tor-browser

ftsdf.c (112099B)

---

/****************************************************************************
*
* ftsdf.c
*
*   Signed Distance Field support for outline fonts (body).
*
* Copyright (C) 2020-2025 by
* David Turner, Robert Wilhelm, and Werner Lemberg.
*
* Written by Anuj Verma.
*
* 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 <freetype/internal/ftobjs.h>
#include <freetype/internal/ftdebug.h>
#include <freetype/ftoutln.h>
#include <freetype/fttrigon.h>
#include <freetype/ftbitmap.h>
#include "ftsdf.h"

#include "ftsdferrs.h"


 /**************************************************************************
  *
  * A brief technical overview of how the SDF rasterizer works
  * ----------------------------------------------------------
  *
  * [Notes]:
  *   * SDF stands for Signed Distance Field everywhere.
  *
  *   * This renderer generates SDF directly from outlines.  There is
  *     another renderer called 'bsdf', which converts bitmaps to SDF; see
  *     file `ftbsdf.c` for more.
  *
  *   * The basic idea of generating the SDF is taken from Viktor Chlumsky's
  *     research paper.  The paper explains both single and multi-channel
  *     SDF, however, this implementation only generates single-channel SDF.
  *
  *       Chlumsky, Viktor: Shape Decomposition for Multi-channel Distance
  *       Fields.  Master's thesis.  Czech Technical University in Prague,
  *       Faculty of InformationTechnology, 2015.
  *
  *     For more information: https://github.com/Chlumsky/msdfgen
  *
  * ========================================================================
  *
  * Generating SDF from outlines is pretty straightforward.
  *
  * (1) We have a set of contours that make the outline of a shape/glyph.
  *     Each contour comprises of several edges, with three types of edges.
  *
  *     * line segments
  *     * conic Bezier curves
  *     * cubic Bezier curves
  *
  * (2) Apart from the outlines we also have a two-dimensional grid, namely
  *     the bitmap that is used to represent the final SDF data.
  *
  * (3) In order to generate SDF, our task is to find shortest signed
  *     distance from each grid point to the outline.  The 'signed
  *     distance' means that if the grid point is filled by any contour
  *     then its sign is positive, otherwise it is negative.  The pseudo
  *     code is as follows.
  *
  *     ```
  *     foreach grid_point (x, y):
  *     {
  *       int min_dist = INT_MAX;
  *
  *       foreach contour in outline:
  *       {
  *         foreach edge in contour:
  *         {
  *           // get shortest distance from point (x, y) to the edge
  *           d = get_min_dist(x, y, edge);
  *
  *           if (d < min_dist)
  *             min_dist = d;
  *         }
  *
  *         bitmap[x, y] = min_dist;
  *       }
  *     }
  *     ```
  *
  * (4) After running this algorithm the bitmap contains information about
  *     the shortest distance from each point to the outline of the shape.
  *     Of course, while this is the most straightforward way of generating
  *     SDF, we use various optimizations in our implementation.  See the
  *     `sdf_generate_*' functions in this file for all details.
  *
  *     The optimization currently used by default is subdivision; see
  *     function `sdf_generate_subdivision` for more.
  *
  *     Also, to see how we compute the shortest distance from a point to
  *     each type of edge, check out the `get_min_distance_*' functions.
  *
  */


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


 /**************************************************************************
  *
  * definitions
  *
  */

 /*
  * If set to 1, the rasterizer uses Newton-Raphson's method for finding
  * the shortest distance from a point to a conic curve.
  *
  * If set to 0, an analytical method gets used instead, which computes the
  * roots of a cubic polynomial to find the shortest distance.  However,
  * the analytical method can currently underflow; we thus use Newton's
  * method by default.
  */
#ifndef USE_NEWTON_FOR_CONIC
#define USE_NEWTON_FOR_CONIC  1
#endif

 /*
  * The number of intervals a Bezier curve gets sampled and checked to find
  * the shortest distance.
  */
#define MAX_NEWTON_DIVISIONS  4

 /*
  * The number of steps of Newton's iterations in each interval of the
  * Bezier curve.  Basically, we run Newton's approximation
  *
  *   x -= Q(t) / Q'(t)
  *
  * for each division to get the shortest distance.
  */
#define MAX_NEWTON_STEPS  4

 /*
  * The epsilon distance (in 16.16 fractional units) used for corner
  * resolving.  If the difference of two distances is less than this value
  * they will be checked for a corner if they are ambiguous.
  */
#define CORNER_CHECK_EPSILON  32

#if 0
 /*
  * Coarse grid dimension.  Will probably be removed in the future because
  * coarse grid optimization is the slowest algorithm.
  */
#define CG_DIMEN  8
#endif


 /**************************************************************************
  *
  * macros
  *
  */

#define MUL_26D6( a, b )  ( ( ( a ) * ( b ) ) / 64 )
#define VEC_26D6_DOT( p, q )  ( MUL_26D6( p.x, q.x ) + \
                               MUL_26D6( p.y, q.y ) )


 /**************************************************************************
  *
  * structures and enums
  *
  */

 /**************************************************************************
  *
  * @Struct:
  *   SDF_TRaster
  *
  * @Description:
  *   This struct is used in place of @FT_Raster and is stored within the
  *   internal FreeType renderer struct.  While rasterizing it is passed to
  *   the @FT_Raster_RenderFunc function, which then can be used however we
  *   want.
  *
  * @Fields:
  *   memory ::
  *     Used internally to allocate intermediate memory while raterizing.
  *
  */
 typedef struct  SDF_TRaster_
 {
   FT_Memory  memory;

 } SDF_TRaster, *SDF_PRaster;


 /**************************************************************************
  *
  * @Enum:
  *   SDF_Edge_Type
  *
  * @Description:
  *   Enumeration of all curve types present in fonts.
  *
  * @Fields:
  *   SDF_EDGE_UNDEFINED ::
  *     Undefined edge, simply used to initialize and detect errors.
  *
  *   SDF_EDGE_LINE ::
  *     Line segment with start and end point.
  *
  *   SDF_EDGE_CONIC ::
  *     A conic/quadratic Bezier curve with start, end, and one control
  *     point.
  *
  *   SDF_EDGE_CUBIC ::
  *     A cubic Bezier curve with start, end, and two control points.
  *
  */
 typedef enum  SDF_Edge_Type_
 {
   SDF_EDGE_UNDEFINED = 0,
   SDF_EDGE_LINE      = 1,
   SDF_EDGE_CONIC     = 2,
   SDF_EDGE_CUBIC     = 3

 } SDF_Edge_Type;


 /**************************************************************************
  *
  * @Enum:
  *   SDF_Contour_Orientation
  *
  * @Description:
  *   Enumeration of all orientation values of a contour.  We determine the
  *   orientation by calculating the area covered by a contour.  Contrary
  *   to values returned by @FT_Outline_Get_Orientation,
  *   `SDF_Contour_Orientation` is independent of the fill rule, which can
  *   be different for different font formats.
  *
  * @Fields:
  *   SDF_ORIENTATION_NONE ::
  *     Undefined orientation, used for initialization and error detection.
  *
  *   SDF_ORIENTATION_CW ::
  *     Clockwise orientation (positive area covered).
  *
  *   SDF_ORIENTATION_CCW ::
  *     Counter-clockwise orientation (negative area covered).
  *
  * @Note:
  *   See @FT_Outline_Get_Orientation for more details.
  *
  */
 typedef enum  SDF_Contour_Orientation_
 {
   SDF_ORIENTATION_NONE = 0,
   SDF_ORIENTATION_CW   = 1,
   SDF_ORIENTATION_CCW  = 2

 } SDF_Contour_Orientation;


 /**************************************************************************
  *
  * @Struct:
  *   SDF_Edge
  *
  * @Description:
  *   Represent an edge of a contour.
  *
  * @Fields:
  *   start_pos ::
  *     Start position of an edge.  Valid for all types of edges.
  *
  *   end_pos ::
  *     Etart position of an edge.  Valid for all types of edges.
  *
  *   control_a ::
  *     A control point of the edge.  Valid only for `SDF_EDGE_CONIC`
  *     and `SDF_EDGE_CUBIC`.
  *
  *   control_b ::
  *     Another control point of the edge.  Valid only for
  *     `SDF_EDGE_CONIC`.
  *
  *   edge_type ::
  *     Type of the edge, see @SDF_Edge_Type for all possible edge types.
  *
  *   next ::
  *     Used to create a singly linked list, which can be interpreted
  *     as a contour.
  *
  */
 typedef struct  SDF_Edge_
 {
   FT_26D6_Vec  start_pos;
   FT_26D6_Vec  end_pos;
   FT_26D6_Vec  control_a;
   FT_26D6_Vec  control_b;

   SDF_Edge_Type  edge_type;

   struct SDF_Edge_*  next;

 } SDF_Edge;


 /**************************************************************************
  *
  * @Struct:
  *   SDF_Contour
  *
  * @Description:
  *   Represent a complete contour, which contains a list of edges.
  *
  * @Fields:
  *   last_pos ::
  *     Contains the value of `end_pos' of the last edge in the list of
  *     edges.  Useful while decomposing the outline with
  *     @FT_Outline_Decompose.
  *
  *   edges ::
  *     Linked list of all the edges that make the contour.
  *
  *   next ::
  *     Used to create a singly linked list, which can be interpreted as a
  *     complete shape or @FT_Outline.
  *
  */
 typedef struct  SDF_Contour_
 {
   FT_26D6_Vec  last_pos;
   SDF_Edge*    edges;

   struct SDF_Contour_*  next;

