_stroke_border_close( stroker
  
  ->borders + 1, TRUE );
  }
  
  Exit:
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_GetBorderCounts( FT_Stroker
  
  stroker,
  
  FT_StrokerBorder border,
          FT_UInt
  
  *anum_points,
  
  FT_UInt *anum_contours
  
  )
  {
  FT_UInt num_points = 0, num_contours = 0;
  
  FT_Error error;
  
  
  if ( !stroker || border > 1 )
  {
  error = FT_Err_Invalid_Argument;
  
  goto
  
  Exit;
  
  }
  
  error = ft_stroke_border_get_counts( stroker->borders + border,
                                       &num_points, &num_contours );
  
  Exit:
  if ( anum_points )
  *
  
  anum_points = num_points;
  
  if ( anum_contours )
  *
  
  anum_contours = num_contours;
  
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_GetCounts( FT_Stroker
  
  stroker,
  
  FT_UInt *anum_points,
          FT_UInt
  
  *anum_contours )
  {
  FT_UInt count1, count2, num_points = 0;
  
  FT_UInt count3, count4, num_contours = 0;
  
  FT_Error error;
  
  
  error = ft_stroke_border_get_counts( stroker->borders + 0,
                                       &count1, &count2 );
  
  if ( error )
  goto
  
  Exit;
  
  error = ft_stroke_border_get_counts( stroker->borders + 1,
                                       &count3, &count4 );
  
  if ( error )
  goto
  
  Exit;
  
  num_points = count1 + count3;
  
  num_contours = count2 + count4;
  
  Exit:
  *
  
  anum_points = num_points;
  
  *
  
  anum_contours = num_contours;
  
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( void )
  
  FT_Stroker_ExportBorder( FT_Stroker
  
  stroker,
  
  FT_StrokerBorder border,
          FT_Outline
  
  *       outline )
  {
  if ( border == FT_STROKER_BORDER_LEFT  ||
  border == FT_STROKER_BORDER_RIGHT )
  {
  FT_StrokeBorder sborder = &stroker->borders[ border ];
  
  
  if ( sborder->valid )
  
  ft_stroke_border_export( sborder, outline
  
  );
  }
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( void )
  
  FT_Stroker_Export( FT_Stroker
  
  stroker,
  
  FT_Outline *outline
  
  )
  {
  FT_Stroker_ExportBorder( stroker, FT_STROKER_BORDER_LEFT, outline
  
  );
  
  FT_Stroker_ExportBorder( stroker, FT_STROKER_BORDER_RIGHT, outline
  
  );
  }
  
  
  /* documentation is in ftstroke.h */
  
  /*
   *  The following is very similar to FT_Outline_Decompose, except
   *  that we do support opened paths, and do not scale the outline.
   */
  FT_EXPORT_DEF( FT_Error )
  
  FT_Stroker_ParseOutline( FT_Stroker
  
  stroker,
  
  FT_Outline *outline,
          FT_Bool
  
  opened )
  {
  FT_Vector v_last;
  
  FT_Vector v_control;
  
  FT_Vector v_start;
  
  FT_Vector *point;
  
  FT_Vector *limit;
  
  char *tags;
  
  FT_Error error;
  
  FT_Int n;         /* index of contour in outline     */
  FT_UInt first;     /* index of first point in contour */
  FT_Int tag;       /* current point's state           */
  
  
  if ( !outline || !stroker )
  return
  
  FT_Err_Invalid_Argument;
  
  FT_Stroker_Rewind( stroker );
  
  first = 0;
  
  for (
  
  n = 0;
  
  n < outline->
  
  n_contours;
  
  n++ )
  {
  FT_UInt last;  /* index of last point in contour */
  
  
  last = outline->contours[ n ];
  
  limit = outline->points + last;
  
  /* skip empty points; we don't stroke these */
  if ( last <= first )
  {
  first = last + 1;
  
  continue;
  }
  
  v_start = outline->points[ first ];
  
  v_last = outline->points[ last ];
  
  v_control = v_start;
  
  point = outline->points + first;
  
  tags = outline->tags + first;
  
  tag = FT_CURVE_TAG( tags[ 0 ] );
  
  /* A contour cannot start with a cubic control point! */
  if ( tag == FT_CURVE_TAG_CUBIC )
  goto
  
  Invalid_Outline;
  
  /* check first point to determine origin */
  if ( tag == FT_CURVE_TAG_CONIC )
  {
  /* First point is conic control.  Yes, this happens. */
  if (
  
