freetype2/src/base/ftstroke.c

/***************************************************************************/
/*                                                                         */
/*  ftstroke.c                                                             */
/*                                                                         */
/*    FreeType path stroker (body).                                        */
/*                                                                         */
/*  Copyright 2002, 2003 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


  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 ;
  }


  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_RIGHT
                                        : FT_STROKER_BORDER_LEFT ;
  }


 /***************************************************************************/
 /***************************************************************************/
 /*****                                                                 *****/
 /*****                       BEZIER COMPUTATIONS                       *****/
 /*****                                                                 *****/
 /***************************************************************************/
 /***************************************************************************/

#define FT_SMALL_CONIC_THRESHOLD  ( FT_ANGLE_PI / 6 )
#define FT_SMALL_CUBIC_THRESHOLD  ( FT_ANGLE_PI / 6 )
#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 )
        *angle_in = *angle_out = 0;
      else
        *angle_in = *angle_out = FT_Atan2( d2.x, d2.y );
    }
    else 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;
  }


  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 || close3 )
    {
      if ( close2 )
      {
        /* basically a point */
        *angle_in = *angle_out = *angle_mid = 0;
      }
      else if ( close1 )
      {
        *angle_in  = *angle_mid = FT_Atan2( d2.x, d2.y );
        *angle_out = FT_Atan2( d3.x, d3.y );
      }
      else  /* close2 */
      {
        *angle_in  = FT_Atan2( d1.x, d1.y );
        *angle_mid = *angle_out = FT_Atan2( d2.x, d2.y );
      }
    }
    else if ( close2 )
    {
      *angle_in  = *angle_mid = FT_Atan2( d1.x, d1.y );
      *angle_out = FT_Atan2( d3.x, d3.y );
    }
    else
    {
      *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_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;


  typedef struct  FT_StrokeBorderRec_
  {
    FT_UInt     num_points;
    FT_UInt     max_points;
    FT_Vector*  points;
    FT_Byte*    tags;
    FT_Bool     movable;
    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   = 0;


    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_ASSERT( border->start >= 0 );

    /* don't record empty paths! */
    if ( border->num_points > (FT_UInt)border->start )
    {
      border->tags[border->start         ] |= FT_STROKE_TAG_BEGIN;
      border->tags[border->num_points - 1] |= FT_STROKE_TAG_END;
    }

    border->start   = -1;
    border->movable = 0;
  }


  static FT_Error
  ft_stroke_border_lineto( FT_StrokeBorder  border,
                           FT_Vector*       to,
                           FT_Bool          movable )
  {
    FT_Error  error = 0;


    FT_ASSERT( border->start >= 0 );

    if ( border->movable )
    {
      /* move last point */
      border->points[border->num_points - 1] = *to;
    }
    else
    {
      /* 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 = 0;
    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 = 0;
    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 = 0;


    /* 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 );

    border->start   = border->num_points;
    border->movable = 0;

    return ft_stroke_border_lineto( border, to, 0 );
  }


  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      = 0;
  }


  static void
  ft_stroke_border_reset( FT_StrokeBorder  border )
  {
    border->num_points = 0;
    border->start      = -1;
    border->valid      = 0;
  }


  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      = 0;
  }


  static FT_Error
  ft_stroke_border_get_counts( FT_StrokeBorder  border,
                               FT_UInt         *anum_points,
                               FT_UInt         *anum_contours )
  {
    FT_Error  error        = 0;
    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 )
      {
        if ( in_contour == 0 )
          goto Fail;

        in_contour = 0;
        num_contours++;
      }
    }

    if ( in_contour != 0 )
      goto Fail;

    border->valid = 1;

  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;
    FT_Angle             angle_out;
    FT_Vector            center;
    FT_Bool              first_point;
    FT_Bool              subpath_open;
    FT_Angle             subpath_angle;
    FT_Vector            subpath_start;

    FT_Stroker_LineCap   line_cap;
    FT_Stroker_LineJoin  line_join;
    FT_Fixed             miter_limit;
    FT_Fixed             radius;