 } SDF_Contour;


 /**************************************************************************
  *
  * @Struct:
  *   SDF_Shape
  *
  * @Description:
  *   Represent a complete shape, which is the decomposition of
  *   @FT_Outline.
  *
  * @Fields:
  *   memory ::
  *     Used internally to allocate memory.
  *
  *   contours ::
  *     Linked list of all the contours that make the shape.
  *
  */
 typedef struct  SDF_Shape_
 {
   FT_Memory     memory;
   SDF_Contour*  contours;

 } SDF_Shape;


 /**************************************************************************
  *
  * @Struct:
  *   SDF_Signed_Distance
  *
  * @Description:
  *   Represent signed distance of a point, i.e., the distance of the edge
  *   nearest to the point.
  *
  * @Fields:
  *   distance ::
  *     Distance of the point from the nearest edge.  Can be squared or
  *     absolute depending on the `USE_SQUARED_DISTANCES` macro defined in
  *     file `ftsdfcommon.h`.
  *
  *   cross ::
  *     Cross product of the shortest distance vector (i.e., the vector
  *     from the point to the nearest edge) and the direction of the edge
  *     at the nearest point.  This is used to resolve ambiguities of
  *     `sign`.
  *
  *   sign ::
  *     A value used to indicate whether the distance vector is outside or
  *     inside the contour corresponding to the edge.
  *
  * @Note:
  *   `sign` may or may not be correct, therefore it must be checked
  *   properly in case there is an ambiguity.
  *
  */
 typedef struct SDF_Signed_Distance_
 {
   FT_16D16  distance;
   FT_16D16  cross;
   FT_Char   sign;

 } SDF_Signed_Distance;


 /**************************************************************************
  *
  * @Struct:
  *   SDF_Params
  *
  * @Description:
  *   Yet another internal parameters required by the rasterizer.
  *
  * @Fields:
  *   orientation ::
  *     This is not the @SDF_Contour_Orientation value but @FT_Orientation,
  *     which determines whether clockwise-oriented outlines are to be
  *     filled or counter-clockwise-oriented ones.
  *
  *   flip_sign ::
  *     If set to true, flip the sign.  By default the points filled by the
  *     outline are positive.
  *
  *   flip_y ::
  *     If set to true the output bitmap is upside-down.  Can be useful
  *     because OpenGL and DirectX use different coordinate systems for
  *     textures.
  *
  *   overload_sign ::
  *     In the subdivision and bounding box optimization, the default
  *     outside sign is taken as -1.  This parameter can be used to modify
  *     that behaviour.  For example, while generating SDF for a single
  *     counter-clockwise contour, the outside sign should be 1.
  *
  */
 typedef struct SDF_Params_
 {
   FT_Orientation  orientation;
   FT_Bool         flip_sign;
   FT_Bool         flip_y;

   FT_Int  overload_sign;

 } SDF_Params;


 /**************************************************************************
  *
  * constants, initializer, and destructor
  *
  */

 static
 const FT_Vector  zero_vector = { 0, 0 };

 static
 const SDF_Edge  null_edge = { { 0, 0 }, { 0, 0 },
                               { 0, 0 }, { 0, 0 },
                               SDF_EDGE_UNDEFINED, NULL };

 static
 const SDF_Contour  null_contour = { { 0, 0 }, NULL, NULL };

 static
 const SDF_Shape  null_shape = { NULL, NULL };

 static
 const SDF_Signed_Distance  max_sdf = { INT_MAX, 0, 0 };


 /* Create a new @SDF_Edge on the heap and assigns the `edge` */
 /* pointer to the newly allocated memory.                    */
 static FT_Error
 sdf_edge_new( FT_Memory   memory,
               SDF_Edge**  edge )
 {
   FT_Error   error = FT_Err_Ok;
   SDF_Edge*  ptr   = NULL;


   if ( !memory || !edge )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( !FT_QNEW( ptr ) )
   {
     *ptr = null_edge;
     *edge = ptr;
   }

 Exit:
   return error;
 }


 /* Free the allocated `edge` variable. */
 static void
 sdf_edge_done( FT_Memory   memory,
                SDF_Edge**  edge )
 {
   if ( !memory || !edge || !*edge )
     return;

   FT_FREE( *edge );
 }


 /* Create a new @SDF_Contour on the heap and assign     */
 /* the `contour` pointer to the newly allocated memory. */
 static FT_Error
 sdf_contour_new( FT_Memory      memory,
                  SDF_Contour**  contour )
 {
   FT_Error      error = FT_Err_Ok;
   SDF_Contour*  ptr   = NULL;


   if ( !memory || !contour )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( !FT_QNEW( ptr ) )
   {
     *ptr     = null_contour;
     *contour = ptr;
   }

 Exit:
   return error;
 }


 /* Free the allocated `contour` variable. */
 /* Also free the list of edges.           */
 static void
 sdf_contour_done( FT_Memory      memory,
                   SDF_Contour**  contour )
 {
   SDF_Edge*  edges;
   SDF_Edge*  temp;


   if ( !memory || !contour || !*contour )
     return;

   edges = (*contour)->edges;

   /* release all edges */
   while ( edges )
   {
     temp  = edges;
     edges = edges->next;

     sdf_edge_done( memory, &temp );
   }

   FT_FREE( *contour );
 }


 /* Create a new @SDF_Shape on the heap and assign     */
 /* the `shape` pointer to the newly allocated memory. */
 static FT_Error
 sdf_shape_new( FT_Memory    memory,
                SDF_Shape**  shape )
 {
   FT_Error    error = FT_Err_Ok;
   SDF_Shape*  ptr   = NULL;


   if ( !memory || !shape )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( !FT_QNEW( ptr ) )
   {
     *ptr        = null_shape;
     ptr->memory = memory;
     *shape      = ptr;
   }

 Exit:
   return error;
 }


 /* Free the allocated `shape` variable. */
 /* Also free the list of contours.      */
 static void
 sdf_shape_done( SDF_Shape**  shape )
 {
   FT_Memory     memory;
   SDF_Contour*  contours;
   SDF_Contour*  temp;


   if ( !shape || !*shape )
     return;

   memory   = (*shape)->memory;
   contours = (*shape)->contours;

   if ( !memory )
     return;

   /* release all contours */
   while ( contours )
   {
     temp     = contours;
     contours = contours->next;

     sdf_contour_done( memory, &temp );
   }

   /* release the allocated shape struct  */
   FT_FREE( *shape );
 }


 /**************************************************************************
  *
  * shape decomposition functions
  *
  */

 /* This function is called when starting a new contour at `to`, */
 /* which gets added to the shape's list.                        */
 static FT_Error
 sdf_move_to( const FT_26D6_Vec* to,
              void*              user )
 {
   SDF_Shape*    shape   = ( SDF_Shape* )user;
   SDF_Contour*  contour = NULL;

   FT_Error   error  = FT_Err_Ok;
   FT_Memory  memory = shape->memory;


   if ( !to || !user )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   FT_CALL( sdf_contour_new( memory, &contour ) );

   contour->last_pos = *to;
   contour->next     = shape->contours;
   shape->contours   = contour;

 Exit:
   return error;
 }


 /* This function is called when there is a line in the      */
 /* contour.  The line starts at the previous edge point and */
 /* stops at `to`.                                           */
 static FT_Error
 sdf_line_to( const FT_26D6_Vec*  to,
              void*               user )
 {
   SDF_Shape*    shape    = ( SDF_Shape* )user;
   SDF_Edge*     edge     = NULL;
   SDF_Contour*  contour  = NULL;

   FT_Error      error    = FT_Err_Ok;
   FT_Memory     memory   = shape->memory;


   if ( !to || !user )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   contour = shape->contours;

   if ( contour->last_pos.x == to->x &&
        contour->last_pos.y == to->y )
     goto Exit;

   FT_CALL( sdf_edge_new( memory, &edge ) );

   edge->edge_type = SDF_EDGE_LINE;
   edge->start_pos = contour->last_pos;
   edge->end_pos   = *to;

   edge->next        = contour->edges;
   contour->edges    = edge;
   contour->last_pos = *to;

 Exit:
   return error;
 }


 /* This function is called when there is a conic Bezier curve   */
 /* in the contour.  The curve starts at the previous edge point */
 /* and stops at `to`, with control point `control_1`.           */
 static FT_Error
 sdf_conic_to( const FT_26D6_Vec*  control_1,
               const FT_26D6_Vec*  to,
               void*               user )
 {
   SDF_Shape*    shape    = ( SDF_Shape* )user;
   SDF_Edge*     edge     = NULL;
   SDF_Contour*  contour  = NULL;

   FT_Error   error  = FT_Err_Ok;
   FT_Memory  memory = shape->memory;


   if ( !control_1 || !to || !user )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   contour = shape->contours;

   /* If the control point coincides with any of the end points */
   /* then it is a line and should be treated as one to avoid   */
   /* unnecessary complexity later in the algorithm.            */
   if ( ( contour->last_pos.x == control_1->x &&
          contour->last_pos.y == control_1->y ) ||
        ( control_1->x == to->x &&
          control_1->y == to->y )               )
   {
     sdf_line_to( to, user );
     goto Exit;
   }