  FT_CURVE_TAG( outline
  
  ->tags[last] ) == FT_CURVE_TAG_ON )
  {
  /* start at last point if it is on the curve */
  v_start = v_last;
  
  limit--;
  }
  else
  {
  /* if both first and last points are conic, */
  /* start at their middle                    */
  v_start.
  
  x = ( v_start.x + v_last.x ) / 2;
  
  v_start.
  
  y = ( v_start.y + v_last.y ) / 2;
  
  }
  point--;
  tags--;
  }
  
  error = FT_Stroker_BeginSubPath( stroker, &v_start, opened );
  
  if ( error )
  goto
  
  Exit;
  
  while ( point < limit )
  {
  point++;
  tags++;
  
  tag = FT_CURVE_TAG( tags[ 0 ] );
  
  switch ( tag )
  {
  case FT_CURVE_TAG_ON:  /* emit a single line_to */
  {
  FT_Vector vec;
  
  
  vec.
  
  x = point->x;
  
  vec.
  
  y = point->y;
  
  error = FT_Stroker_LineTo( stroker, &vec );
  
  if ( error )
  goto
  
  Exit;
  
  continue;
  }
  
  case FT_CURVE_TAG_CONIC:  /* consume conic arcs */
  v_control.
  
  x = point->x;
  
  v_control.
  
  y = point->y;
  
  Do_Conic:
  if ( point < limit )
  {
  FT_Vector vec;
  
  FT_Vector v_middle;
  
  
  point++;
  tags++;
  
  tag = FT_CURVE_TAG( tags[ 0 ] );
  
  vec = point[ 0 ];
  
  if ( tag == FT_CURVE_TAG_ON )
  {
  error = FT_Stroker_ConicTo( stroker, &v_control, &vec );
  
  if ( error )
  goto
  
  Exit;
  
  continue;
  }
  
  if ( tag != FT_CURVE_TAG_CONIC )
  goto
  
  Invalid_Outline;
  
  v_middle.
  
  x = ( v_control.x + vec.x ) / 2;
  
  v_middle.
  
  y = ( v_control.y + vec.y ) / 2;
  
  error = FT_Stroker_ConicTo( stroker, &v_control, &v_middle );
  
  if ( error )
  goto
  
  Exit;
  
  v_control = vec;
  
  goto
  
  Do_Conic;
  
  }
  
  error = FT_Stroker_ConicTo( stroker, &v_control, &v_start );
  
  goto
  
  Close;
  
  default:  /* FT_CURVE_TAG_CUBIC */
  {
  FT_Vector vec1, vec2;
  
  
  if ( point + 1 > limit                             ||
  
  FT_CURVE_TAG( tags[1] )
  
  != FT_CURVE_TAG_CUBIC )
  goto
  
  Invalid_Outline;
  
  point += 2;
  tags  += 2;
  
  vec1 = point[ -2 ];
  
  vec2 = point[ -1 ];
  
  if ( point <= limit )
  {
  FT_Vector vec;
  
  
  vec = point[ 0 ];
  
  error = FT_Stroker_CubicTo( stroker, &vec1, &vec2, &vec );
  
  if ( error )
  goto
  
  Exit;
  
  continue;
  }
  
  error = FT_Stroker_CubicTo( stroker, &vec1, &vec2, &v_start );
  
  goto
  
  Close;
  
  }
  }
  }
  
  Close:
  if ( error )
  goto
  
  Exit;
  
  /* don't try to end the path if no segments have been generated */
  if ( !stroker->first_point )
  {
  error = FT_Stroker_EndSubPath( stroker );
  
  if ( error )
  goto
  
  Exit;
  
  }
  
  first = last + 1;
  
  }
  
  return
  
  FT_Err_Ok;
  
  Exit:
  return
  
  error;
  
  Invalid_Outline:
  return
  
  FT_Err_Invalid_Outline;
  
  }
  
  
  /* declare an extern to access `ft_outline_glyph_class' globally     */
  /* allocated  in `ftglyph.c', and use the FT_OUTLINE_GLYPH_CLASS_GET */
  /* macro to access it when FT_CONFIG_OPTION_PIC is defined           */
  #ifndef FT_CONFIG_OPTION_PIC
  
  extern const FT_Glyph_Class ft_outline_glyph_class;
  
  #endif
  
  #include "basepic.h"
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Glyph_Stroke( FT_Glyph
  
  *pglyph,
  
  FT_Stroker stroker,
          FT_Bool
  
  destroy )
  {
  FT_Error error = FT_Err_Invalid_Argument;
  