    FT_Bool              valid;
    FT_StrokeBorderRec   borders[2];
    FT_Memory            memory;

  } FT_StrokerRec;


  FT_EXPORT_DEF( FT_Error )
  FT_Stroker_New( FT_Memory    memory,
                  FT_Stroker  *astroker )
  {
    FT_Error    error;
    FT_Stroker  stroker;


    if ( !FT_NEW( stroker ) )
    {
      stroker->memory = memory;

      ft_stroke_border_init( &stroker->borders[0], memory );
      ft_stroke_border_init( &stroker->borders[1], memory );
    }
    *astroker = stroker;
    return error;
  }


  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;

    ft_stroke_border_reset( &stroker->borders[0] );
    ft_stroke_border_reset( &stroker->borders[1] );
  }


  FT_EXPORT_DEF( void )
  FT_Stroker_Done( FT_Stroker  stroker )
  {
    if ( stroker )
    {
      FT_Memory  memory = stroker->memory;


      ft_stroke_border_done( &stroker->borders[0] );
      ft_stroke_border_done( &stroker->borders[1] );

      stroker->memory = NULL;
      FT_FREE( stroker );
    }
  }


  /* creates 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  = 0;
    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 = 0;
    return error;
  }


  /* adds 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  = 0;


    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, 0 );
      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, 0 );
    }

  Exit:
    return error;
  }


  /* process an inside corner, i.e. compute intersection */
  static FT_Error
  ft_stroker_inside( FT_Stroker  stroker,
                     FT_Int      side)
  {
    FT_StrokeBorder  border = stroker->borders + side;
    FT_Angle         phi, theta, rotate;
    FT_Fixed         length, thcos, sigma;
    FT_Vector        delta;
    FT_Error         error = 0;


    rotate = FT_SIDE_TO_ROTATE( side );

    /* compute median angle */
    theta = FT_Angle_Diff( stroker->angle_in, stroker->angle_out );
    if ( theta == FT_ANGLE_PI )
      theta = rotate;
    else
      theta = theta / 2;

    phi = stroker->angle_in + theta;

    thcos  = FT_Cos( theta );
    sigma  = FT_MulFix( stroker->miter_limit, thcos );

    if ( sigma < 0x10000L )
    {
      FT_Vector_From_Polar( &delta, stroker->radius,
                            stroker->angle_out + rotate );
      delta.x += stroker->center.x;
      delta.y += stroker->center.y;
      border->movable = 0;
    }
    else
    {
      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, 0 );

    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_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 or beveled corner */
      FT_Fixed  sigma, radius = stroker->radius;
      FT_Angle  theta, phi;
      FT_Fixed  thcos;
      FT_Bool   miter;


      rotate = FT_SIDE_TO_ROTATE( side );
      miter  = FT_BOOL( stroker->line_join == FT_STROKER_LINEJOIN_MITER );

      theta  = FT_Angle_Diff( stroker->angle_in, stroker->angle_out );
      if ( theta == FT_ANGLE_PI )
      {
        theta = rotate;
        phi   = stroker->angle_in;
      }
      else
      {
        theta = theta / 2;
        phi   = stroker->angle_in + theta + rotate;
      }

      thcos = FT_Cos( theta );
      sigma = FT_MulFix( stroker->miter_limit, thcos );

      if ( sigma >= 0x10000L )
        miter = 0;

      if ( miter )  /* this is a miter (broken angle) */
      {
        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, 0 );
        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, 0 );
        if ( error )
          goto Exit;

        /* finally, add a movable end point */
        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, 1 );
      }

      else /* this is a bevel (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, 0 );
        if (error) goto Exit;

        /* now add end point */
        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, 1 );
      }
    }

  Exit:
    return error;
  }


  static FT_Error
  ft_stroker_process_corner( FT_Stroker  stroker )
  {
    FT_Error  error = 0;
    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 );
    if ( error )
      goto Exit;

    /* process the outside side */
    error = ft_stroker_outside( stroker, 1 - inside_side );