   FT_CALL( sdf_edge_new( memory, &edge ) );

   edge->edge_type = SDF_EDGE_CONIC;
   edge->start_pos = contour->last_pos;
   edge->control_a = *control_1;
   edge->end_pos   = *to;

   edge->next        = contour->edges;
   contour->edges    = edge;
   contour->last_pos = *to;

 Exit:
   return error;
 }


 /* This function is called when there is a cubic Bezier curve   */
 /* in the contour.  The curve starts at the previous edge point */
 /* and stops at `to`, with two control points `control_1` and   */
 /* `control_2`.                                                 */
 static FT_Error
 sdf_cubic_to( const FT_26D6_Vec*  control_1,
               const FT_26D6_Vec*  control_2,
               const FT_26D6_Vec*  to,
               void*               user )
 {
   SDF_Shape*    shape   = ( SDF_Shape* )user;
   SDF_Edge*     edge    = NULL;
   SDF_Contour*  contour = NULL;

   FT_Error   error  = FT_Err_Ok;
   FT_Memory  memory = shape->memory;


   if ( !control_2 || !control_1 || !to || !user )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   contour = shape->contours;

   FT_CALL( sdf_edge_new( memory, &edge ) );

   edge->edge_type = SDF_EDGE_CUBIC;
   edge->start_pos = contour->last_pos;
   edge->control_a = *control_1;
   edge->control_b = *control_2;
   edge->end_pos   = *to;

   edge->next        = contour->edges;
   contour->edges    = edge;
   contour->last_pos = *to;

 Exit:
   return error;
 }


 /* Construct the structure to hold all four outline */
 /* decomposition functions.                         */
 FT_DEFINE_OUTLINE_FUNCS(
   sdf_decompose_funcs,

   (FT_Outline_MoveTo_Func) sdf_move_to,   /* move_to  */
   (FT_Outline_LineTo_Func) sdf_line_to,   /* line_to  */
   (FT_Outline_ConicTo_Func)sdf_conic_to,  /* conic_to */
   (FT_Outline_CubicTo_Func)sdf_cubic_to,  /* cubic_to */

   0,                                      /* shift    */
   0                                       /* delta    */
 )


 /* Decompose `outline` and put it into the `shape` structure.  */
 static FT_Error
 sdf_outline_decompose( FT_Outline*  outline,
                        SDF_Shape*   shape )
 {
   FT_Error  error = FT_Err_Ok;


   if ( !outline || !shape )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   error = FT_Outline_Decompose( outline,
                                 &sdf_decompose_funcs,
                                 (void*)shape );

 Exit:
   return error;
 }


 /**************************************************************************
  *
  * utility functions
  *
  */

 /* Return the control box of an edge.  The control box is a rectangle */
 /* in which all the control points can fit tightly.                   */
 static FT_CBox
 get_control_box( SDF_Edge  edge )
 {
   FT_CBox  cbox   = { 0, 0, 0, 0 };
   FT_Bool  is_set = 0;


   switch ( edge.edge_type )
   {
   case SDF_EDGE_CUBIC:
     cbox.xMin = edge.control_b.x;
     cbox.xMax = edge.control_b.x;
     cbox.yMin = edge.control_b.y;
     cbox.yMax = edge.control_b.y;

     is_set = 1;
     FALL_THROUGH;

   case SDF_EDGE_CONIC:
     if ( is_set )
     {
       cbox.xMin = edge.control_a.x < cbox.xMin
                   ? edge.control_a.x
                   : cbox.xMin;
       cbox.xMax = edge.control_a.x > cbox.xMax
                   ? edge.control_a.x
                   : cbox.xMax;

       cbox.yMin = edge.control_a.y < cbox.yMin
                   ? edge.control_a.y
                   : cbox.yMin;
       cbox.yMax = edge.control_a.y > cbox.yMax
                   ? edge.control_a.y
                   : cbox.yMax;
     }
     else
     {
       cbox.xMin = edge.control_a.x;
       cbox.xMax = edge.control_a.x;
       cbox.yMin = edge.control_a.y;
       cbox.yMax = edge.control_a.y;

       is_set = 1;
     }
     FALL_THROUGH;

   case SDF_EDGE_LINE:
     if ( is_set )
     {
       cbox.xMin = edge.start_pos.x < cbox.xMin
                   ? edge.start_pos.x
                   : cbox.xMin;
       cbox.xMax = edge.start_pos.x > cbox.xMax
                   ? edge.start_pos.x
                   : cbox.xMax;

       cbox.yMin = edge.start_pos.y < cbox.yMin
                   ? edge.start_pos.y
                   : cbox.yMin;
       cbox.yMax = edge.start_pos.y > cbox.yMax
                   ? edge.start_pos.y
                   : cbox.yMax;
     }
     else
     {
       cbox.xMin = edge.start_pos.x;
       cbox.xMax = edge.start_pos.x;
       cbox.yMin = edge.start_pos.y;
       cbox.yMax = edge.start_pos.y;
     }

     cbox.xMin = edge.end_pos.x < cbox.xMin
                 ? edge.end_pos.x
                 : cbox.xMin;
     cbox.xMax = edge.end_pos.x > cbox.xMax
                 ? edge.end_pos.x
                 : cbox.xMax;

     cbox.yMin = edge.end_pos.y < cbox.yMin
                 ? edge.end_pos.y
                 : cbox.yMin;
     cbox.yMax = edge.end_pos.y > cbox.yMax
                 ? edge.end_pos.y
                 : cbox.yMax;

     break;

   default:
     break;
   }

   return cbox;
 }


 /* Return orientation of a single contour.                    */
 /* Note that the orientation is independent of the fill rule! */
 /* So, for TTF a clockwise-oriented contour has to be filled  */
 /* and the opposite for OTF fonts.                            */
 static SDF_Contour_Orientation
 get_contour_orientation ( SDF_Contour*  contour )
 {
   SDF_Edge*  head = NULL;
   FT_26D6    area = 0;


   /* return none if invalid parameters */
   if ( !contour || !contour->edges )
     return SDF_ORIENTATION_NONE;

   head = contour->edges;

   /* Calculate the area of the control box for all edges. */
   while ( head )
   {
     switch ( head->edge_type )
     {
     case SDF_EDGE_LINE:
       area += MUL_26D6( ( head->end_pos.x - head->start_pos.x ),
                         ( head->end_pos.y + head->start_pos.y ) );
       break;

     case SDF_EDGE_CONIC:
       area += MUL_26D6( head->control_a.x - head->start_pos.x,
                         head->control_a.y + head->start_pos.y );
       area += MUL_26D6( head->end_pos.x - head->control_a.x,
                         head->end_pos.y + head->control_a.y );
       break;

     case SDF_EDGE_CUBIC:
       area += MUL_26D6( head->control_a.x - head->start_pos.x,
                         head->control_a.y + head->start_pos.y );
       area += MUL_26D6( head->control_b.x - head->control_a.x,
                         head->control_b.y + head->control_a.y );
       area += MUL_26D6( head->end_pos.x - head->control_b.x,
                         head->end_pos.y + head->control_b.y );
       break;

     default:
       return SDF_ORIENTATION_NONE;
     }

     head = head->next;
   }

   /* Clockwise contours cover a positive area, and counter-clockwise */
   /* contours cover a negative area.                                 */
   if ( area > 0 )
     return SDF_ORIENTATION_CW;
   else
     return SDF_ORIENTATION_CCW;
 }


 /* This function is exactly the same as the one */
 /* in the smooth renderer.  It splits a conic   */
 /* into two conics exactly half way at t = 0.5. */
 static void
 split_conic( FT_26D6_Vec*  base )
 {
   FT_26D6  a, b;


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

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


 /* This function is exactly the same as the one */
 /* in the smooth renderer.  It splits a cubic   */
 /* into two cubics exactly half way at t = 0.5. */
 static void
 split_cubic( FT_26D6_Vec*  base )
 {
   FT_26D6  a, b, c;


   base[6].x = base[3].x;
   a         = base[0].x + base[1].x;
   b         = base[1].x + base[2].x;
   c         = base[2].x + base[3].x;
   base[5].x = c / 2;
   c        += b;
   base[4].x = c / 4;
   base[1].x = a / 2;
   a        += b;
   base[2].x = a / 4;
   base[3].x = ( a + c ) / 8;

   base[6].y = base[3].y;
   a         = base[0].y + base[1].y;
   b         = base[1].y + base[2].y;
   c         = base[2].y + base[3].y;
   base[5].y = c / 2;
   c        += b;
   base[4].y = c / 4;
   base[1].y = a / 2;
   a        += b;
   base[2].y = a / 4;
   base[3].y = ( a + c ) / 8;
 }


 /* Split a conic Bezier curve into a number of lines */
 /* and add them to `out'.                            */
 /*                                                   */
 /* This function uses recursion; we thus need        */
 /* parameter `max_splits' for stopping.              */
 static FT_Error
 split_sdf_conic( FT_Memory     memory,
                  FT_26D6_Vec*  control_points,
                  FT_UInt       max_splits,
                  SDF_Edge**    out )
 {
   FT_Error     error = FT_Err_Ok;
   FT_26D6_Vec  cpos[5];
   SDF_Edge*    left,*  right;


   if ( !memory || !out  )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   /* split conic outline */
   cpos[0] = control_points[0];
   cpos[1] = control_points[1];
   cpos[2] = control_points[2];

   split_conic( cpos );

   /* If max number of splits is done */
   /* then stop and add the lines to  */
   /* the list.                       */
   if ( max_splits <= 2 )
     goto Append;