  FT_Glyph glyph = NULL;
  
  FT_Library library = stroker->library;
  
  FT_UNUSED( library );
  
  
  if ( pglyph == NULL )
  goto
  
  Exit;
  
  glyph = *pglyph;
  
  if ( glyph == NULL || glyph->clazz != FT_OUTLINE_GLYPH_CLASS_GET )
  goto
  
  Exit;
  
  {
  FT_Glyph copy;
  
  
  error = FT_Glyph_Copy( glyph, &copy );
  
  if ( error )
  goto
  
  Exit;
  
  glyph = copy;
  
  }
  
  {
  FT_OutlineGlyph oglyph = ( FT_OutlineGlyph ) glyph;
  
  FT_Outline *outline = &oglyph->outline;
  
  FT_UInt num_points, num_contours;
  
  
  error = FT_Stroker_ParseOutline( stroker, outline, FALSE );
  
  if ( error )
  goto
  
  Fail;
  
  (void)
  
  FT_Stroker_GetCounts( stroker,
  
  &num_points, &num_contours );
  
  FT_Outline_Done( glyph
  
  ->library, outline );
  
  error = FT_Outline_New( glyph->library,
                          num_points, num_contours, outline );
  
  if ( error )
  goto
  
  Fail;
  
  outline->
  
  n_points = 0;
  
  outline->
  
  n_contours = 0;
  
  FT_Stroker_Export( stroker, outline
  
  );
  }
  
  if ( destroy )
  
  FT_Done_Glyph( *pglyph );
  
  *
  
  pglyph = glyph;
  
  goto
  
  Exit;
  
  Fail:
  
  FT_Done_Glyph( glyph );
  
  glyph = NULL;
  
  if ( !destroy )
  *
  
  pglyph = NULL;
  
  Exit:
  return
  
  error;
  
  }
  
  
  /* documentation is in ftstroke.h */
  
  FT_EXPORT_DEF( FT_Error )
  
  FT_Glyph_StrokeBorder( FT_Glyph
  
  *pglyph,
  
  FT_Stroker stroker,
          FT_Bool
  
  inside,
  
  FT_Bool destroy
  
  )
  {
  FT_Error error = FT_Err_Invalid_Argument;
  
  FT_Glyph glyph = NULL;
  
  FT_Library library = stroker->library;
  
  FT_UNUSED( library );
  
  
  if ( pglyph == NULL )
  goto
  
  Exit;
  
  glyph = *pglyph;
  
  if ( glyph == NULL || glyph->clazz != FT_OUTLINE_GLYPH_CLASS_GET )
  goto
  
  Exit;
  
  {
  FT_Glyph copy;
  
  
  error = FT_Glyph_Copy( glyph, &copy );
  
  if ( error )
  goto
  
  Exit;
  
  glyph = copy;
  
  }
  
  {
  FT_OutlineGlyph oglyph = ( FT_OutlineGlyph ) glyph;
  
  FT_StrokerBorder border;
  
  FT_Outline *outline = &oglyph->outline;
  
  FT_UInt num_points, num_contours;
  
  
  border = FT_Outline_GetOutsideBorder( outline );
  
  if ( inside )
  {
  if ( border == FT_STROKER_BORDER_LEFT )
  
  border = FT_STROKER_BORDER_RIGHT;
  
  else
  
  border = FT_STROKER_BORDER_LEFT;
  
  }
  
  error = FT_Stroker_ParseOutline( stroker, outline, FALSE );
  
  if ( error )
  goto
  
  Fail;
  
  (void)
  
  FT_Stroker_GetBorderCounts( stroker, border,
  
  &num_points, &num_contours );
  
  FT_Outline_Done( glyph
  
  ->library, outline );
  
  error = FT_Outline_New( glyph->library,
                          num_points,
                          num_contours,
                          outline );
  
  if ( error )
  goto
  
  Fail;
  
  outline->
  
  n_points = 0;
  
  outline->
  
  n_contours = 0;
  
  FT_Stroker_ExportBorder( stroker, border, outline
  
  );
  }
  
  if ( destroy )
  
  FT_Done_Glyph( *pglyph );
  
  *
  
  pglyph = glyph;
  
  goto
  
  Exit;
  
  Fail:
  
  FT_Done_Glyph( glyph );
  
  glyph = NULL;
  
  if ( !destroy )
  *
  
  pglyph = NULL;
  
  Exit:
  return
  
  error;
  
  }
  
  
  /* END */
  