  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_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 for last cap */
    stroker->subpath_angle = start_angle;
    stroker->first_point   = 0;

  Exit:
    return error;
  }


  FT_EXPORT_DEF( FT_Error )
  FT_Stroker_LineTo( FT_Stroker  stroker,
                     FT_Vector*  to )
  {
    FT_Error         error = 0;
    FT_StrokeBorder  border;
    FT_Vector        delta;
    FT_Angle         angle;
    FT_Int           side;

    delta.x = to->x - stroker->center.x;
    delta.y = to->y - stroker->center.y;

    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 );
      if ( error )
        goto Exit;
    }
    else
    {
      /* process the current corner */
      stroker->angle_out = angle;
      error = ft_stroker_process_corner( stroker );
      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;

      error = ft_stroke_border_lineto( border, &point, 1 );
      if ( error )
        goto Exit;

      delta.x = -delta.x;
      delta.y = -delta.y;
    }

    stroker->angle_in = angle;
    stroker->center   = *to;

  Exit:
    return error;
  }


  FT_EXPORT_DEF( FT_Error )
  FT_Stroker_ConicTo( FT_Stroker  stroker,
                      FT_Vector*  control,
                      FT_Vector*  to )
  {
    FT_Error    error = 0;
    FT_Vector   bez_stack[34];
    FT_Vector*  arc;
    FT_Vector*  limit = bez_stack + 30;
    FT_Angle    start_angle;
    FT_Bool     first_arc = 1;


    arc    = bez_stack;
    arc[0] = *to;
    arc[1] = *control;
    arc[2] = stroker->center;

    while ( arc >= bez_stack )
    {
      FT_Angle  angle_in, angle_out;


      angle_in = angle_out = 0;  /* remove compiler warnings */

      if ( arc < limit                                             &&
           !ft_conic_is_small_enough( arc, &angle_in, &angle_out ) )
      {
        ft_conic_split( arc );
        arc += 2;
        continue;
      }

      if ( first_arc )
      {
        first_arc = 0;

        start_angle = angle_in;

        /* process corner if necessary */
        if ( stroker->first_point )
          error = ft_stroker_subpath_start( stroker, start_angle );
        else
        {
          stroker->angle_out = start_angle;
          error = ft_stroker_process_corner( stroker );
        }
      }

      /* the arc's angle is small enough; we can add it directly to each */
      /* border                                                          */
      {
        FT_Vector  ctrl, end;
        FT_Angle   theta, phi, rotate;
        FT_Fixed   length;
        FT_Int     side;


        theta  = FT_Angle_Diff( angle_in, angle_out ) / 2;
        phi    = angle_in + theta;
        length = FT_DivFix( stroker->radius, FT_Cos( theta ) );

        for ( side = 0; side <= 1; side++ )
        {
          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;

          error = ft_stroke_border_conicto( stroker->borders + side,
                                            &ctrl, &end );
          if ( error )
            goto Exit;
        }
      }

      arc -= 2;

      if ( arc < bez_stack )
        stroker->angle_in = angle_out;
    }

    stroker->center = *to;

  Exit:
    return error;
  }


  FT_EXPORT_DEF( FT_Error )
  FT_Stroker_CubicTo( FT_Stroker  stroker,
                      FT_Vector*  control1,
                      FT_Vector*  control2,
                      FT_Vector*  to )
  {
    FT_Error    error = 0;
    FT_Vector   bez_stack[37];
    FT_Vector*  arc;
    FT_Vector*  limit = bez_stack + 32;
    FT_Angle    start_angle;
    FT_Bool     first_arc = 1;


    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;


      /* remove compiler warnings */
      angle_in = angle_out = angle_mid = 0;

      if ( arc < limit                                         &&
           !ft_cubic_is_small_enough( arc, &angle_in,
                                      &angle_mid, &angle_out ) )
      {
        ft_cubic_split( arc );
        arc += 3;
        continue;
      }

      if ( first_arc )
      {
        first_arc = 0;