   /* Otherwise keep splitting. */
   FT_CALL( split_sdf_conic( memory, &cpos[0], max_splits / 2, out ) );
   FT_CALL( split_sdf_conic( memory, &cpos[2], max_splits / 2, out ) );

   /* [NOTE]: This is not an efficient way of   */
   /* splitting the curve.  Check the deviation */
   /* instead and stop if the deviation is less */
   /* than a pixel.                             */

   goto Exit;

 Append:
   /* Do allocation and add the lines to the list. */

   FT_CALL( sdf_edge_new( memory, &left ) );
   FT_CALL( sdf_edge_new( memory, &right ) );

   left->start_pos  = cpos[0];
   left->end_pos    = cpos[2];
   left->edge_type  = SDF_EDGE_LINE;

   right->start_pos = cpos[2];
   right->end_pos   = cpos[4];
   right->edge_type = SDF_EDGE_LINE;

   left->next  = right;
   right->next = (*out);
   *out        = left;

 Exit:
   return error;
 }


 /* Split a cubic Bezier curve into a number of lines */
 /* and add them to `out`.                            */
 /*                                                   */
 /* This function uses recursion; we thus need        */
 /* parameter `max_splits' for stopping.              */
 static FT_Error
 split_sdf_cubic( FT_Memory     memory,
                  FT_26D6_Vec*  control_points,
                  FT_UInt       max_splits,
                  SDF_Edge**    out )
 {
   FT_Error       error = FT_Err_Ok;
   FT_26D6_Vec    cpos[7];
   SDF_Edge*      left, *right;
   const FT_26D6  threshold = ONE_PIXEL / 4;


   if ( !memory || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   /* split the cubic */
   cpos[0] = control_points[0];
   cpos[1] = control_points[1];
   cpos[2] = control_points[2];
   cpos[3] = control_points[3];

   /* If the segment is flat enough we won't get any benefit by */
   /* splitting it further, so we can just stop splitting.      */
   /*                                                           */
   /* Check the deviation of the Bezier curve and stop if it is */
   /* smaller than the pre-defined `threshold` value.           */
   if ( FT_ABS( 2 * cpos[0].x - 3 * cpos[1].x + cpos[3].x ) < threshold &&
        FT_ABS( 2 * cpos[0].y - 3 * cpos[1].y + cpos[3].y ) < threshold &&
        FT_ABS( cpos[0].x - 3 * cpos[2].x + 2 * cpos[3].x ) < threshold &&
        FT_ABS( cpos[0].y - 3 * cpos[2].y + 2 * cpos[3].y ) < threshold )
   {
     split_cubic( cpos );
     goto Append;
   }

   split_cubic( cpos );

   /* If max number of splits is done */
   /* then stop and add the lines to  */
   /* the list.                       */
   if ( max_splits <= 2 )
     goto Append;

   /* Otherwise keep splitting. */
   FT_CALL( split_sdf_cubic( memory, &cpos[0], max_splits / 2, out ) );
   FT_CALL( split_sdf_cubic( memory, &cpos[3], max_splits / 2, out ) );

   /* [NOTE]: This is not an efficient way of   */
   /* splitting the curve.  Check the deviation */
   /* instead and stop if the deviation is less */
   /* than a pixel.                             */

   goto Exit;

 Append:
   /* Do allocation and add the lines to the list. */

   FT_CALL( sdf_edge_new( memory, &left) );
   FT_CALL( sdf_edge_new( memory, &right) );

   left->start_pos  = cpos[0];
   left->end_pos    = cpos[3];
   left->edge_type  = SDF_EDGE_LINE;

   right->start_pos = cpos[3];
   right->end_pos   = cpos[6];
   right->edge_type = SDF_EDGE_LINE;

   left->next  = right;
   right->next = (*out);
   *out        = left;

 Exit:
   return error;
 }


 /* Subdivide an entire shape into line segments */
 /* such that it doesn't look visually different */
 /* from the original curve.                     */
 static FT_Error
 split_sdf_shape( SDF_Shape*  shape )
 {
   FT_Error   error = FT_Err_Ok;
   FT_Memory  memory;

   SDF_Contour*  contours;
   SDF_Contour*  new_contours = NULL;


   if ( !shape || !shape->memory )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   contours = shape->contours;
   memory   = shape->memory;

   /* for each contour */
   while ( contours )
   {
     SDF_Edge*  edges     = contours->edges;
     SDF_Edge*  new_edges = NULL;

     SDF_Contour*  tempc;


     /* for each edge */
     while ( edges )
     {
       SDF_Edge*  edge = edges;
       SDF_Edge*  temp;

       switch ( edge->edge_type )
       {
       case SDF_EDGE_LINE:
         /* Just create a duplicate edge in case     */
         /* it is a line.  We can use the same edge. */
         FT_CALL( sdf_edge_new( memory, &temp ) );

         ft_memcpy( temp, edge, sizeof ( *edge ) );

         temp->next = new_edges;
         new_edges  = temp;
         break;

       case SDF_EDGE_CONIC:
         /* Subdivide the curve and add it to the list. */
         {
           FT_26D6_Vec  ctrls[3];
           FT_26D6      dx, dy;
           FT_UInt      num_splits;


           ctrls[0] = edge->start_pos;
           ctrls[1] = edge->control_a;
           ctrls[2] = edge->end_pos;

           dx = FT_ABS( ctrls[2].x + ctrls[0].x - 2 * ctrls[1].x );
           dy = FT_ABS( ctrls[2].y + ctrls[0].y - 2 * ctrls[1].y );
           if ( dx < dy )
             dx = dy;

           /* Calculate the number of necessary bisections.  Each      */
           /* bisection causes a four-fold reduction of the deviation, */
           /* hence we bisect the Bezier curve until the deviation     */
           /* becomes less than 1/8 of a pixel.  For more details      */
           /* check file `ftgrays.c`.                                  */
           num_splits = 1;
           while ( dx > ONE_PIXEL / 8 )
           {
             dx         >>= 2;
             num_splits <<= 1;
           }

           error = split_sdf_conic( memory, ctrls, num_splits, &new_edges );
         }
         break;

       case SDF_EDGE_CUBIC:
         /* Subdivide the curve and add it to the list. */
         {
           FT_26D6_Vec  ctrls[4];


           ctrls[0] = edge->start_pos;
           ctrls[1] = edge->control_a;
           ctrls[2] = edge->control_b;
           ctrls[3] = edge->end_pos;

           error = split_sdf_cubic( memory, ctrls, 32, &new_edges );
         }
         break;

       default:
         error = FT_THROW( Invalid_Argument );
       }

       if ( error != FT_Err_Ok )
         goto Exit;

       edges = edges->next;
     }

     /* add to the contours list */
     FT_CALL( sdf_contour_new( memory, &tempc ) );

     tempc->next  = new_contours;
     tempc->edges = new_edges;
     new_contours = tempc;
     new_edges    = NULL;

     /* deallocate the contour */
     tempc    = contours;
     contours = contours->next;

     sdf_contour_done( memory, &tempc );
   }

   shape->contours = new_contours;

 Exit:
   return error;
 }


 /**************************************************************************
  *
  * for debugging
  *
  */

#ifdef FT_DEBUG_LEVEL_TRACE

 static void
 sdf_shape_dump( SDF_Shape*  shape )
 {
   FT_UInt  num_contours = 0;

   FT_UInt  total_edges = 0;
   FT_UInt  total_lines = 0;
   FT_UInt  total_conic = 0;
   FT_UInt  total_cubic = 0;

   SDF_Contour*  contour_list;


   if ( !shape )
   {
     FT_TRACE5(( "sdf_shape_dump: null shape\n" ));
     return;
   }

   contour_list = shape->contours;

   FT_TRACE5(( "sdf_shape_dump (values are in 26.6 format):\n" ));

   while ( contour_list )
   {
     FT_UInt       num_edges = 0;
     SDF_Edge*     edge_list;
     SDF_Contour*  contour = contour_list;


     FT_TRACE5(( "  Contour %u\n", num_contours ));

     edge_list = contour->edges;

     while ( edge_list )
     {
       SDF_Edge*  edge = edge_list;


       FT_TRACE5(( "  %3u: ", num_edges ));

       switch ( edge->edge_type )
       {
       case SDF_EDGE_LINE:
         FT_TRACE5(( "Line:  (%ld, %ld) -- (%ld, %ld)\n",
                     edge->start_pos.x, edge->start_pos.y,
                     edge->end_pos.x, edge->end_pos.y ));
         total_lines++;
         break;

       case SDF_EDGE_CONIC:
         FT_TRACE5(( "Conic: (%ld, %ld) .. (%ld, %ld) .. (%ld, %ld)\n",
                     edge->start_pos.x, edge->start_pos.y,
                     edge->control_a.x, edge->control_a.y,
                     edge->end_pos.x, edge->end_pos.y ));
         total_conic++;
         break;

       case SDF_EDGE_CUBIC:
         FT_TRACE5(( "Cubic: (%ld, %ld) .. (%ld, %ld)"
                     " .. (%ld, %ld) .. (%ld %ld)\n",
                     edge->start_pos.x, edge->start_pos.y,
                     edge->control_a.x, edge->control_a.y,
                     edge->control_b.x, edge->control_b.y,
                     edge->end_pos.x, edge->end_pos.y ));
         total_cubic++;
         break;

       default:
         break;
       }

       num_edges++;
       total_edges++;
       edge_list = edge_list->next;
     }

     num_contours++;
     contour_list = contour_list->next;
   }

   FT_TRACE5(( "\n" ));
   FT_TRACE5(( "  total number of contours = %u\n", num_contours ));
   FT_TRACE5(( "  total number of edges    = %u\n", total_edges ));
   FT_TRACE5(( "    |__lines = %u\n", total_lines ));
   FT_TRACE5(( "    |__conic = %u\n", total_conic ));
   FT_TRACE5(( "    |__cubic = %u\n", total_cubic ));
 }

#endif /* FT_DEBUG_LEVEL_TRACE */


 /**************************************************************************
  *
  * math functions
  *
  */

#if !USE_NEWTON_FOR_CONIC

 /* [NOTE]: All the functions below down until rasterizer */
 /*         can be avoided if we decide to subdivide the  */
 /*         curve into lines.                             */

 /* This function uses Newton's iteration to find */
 /* the cube root of a fixed-point integer.       */
 static FT_16D16
 cube_root( FT_16D16  val )
 {
   /* [IMPORTANT]: This function is not good as it may */
   /* not break, so use a lookup table instead.  Or we */
   /* can use an algorithm similar to `square_root`.   */

   FT_Int  v, g, c;


   if ( val == 0                  ||
        val == -FT_INT_16D16( 1 ) ||
        val ==  FT_INT_16D16( 1 ) )
     return val;

   v = val < 0 ? -val : val;
   g = square_root( v );
   c = 0;

   while ( 1 )
   {
     c = FT_MulFix( FT_MulFix( g, g ), g ) - v;
     c = FT_DivFix( c, 3 * FT_MulFix( g, g ) );

     g -= c;

     if ( ( c < 0 ? -c : c ) < 30 )
       break;
   }

   return val < 0 ? -g : g;
 }


 /* Calculate the perpendicular by using '1 - base^2'. */
 /* Then use arctan to compute the angle.              */
 static FT_16D16
 arc_cos( FT_16D16  val )
 {
   FT_16D16  p;
   FT_16D16  b   = val;
   FT_16D16  one = FT_INT_16D16( 1 );


   if ( b > one )
     b = one;
   if ( b < -one )
     b = -one;

   p = one - FT_MulFix( b, b );
   p = square_root( p );

   return FT_Atan2( b, p );
 }


 /* Compute roots of a quadratic polynomial, assign them to `out`, */
 /* and return number of real roots.                               */
 /*                                                                */
 /* The procedure can be found at                                  */
 /*                                                                */
 /*   https://mathworld.wolfram.com/QuadraticFormula.html          */
 static FT_UShort
 solve_quadratic_equation( FT_26D6   a,
                           FT_26D6   b,
                           FT_26D6   c,
                           FT_16D16  out[2] )
 {
   FT_16D16  discriminant = 0;


   a = FT_26D6_16D16( a );
   b = FT_26D6_16D16( b );
   c = FT_26D6_16D16( c );

   if ( a == 0 )
   {
     if ( b == 0 )
       return 0;
     else
     {
       out[0] = FT_DivFix( -c, b );

       return 1;
     }
   }

   discriminant = FT_MulFix( b, b ) - 4 * FT_MulFix( a, c );

   if ( discriminant < 0 )
     return 0;
   else if ( discriminant == 0 )
   {
     out[0] = FT_DivFix( -b, 2 * a );

     return 1;
   }
   else
   {
     discriminant = square_root( discriminant );

     out[0] = FT_DivFix( -b + discriminant, 2 * a );
     out[1] = FT_DivFix( -b - discriminant, 2 * a );

     return 2;
   }
 }


 /* Compute roots of a cubic polynomial, assign them to `out`, */
 /* and return number of real roots.                           */
 /*                                                            */
 /* The procedure can be found at                              */
 /*                                                            */
 /*   https://mathworld.wolfram.com/CubicFormula.html          */
 static FT_UShort
 solve_cubic_equation( FT_26D6   a,
                       FT_26D6   b,
                       FT_26D6   c,
                       FT_26D6   d,
                       FT_16D16  out[3] )
 {
   FT_16D16  q = 0;      /* intermediate */
   FT_16D16  r = 0;      /* intermediate */

   FT_16D16  a2 = b;     /* x^2 coefficients */
   FT_16D16  a1 = c;     /* x coefficients   */
   FT_16D16  a0 = d;     /* constant         */

   FT_16D16  q3   = 0;
   FT_16D16  r2   = 0;
   FT_16D16  a23  = 0;
   FT_16D16  a22  = 0;
   FT_16D16  a1x2 = 0;


   /* cutoff value for `a` to be a cubic, otherwise solve quadratic */
   if ( a == 0 || FT_ABS( a ) < 16 )
     return solve_quadratic_equation( b, c, d, out );

   if ( d == 0 )
   {
     out[0] = 0;

     return solve_quadratic_equation( a, b, c, out + 1 ) + 1;
   }

   /* normalize the coefficients; this also makes them 16.16 */
   a2 = FT_DivFix( a2, a );
   a1 = FT_DivFix( a1, a );
   a0 = FT_DivFix( a0, a );

   /* compute intermediates */
   a1x2 = FT_MulFix( a1, a2 );
   a22  = FT_MulFix( a2, a2 );
   a23  = FT_MulFix( a22, a2 );

   q = ( 3 * a1 - a22 ) / 9;
   r = ( 9 * a1x2 - 27 * a0 - 2 * a23 ) / 54;

   /* [BUG]: `q3` and `r2` still cause underflow. */

   q3 = FT_MulFix( q, q );
   q3 = FT_MulFix( q3, q );

   r2 = FT_MulFix( r, r );

   if ( q3 < 0 && r2 < -q3 )
   {
     FT_16D16  t = 0;


     q3 = square_root( -q3 );
     t  = FT_DivFix( r, q3 );

     if ( t > ( 1 << 16 ) )
       t =  ( 1 << 16 );
     if ( t < -( 1 << 16 ) )
       t = -( 1 << 16 );

     t   = arc_cos( t );
     a2 /= 3;
     q   = 2 * square_root( -q );

     out[0] = FT_MulFix( q, FT_Cos( t / 3 ) ) - a2;
     out[1] = FT_MulFix( q, FT_Cos( ( t + FT_ANGLE_PI * 2 ) / 3 ) ) - a2;
     out[2] = FT_MulFix( q, FT_Cos( ( t + FT_ANGLE_PI * 4 ) / 3 ) ) - a2;

     return 3;
   }

   else if ( r2 == -q3 )
   {
     FT_16D16  s = 0;


     s   = cube_root( r );
     a2 /= -3;

     out[0] = a2 + ( 2 * s );
     out[1] = a2 - s;

     return 2;
   }

   else
   {
     FT_16D16  s    = 0;
     FT_16D16  t    = 0;
     FT_16D16  dis  = 0;


     if ( q3 == 0 )
       dis = FT_ABS( r );
     else
       dis = square_root( q3 + r2 );

     s = cube_root( r + dis );
     t = cube_root( r - dis );
     a2 /= -3;
     out[0] = ( a2 + ( s + t ) );

     return 1;
   }
 }

#endif /* !USE_NEWTON_FOR_CONIC */


 /*************************************************************************/
 /*************************************************************************/
 /**                                                                     **/
 /**  RASTERIZER                                                         **/
 /**                                                                     **/
 /*************************************************************************/
 /*************************************************************************/

 /**************************************************************************
  *
  * @Function:
  *   resolve_corner
  *
  * @Description:
  *   At some places on the grid two edges can give opposite directions;
  *   this happens when the closest point is on one of the endpoint.  In
  *   that case we need to check the proper sign.
  *
  *   This can be visualized by an example:
  *
  *   ```
  *                x
  *
  *                   o
  *                  ^ \
  *                 /   \
  *                /     \
  *           (a) /       \  (b)
  *              /         \
  *             /           \
  *            /             v
  *   ```
  *
  *   Suppose `x` is the point whose shortest distance from an arbitrary
  *   contour we want to find out.  It is clear that `o` is the nearest
  *   point on the contour.  Now to determine the sign we do a cross
  *   product of the shortest distance vector and the edge direction, i.e.,
  *
  *   ```
  *   => sign = cross(x - o, direction(a))
  *   ```
  *
  *   Using the right hand thumb rule we can see that the sign will be
  *   positive.
  *
  *   If we use `b', however, we have
  *
  *   ```
  *   => sign = cross(x - o, direction(b))
  *   ```
  *
  *   In this case the sign will be negative.  To determine the correct
  *   sign we thus divide the plane in two halves and check which plane the
  *   point lies in.
  *
  *   ```
  *                   |
  *                x  |
  *                   |
  *                   o
  *                  ^|\
  *                 / | \
  *                /  |  \
  *           (a) /   |   \  (b)
  *              /    |    \
  *             /           \
  *            /             v
  *   ```
  *
  *   We can see that `x` lies in the plane of `a`, so we take the sign
  *   determined by `a`.  This test can be easily done by calculating the
  *   orthogonality and taking the greater one.
  *
  *   The orthogonality is simply the sinus of the two vectors (i.e.,
  *   x - o) and the corresponding direction.  We efficiently pre-compute
  *   the orthogonality with the corresponding `get_min_distance_*`
  *   functions.
  *
  * @Input:
  *   sdf1 ::
  *     First signed distance (can be any of `a` or `b`).
  *
  *   sdf1 ::
  *     Second signed distance (can be any of `a` or `b`).
  *
  * @Return:
  *   The correct signed distance, which is computed by using the above
  *   algorithm.
  *
  * @Note:
  *   The function does not care about the actual distance, it simply
  *   returns the signed distance which has a larger cross product.  As a
  *   consequence, this function should not be used if the two distances
  *   are fairly apart.  In that case simply use the signed distance with
  *   a shorter absolute distance.
  *
  */
 static SDF_Signed_Distance
 resolve_corner( SDF_Signed_Distance  sdf1,
                 SDF_Signed_Distance  sdf2 )
 {
   return FT_ABS( sdf1.cross ) > FT_ABS( sdf2.cross ) ? sdf1 : sdf2;
 }