        /* process corner if necessary */
        start_angle = angle_in;

        if ( stroker->first_point )
          error = ft_stroker_subpath_start( stroker, start_angle );
        else
        {
          stroker->angle_out = start_angle;
          error = ft_stroker_process_corner( stroker );
        }
        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;
        FT_Fixed   length1, length2;
        FT_Int     side;


        theta1  = ft_pos_abs( angle_mid - angle_in ) / 2;
        theta2  = ft_pos_abs( angle_out - angle_mid ) / 2;
        phi1    = (angle_mid + angle_in ) / 2;
        phi2    = (angle_mid + angle_out ) / 2;
        length1 = FT_DivFix( stroker->radius, FT_Cos( theta1 ) );
        length2 = FT_DivFix( stroker->radius, FT_Cos(theta2) );

        for ( side = 0; side <= 1; side++ )
        {
          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;

          error = ft_stroke_border_cubicto( stroker->borders + side,
                                            &ctrl1, &ctrl2, &end );
          if ( error )
            goto Exit;
        }
      }

      arc -= 3;
      if ( arc < bez_stack )
        stroker->angle_in = angle_out;
    }

    stroker->center = *to;

  Exit:
    return error;
  }


  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 "end_subpath" routine.                                       */
    /*                                                                     */
    stroker->first_point   = 1;
    stroker->center        = *to;
    stroker->subpath_open  = open;

    /* record the subpath start point index for each border */
    stroker->subpath_start = *to;
    return 0;
  }


  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  = 0;


    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 | FT_STROKE_TAG_END );
          else
          {
            /* switch begin/end tags if necessary.. */
            if ( dst_tag[0] & ( FT_STROKE_TAG_BEGIN | FT_STROKE_TAG_END ) )
              dst_tag[0] ^= ( FT_STROKE_TAG_BEGIN | FT_STROKE_TAG_END );
          }

          src_point--;
          src_tag--;
          dst_point++;
          dst_tag++;
        }
      }

      left->num_points   = left->start;
      right->num_points += new_points;

      right->movable = 0;
      left->movable  = 0;
    }

  Exit:
    return error;
  }


  /* there's a lot of magic in this function! */
  FT_EXPORT_DEF( FT_Error )
  FT_Stroker_EndSubPath( FT_Stroker  stroker )
  {
    FT_Error  error  = 0;


    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, 1 );
      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 );
    }
    else
    {
      FT_Angle  turn;
      FT_Int    inside_side;


      /* 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;

        /* IMPORTANT: WE DO NOT PROCESS THE INSIDE BORDER HERE! */
        /* process the inside side                              */
        /* error = ft_stroker_inside( stroker, inside_side );   */
        /* if ( error )                                         */
        /*   goto Exit;                                         */

        /* process the outside side */
        error = ft_stroker_outside( stroker, 1 - inside_side );
        if ( error )
          goto Exit;
      }

      /* we will first end our two subpaths */
      ft_stroke_border_close( stroker->borders + 0 );
      ft_stroke_border_close( stroker->borders + 1 );

      /* now, add the reversed left subpath to "right" */
      error = ft_stroker_add_reverse_left( stroker, 0 );
      if ( error )
        goto Exit;
    }

  Exit:
    return error;
  }


  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;
  }


  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;
  }


  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 );
    }
  }


  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 );
  }


  /*
   *  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;

    first = 0;

    for ( n = 0; n < outline->n_contours; n++ )
    {
      FT_Int  last;  /* index of last point in contour */


      last  = outline->contours[n];
      limit = outline->points + last;

      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 and record its position    */
          /* for closure                                      */
          v_start.x = ( v_start.x + v_last.x ) / 2;
          v_start.y = ( v_start.y + v_last.y ) / 2;

          v_last = v_start;
        }
        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;

      error = FT_Stroker_EndSubPath( stroker );
      if ( error )
        goto Exit;

      first = last + 1;
    }

    return 0;

  Exit:
    return error;

  Invalid_Outline:
    return FT_Err_Invalid_Outline;
  }


/* END */