 /**************************************************************************
  *
  * @Function:
  *   get_min_distance_line
  *
  * @Description:
  *   Find the shortest distance from the `line` segment to a given `point`
  *   and assign it to `out`.  Use it for line segments only.
  *
  * @Input:
  *   line ::
  *     The line segment to which the shortest distance is to be computed.
  *
  *   point ::
  *     Point from which the shortest distance is to be computed.
  *
  * @Output:
  *   out ::
  *     Signed distance from `point` to `line`.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  * @Note:
  *   The `line' parameter must have an edge type of `SDF_EDGE_LINE`.
  *
  */
 static FT_Error
 get_min_distance_line( SDF_Edge*             line,
                        FT_26D6_Vec           point,
                        SDF_Signed_Distance*  out )
 {
   /*
    * In order to calculate the shortest distance from a point to
    * a line segment, we do the following.  Let's assume that
    *
    * ```
    * a = start point of the line segment
    * b = end point of the line segment
    * p = point from which shortest distance is to be calculated
    * ```
    *
    * (1) Write the parametric equation of the line.
    *
    *     ```
    *     point_on_line = a + (b - a) * t   (t is the factor)
    *     ```
    *
    * (2) Find the projection of point `p` on the line.  The projection
    *     will be perpendicular to the line, which allows us to get the
    *     solution by making the dot product zero.
    *
    *     ```
    *     (point_on_line - a) . (p - point_on_line) = 0
    *
    *                (point_on_line)
    *      (a) x-------o----------------x (b)
    *                |_|
    *                  |
    *                  |
    *                 (p)
    *     ```
    *
    * (3) Simplification of the above equation yields the factor of
    *     `point_on_line`:
    *
    *     ```
    *     t = ((p - a) . (b - a)) / |b - a|^2
    *     ```
    *
    * (4) We clamp factor `t` between [0.0f, 1.0f] because `point_on_line`
    *     can be outside of the line segment:
    *
    *     ```
    *                                          (point_on_line)
    *     (a) x------------------------x (b) -----o---
    *                                           |_|
    *                                             |
    *                                             |
    *                                            (p)
    *     ```
    *
    * (5) Finally, the distance we are interested in is
    *
    *     ```
    *     |point_on_line - p|
    *     ```
    */

   FT_Error  error = FT_Err_Ok;

   FT_Vector  a;                   /* start position */
   FT_Vector  b;                   /* end position   */
   FT_Vector  p;                   /* current point  */

   FT_26D6_Vec  line_segment;      /* `b` - `a` */
   FT_26D6_Vec  p_sub_a;           /* `p` - `a` */

   FT_26D6   sq_line_length;       /* squared length of `line_segment` */
   FT_16D16  factor;               /* factor of the nearest point      */
   FT_26D6   cross;                /* used to determine sign           */

   FT_16D16_Vec  nearest_point;    /* `point_on_line`       */
   FT_16D16_Vec  nearest_vector;   /* `p` - `nearest_point` */


   if ( !line || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( line->edge_type != SDF_EDGE_LINE )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   a = line->start_pos;
   b = line->end_pos;
   p = point;

   line_segment.x = b.x - a.x;
   line_segment.y = b.y - a.y;

   p_sub_a.x = p.x - a.x;
   p_sub_a.y = p.y - a.y;

   sq_line_length = ( line_segment.x * line_segment.x ) / 64 +
                    ( line_segment.y * line_segment.y ) / 64;

   /* currently factor is 26.6 */
   factor = ( p_sub_a.x * line_segment.x ) / 64 +
            ( p_sub_a.y * line_segment.y ) / 64;

   /* now factor is 16.16 */
   factor = FT_DivFix( factor, sq_line_length );

   /* clamp the factor between 0.0 and 1.0 in fixed-point */
   if ( factor > FT_INT_16D16( 1 ) )
     factor = FT_INT_16D16( 1 );
   if ( factor < 0 )
     factor = 0;

   nearest_point.x = FT_MulFix( FT_26D6_16D16( line_segment.x ),
                                factor );
   nearest_point.y = FT_MulFix( FT_26D6_16D16( line_segment.y ),
                                factor );

   nearest_point.x = FT_26D6_16D16( a.x ) + nearest_point.x;
   nearest_point.y = FT_26D6_16D16( a.y ) + nearest_point.y;

   nearest_vector.x = nearest_point.x - FT_26D6_16D16( p.x );
   nearest_vector.y = nearest_point.y - FT_26D6_16D16( p.y );

   cross = FT_MulFix( nearest_vector.x, line_segment.y ) -
           FT_MulFix( nearest_vector.y, line_segment.x );

   /* assign the output */
   out->sign     = cross < 0 ? 1 : -1;
   out->distance = VECTOR_LENGTH_16D16( nearest_vector );

   /* Instead of finding `cross` for checking corner we */
   /* directly set it here.  This is more efficient     */
   /* because if the distance is perpendicular we can   */
   /* directly set it to 1.                             */
   if ( factor != 0 && factor != FT_INT_16D16( 1 ) )
     out->cross = FT_INT_16D16( 1 );
   else
   {
     /* [OPTIMIZATION]: Pre-compute this direction. */
     /* If not perpendicular then compute `cross`.  */
     FT_Vector_NormLen( &line_segment );
     FT_Vector_NormLen( &nearest_vector );

     out->cross = FT_MulFix( line_segment.x, nearest_vector.y ) -
                  FT_MulFix( line_segment.y, nearest_vector.x );
   }

 Exit:
   return error;
 }


 /**************************************************************************
  *
  * @Function:
  *   get_min_distance_conic
  *
  * @Description:
  *   Find the shortest distance from the `conic` Bezier curve to a given
  *   `point` and assign it to `out`.  Use it for conic/quadratic curves
  *   only.
  *
  * @Input:
  *   conic ::
  *     The conic Bezier curve to which the shortest distance is to be
  *     computed.
  *
  *   point ::
  *     Point from which the shortest distance is to be computed.
  *
  * @Output:
  *   out ::
  *     Signed distance from `point` to `conic`.
  *
  * @Return:
  *     FreeType error, 0 means success.
  *
  * @Note:
  *   The `conic` parameter must have an edge type of `SDF_EDGE_CONIC`.
  *
  */

#if !USE_NEWTON_FOR_CONIC

 /*
  * The function uses an analytical method to find the shortest distance
  * which is faster than the Newton-Raphson method, but has underflows at
  * the moment.  Use Newton's method if you can see artifacts in the SDF.
  */
 static FT_Error
 get_min_distance_conic( SDF_Edge*             conic,
                         FT_26D6_Vec           point,
                         SDF_Signed_Distance*  out )
 {
   /*
    * The procedure to find the shortest distance from a point to a
    * quadratic Bezier curve is similar to the line segment algorithm.  The
    * shortest distance is perpendicular to the Bezier curve; the only
    * difference from line is that there can be more than one
    * perpendicular, and we also have to check the endpoints, because the
    * perpendicular may not be the shortest.
    *
    * Let's assume that
    * ```
    * p0 = first endpoint
    * p1 = control point
    * p2 = second endpoint
    * p  = point from which shortest distance is to be calculated
    * ```
    *
    * (1) The equation of a quadratic Bezier curve can be written as
    *
    *     ```
    *     B(t) = (1 - t)^2 * p0 + 2(1 - t)t * p1 + t^2 * p2
    *     ```
    *
    *     with `t` a factor in the range [0.0f, 1.0f].  This equation can
    *     be rewritten as
    *
    *     ```
    *     B(t) = t^2 * (p0 - 2p1 + p2) + 2t * (p1 - p0) + p0
    *     ```
    *
    *     With
    *
    *     ```
    *     A = p0 - 2p1 + p2
    *     B = p1 - p0
    *     ```
    *
    *     we have
    *
    *     ```
    *     B(t) = t^2 * A + 2t * B + p0
    *     ```
    *
    * (2) The derivative of the last equation above is
    *
    *     ```
    *     B'(t) = 2 *(tA + B)
    *     ```
    *
    * (3) To find the shortest distance from `p` to `B(t)` we find the
    *     point on the curve at which the shortest distance vector (i.e.,
    *     `B(t) - p`) and the direction (i.e., `B'(t)`) make 90 degrees.
    *     In other words, we make the dot product zero.
    *
    *     ```
    *     (B(t) - p) . (B'(t)) = 0
    *     (t^2 * A + 2t * B + p0 - p) . (2 * (tA + B)) = 0
    *     ```
    *
    *     After simplifying we get a cubic equation
    *
    *     ```
    *     at^3 + bt^2 + ct + d = 0
    *     ```
    *
    *     with
    *
    *     ```
    *     a = A.A
    *     b = 3A.B
    *     c = 2B.B + A.p0 - A.p
    *     d = p0.B - p.B
    *     ```
    *
    * (4) Now the roots of the equation can be computed using 'Cardano's
    *     Cubic formula'; we clamp the roots in the range [0.0f, 1.0f].
    *
    * [note]: `B` and `B(t)` are different in the above equations.
    */

   FT_Error  error = FT_Err_Ok;

   FT_26D6_Vec  aA, bB;         /* A, B in the above comment             */
   FT_26D6_Vec  nearest_point = { 0, 0 };
                                /* point on curve nearest to `point`     */
   FT_26D6_Vec  direction;      /* direction of curve at `nearest_point` */

   FT_26D6_Vec  p0, p1, p2;     /* control points of a conic curve       */
   FT_26D6_Vec  p;              /* `point` to which shortest distance    */

   FT_26D6  a, b, c, d;         /* cubic coefficients                    */

   FT_16D16  roots[3] = { 0, 0, 0 };    /* real roots of the cubic eq.   */
   FT_16D16  min_factor;                /* factor at `nearest_point`     */
   FT_16D16  cross;                     /* to determine the sign         */
   FT_16D16  min      = FT_INT_MAX;     /* shortest squared distance     */

   FT_UShort  num_roots;                /* number of real roots of cubic */
   FT_UShort  i;


   if ( !conic || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( conic->edge_type != SDF_EDGE_CONIC )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   p0 = conic->start_pos;
   p1 = conic->control_a;
   p2 = conic->end_pos;
   p  = point;

   /* compute substitution coefficients */
   aA.x = p0.x - 2 * p1.x + p2.x;
   aA.y = p0.y - 2 * p1.y + p2.y;

   bB.x = p1.x - p0.x;
   bB.y = p1.y - p0.y;

   /* compute cubic coefficients */
   a = VEC_26D6_DOT( aA, aA );

   b = 3 * VEC_26D6_DOT( aA, bB );

   c = 2 * VEC_26D6_DOT( bB, bB ) +
           VEC_26D6_DOT( aA, p0 ) -
           VEC_26D6_DOT( aA, p );

   d = VEC_26D6_DOT( p0, bB ) -
       VEC_26D6_DOT( p, bB );

   /* find the roots */
   num_roots = solve_cubic_equation( a, b, c, d, roots );

   if ( num_roots == 0 )
   {
     roots[0]  = 0;
     roots[1]  = FT_INT_16D16( 1 );
     num_roots = 2;
   }

   /* [OPTIMIZATION]: Check the roots, clamp them and discard */
   /*                 duplicate roots.                        */

   /* convert these values to 16.16 for further computation */
   aA.x = FT_26D6_16D16( aA.x );
   aA.y = FT_26D6_16D16( aA.y );

   bB.x = FT_26D6_16D16( bB.x );
   bB.y = FT_26D6_16D16( bB.y );

   p0.x = FT_26D6_16D16( p0.x );
   p0.y = FT_26D6_16D16( p0.y );

   p.x = FT_26D6_16D16( p.x );
   p.y = FT_26D6_16D16( p.y );

   for ( i = 0; i < num_roots; i++ )
   {
     FT_16D16  t    = roots[i];
     FT_16D16  t2   = 0;
     FT_16D16  dist = 0;

     FT_16D16_Vec  curve_point;
     FT_16D16_Vec  dist_vector;

     /*
      * Ideally we should discard the roots which are outside the range
      * [0.0, 1.0] and check the endpoints of the Bezier curve, but Behdad
      * Esfahbod proved the following lemma.
      *
      * Lemma:
      *
      * (1) If the closest point on the curve [0, 1] is to the endpoint at
      *     `t` = 1 and the cubic has no real roots at `t` = 1 then the
      *     cubic must have a real root at some `t` > 1.
      *
      * (2) Similarly, if the closest point on the curve [0, 1] is to the
      *     endpoint at `t` = 0 and the cubic has no real roots at `t` = 0
      *     then the cubic must have a real root at some `t` < 0.
      *
      * Now because of this lemma we only need to clamp the roots and that
      * will take care of the endpoints.
      *
      * For more details see
      *
      *   https://lists.nongnu.org/archive/html/freetype-devel/2020-06/msg00147.html
      */

     if ( t < 0 )
       t = 0;
     if ( t > FT_INT_16D16( 1 ) )
       t = FT_INT_16D16( 1 );

     t2 = FT_MulFix( t, t );

     /* B(t) = t^2 * A + 2t * B + p0 - p */
     curve_point.x = FT_MulFix( aA.x, t2 ) +
                     2 * FT_MulFix( bB.x, t ) + p0.x;
     curve_point.y = FT_MulFix( aA.y, t2 ) +
                     2 * FT_MulFix( bB.y, t ) + p0.y;

     /* `curve_point` - `p` */
     dist_vector.x = curve_point.x - p.x;
     dist_vector.y = curve_point.y - p.y;

     dist = VECTOR_LENGTH_16D16( dist_vector );

     if ( dist < min )
     {
       min           = dist;
       nearest_point = curve_point;
       min_factor    = t;
     }
   }

   /* B'(t) = 2 * (tA + B) */
   direction.x = 2 * FT_MulFix( aA.x, min_factor ) + 2 * bB.x;
   direction.y = 2 * FT_MulFix( aA.y, min_factor ) + 2 * bB.y;

   /* determine the sign */
   cross = FT_MulFix( nearest_point.x - p.x, direction.y ) -
           FT_MulFix( nearest_point.y - p.y, direction.x );

   /* assign the values */
   out->distance = min;
   out->sign     = cross < 0 ? 1 : -1;

   if ( min_factor != 0 && min_factor != FT_INT_16D16( 1 ) )
     out->cross = FT_INT_16D16( 1 );   /* the two are perpendicular */
   else
   {
     /* convert to nearest vector */
     nearest_point.x -= FT_26D6_16D16( p.x );
     nearest_point.y -= FT_26D6_16D16( p.y );

     /* compute `cross` if not perpendicular */
     FT_Vector_NormLen( &direction );
     FT_Vector_NormLen( &nearest_point );

     out->cross = FT_MulFix( direction.x, nearest_point.y ) -
                  FT_MulFix( direction.y, nearest_point.x );
   }

 Exit:
   return error;
 }

#else /* USE_NEWTON_FOR_CONIC */

 /*
  * The function uses Newton's approximation to find the shortest distance,
  * which is a bit slower than the analytical method but doesn't cause
  * underflow.
  */
 static FT_Error
 get_min_distance_conic( SDF_Edge*             conic,
                         FT_26D6_Vec           point,
                         SDF_Signed_Distance*  out )
 {
   /*
    * This method uses Newton-Raphson's approximation to find the shortest
    * distance from a point to a conic curve.  It does not involve solving
    * any cubic equation, that is why there is no risk of underflow.
    *
    * Let's assume that
    *
    * ```
    * p0 = first endpoint
    * p1 = control point
    * p3 = second endpoint
    * p  = point from which shortest distance is to be calculated
    * ```
    *
    * (1) The equation of a quadratic Bezier curve can be written as
    *
    *     ```
    *     B(t) = (1 - t)^2 * p0 + 2(1 - t)t * p1 + t^2 * p2
    *     ```
    *
    *     with `t` the factor in the range [0.0f, 1.0f].  The above
    *     equation can be rewritten as
    *
    *     ```
    *     B(t) = t^2 * (p0 - 2p1 + p2) + 2t * (p1 - p0) + p0
    *     ```
    *
    *     With
    *
    *     ```
    *     A = p0 - 2p1 + p2
    *     B = 2 * (p1 - p0)
    *     ```
    *
    *     we have
    *
    *     ```
    *     B(t) = t^2 * A + t * B + p0
    *     ```
    *
    * (2) The derivative of the above equation is
    *
    *     ```
    *     B'(t) = 2t * A + B
    *     ```
    *
    * (3) The second derivative of the above equation is
    *
    *     ```
    *     B''(t) = 2A
    *     ```
    *
    * (4) The equation `P(t)` of the distance from point `p` to the curve
    *     can be written as
    *
    *     ```
    *     P(t) = t^2 * A + t^2 * B + p0 - p
    *     ```
    *
    *     With
    *
    *     ```
    *     C = p0 - p
    *     ```
    *
    *     we have
    *
    *     ```
    *     P(t) = t^2 * A + t * B + C
    *     ```
    *
    * (5) Finally, the equation of the angle between `B(t)` and `P(t)` can
    *     be written as
    *
    *     ```
    *     Q(t) = P(t) . B'(t)
    *     ```
    *
    * (6) Our task is to find a value of `t` such that the above equation
    *     `Q(t)` becomes zero, that is, the point-to-curve vector makes
    *     90~degrees with the curve.  We solve this with the Newton-Raphson
    *     method.
    *
    * (7) We first assume an arbitrary value of factor `t`, which we then
    *     improve.
    *
    *     ```
    *     t := Q(t) / Q'(t)
    *     ```
    *
    *     Putting the value of `Q(t)` from the above equation gives
    *
    *     ```
    *     t := P(t) . B'(t) / derivative(P(t) . B'(t))
    *     t := P(t) . B'(t) /
    *            (P'(t) . B'(t) + P(t) . B''(t))
    *     ```
    *
    *     Note that `P'(t)` is the same as `B'(t)` because the constant is
    *     gone due to the derivative.
    *
    * (8) Finally we get the equation to improve the factor as
    *
    *     ```
    *     t := P(t) . B'(t) /
    *            (B'(t) . B'(t) + P(t) . B''(t))
    *     ```
    *
    * [note]: `B` and `B(t)` are different in the above equations.
    */

   FT_Error  error = FT_Err_Ok;

   FT_26D6_Vec  aA, bB, cC;     /* A, B, C in the above comment          */
   FT_26D6_Vec  nearest_point = { 0, 0 };
                                /* point on curve nearest to `point`     */
   FT_26D6_Vec  direction;      /* direction of curve at `nearest_point` */

   FT_26D6_Vec  p0, p1, p2;     /* control points of a conic curve       */
   FT_26D6_Vec  p;              /* `point` to which shortest distance    */

   FT_16D16  min_factor = 0;            /* factor at `nearest_point'     */
   FT_16D16  cross;                     /* to determine the sign         */
   FT_16D16  min        = FT_INT_MAX;   /* shortest squared distance     */

   FT_UShort  iterations;
   FT_UShort  steps;


   if ( !conic || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( conic->edge_type != SDF_EDGE_CONIC )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   p0 = conic->start_pos;
   p1 = conic->control_a;
   p2 = conic->end_pos;
   p  = point;

   /* compute substitution coefficients */
   aA.x = p0.x - 2 * p1.x + p2.x;
   aA.y = p0.y - 2 * p1.y + p2.y;

   bB.x = 2 * ( p1.x - p0.x );
   bB.y = 2 * ( p1.y - p0.y );

   cC.x = p0.x;
   cC.y = p0.y;

   /* do Newton's iterations */
   for ( iterations = 0; iterations <= MAX_NEWTON_DIVISIONS; iterations++ )
   {
     FT_16D16  factor = FT_INT_16D16( iterations ) / MAX_NEWTON_DIVISIONS;
     FT_16D16  factor2;
     FT_16D16  length;

     FT_16D16_Vec  curve_point; /* point on the curve  */
     FT_16D16_Vec  dist_vector; /* `curve_point` - `p` */

     FT_26D6_Vec  d1;           /* first  derivative   */
     FT_26D6_Vec  d2;           /* second derivative   */

     FT_16D16  temp1;
     FT_16D16  temp2;


     for ( steps = 0; steps < MAX_NEWTON_STEPS; steps++ )
     {
       factor2 = FT_MulFix( factor, factor );

       /* B(t) = t^2 * A + t * B + p0 */
       curve_point.x = FT_MulFix( aA.x, factor2 ) +
                       FT_MulFix( bB.x, factor ) + cC.x;
       curve_point.y = FT_MulFix( aA.y, factor2 ) +
                       FT_MulFix( bB.y, factor ) + cC.y;

       /* convert to 16.16 */
       curve_point.x = FT_26D6_16D16( curve_point.x );
       curve_point.y = FT_26D6_16D16( curve_point.y );

       /* P(t) in the comment */
       dist_vector.x = curve_point.x - FT_26D6_16D16( p.x );
       dist_vector.y = curve_point.y - FT_26D6_16D16( p.y );

       length = VECTOR_LENGTH_16D16( dist_vector );

       if ( length < min )
       {
         min           = length;
         min_factor    = factor;
         nearest_point = curve_point;
       }

       /* This is Newton's approximation.          */
       /*                                          */
       /*   t := P(t) . B'(t) /                    */
       /*          (B'(t) . B'(t) + P(t) . B''(t)) */

       /* B'(t) = 2tA + B */
       d1.x = FT_MulFix( aA.x, 2 * factor ) + bB.x;
       d1.y = FT_MulFix( aA.y, 2 * factor ) + bB.y;

       /* B''(t) = 2A */
       d2.x = 2 * aA.x;
       d2.y = 2 * aA.y;

       dist_vector.x /= 1024;
       dist_vector.y /= 1024;

       /* temp1 = P(t) . B'(t) */
       temp1 = VEC_26D6_DOT( dist_vector, d1 );

       /* temp2 = B'(t) . B'(t) + P(t) . B''(t) */
       temp2 = VEC_26D6_DOT( d1, d1 ) +
               VEC_26D6_DOT( dist_vector, d2 );

       factor -= FT_DivFix( temp1, temp2 );

       if ( factor < 0 || factor > FT_INT_16D16( 1 ) )
         break;
     }
   }

   /* B'(t) = 2t * A + B */
   direction.x = 2 * FT_MulFix( aA.x, min_factor ) + bB.x;
   direction.y = 2 * FT_MulFix( aA.y, min_factor ) + bB.y;

   /* determine the sign */
   cross = FT_MulFix( nearest_point.x - FT_26D6_16D16( p.x ),
                      direction.y )                           -
           FT_MulFix( nearest_point.y - FT_26D6_16D16( p.y ),
                      direction.x );

   /* assign the values */
   out->distance = min;
   out->sign     = cross < 0 ? 1 : -1;

   if ( min_factor != 0 && min_factor != FT_INT_16D16( 1 ) )
     out->cross = FT_INT_16D16( 1 );   /* the two are perpendicular */
   else
   {
     /* convert to nearest vector */
     nearest_point.x -= FT_26D6_16D16( p.x );
     nearest_point.y -= FT_26D6_16D16( p.y );

     /* compute `cross` if not perpendicular */
     FT_Vector_NormLen( &direction );
     FT_Vector_NormLen( &nearest_point );

     out->cross = FT_MulFix( direction.x, nearest_point.y ) -
                  FT_MulFix( direction.y, nearest_point.x );
   }

 Exit:
   return error;
 }


#endif /* USE_NEWTON_FOR_CONIC */


 /**************************************************************************
  *
  * @Function:
  *   get_min_distance_cubic
  *
  * @Description:
  *   Find the shortest distance from the `cubic` Bezier curve to a given
  *   `point` and assigns it to `out`.  Use it for cubic curves only.
  *
  * @Input:
  *   cubic ::
  *     The cubic Bezier curve to which the shortest distance is to be
  *     computed.
  *
  *   point ::
  *     Point from which the shortest distance is to be computed.
  *
  * @Output:
  *   out ::
  *     Signed distance from `point` to `cubic`.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  * @Note:
  *   The function uses Newton's approximation to find the shortest
  *   distance.  Another way would be to divide the cubic into conic or
  *   subdivide the curve into lines, but that is not implemented.
  *
  *   The `cubic` parameter must have an edge type of `SDF_EDGE_CUBIC`.
  *
  */
 static FT_Error
 get_min_distance_cubic( SDF_Edge*             cubic,
                         FT_26D6_Vec           point,
                         SDF_Signed_Distance*  out )
 {
   /*
    * The procedure to find the shortest distance from a point to a cubic
    * Bezier curve is similar to quadratic curve algorithm.  The only
    * difference is that while calculating factor `t`, instead of a cubic
    * polynomial equation we have to find the roots of a 5th degree
    * polynomial equation.  Solving this would require a significant amount
    * of time, and still the results may not be accurate.  We are thus
    * going to directly approximate the value of `t` using the Newton-Raphson
    * method.
    *
    * Let's assume that
    *
    * ```
    * p0 = first endpoint
    * p1 = first control point
    * p2 = second control point
    * p3 = second endpoint
    * p  = point from which shortest distance is to be calculated
    * ```
    *
    * (1) The equation of a cubic Bezier curve can be written as
    *
    *     ```
    *     B(t) = (1 - t)^3 * p0 + 3(1 - t)^2 t * p1 +
    *              3(1 - t)t^2 * p2 + t^3 * p3
    *     ```
    *
    *     The equation can be expanded and written as
    *
    *     ```
    *     B(t) = t^3 * (-p0 + 3p1 - 3p2 + p3) +
    *              3t^2 * (p0 - 2p1 + p2) + 3t * (-p0 + p1) + p0
    *     ```
    *
    *     With
    *
    *     ```
    *     A = -p0 + 3p1 - 3p2 + p3
    *     B = 3(p0 - 2p1 + p2)
    *     C = 3(-p0 + p1)
    *     ```
    *
    *     we have
    *
    *     ```
    *     B(t) = t^3 * A + t^2 * B + t * C + p0
    *     ```
    *
    * (2) The derivative of the above equation is
    *
    *     ```
    *     B'(t) = 3t^2 * A + 2t * B + C
    *     ```
    *
    * (3) The second derivative of the above equation is
    *
    *     ```
    *     B''(t) = 6t * A + 2B
    *     ```
    *
    * (4) The equation `P(t)` of the distance from point `p` to the curve
    *     can be written as
    *
    *     ```
    *     P(t) = t^3 * A + t^2 * B + t * C + p0 - p
    *     ```
    *
    *     With
    *
    *     ```
    *     D = p0 - p
    *     ```
    *
    *     we have
    *
    *     ```
    *     P(t) = t^3 * A + t^2 * B + t * C + D
    *     ```
    *
    * (5) Finally the equation of the angle between `B(t)` and `P(t)` can
    *     be written as
    *
    *     ```
    *     Q(t) = P(t) . B'(t)
    *     ```
    *
    * (6) Our task is to find a value of `t` such that the above equation
    *     `Q(t)` becomes zero, that is, the point-to-curve vector makes
    *     90~degree with curve.  We solve this with the Newton-Raphson
    *     method.
    *
    * (7) We first assume an arbitrary value of factor `t`, which we then
    *     improve.
    *
    *     ```
    *     t := Q(t) / Q'(t)
    *     ```
    *
    *     Putting the value of `Q(t)` from the above equation gives
    *
    *     ```
    *     t := P(t) . B'(t) / derivative(P(t) . B'(t))
    *     t := P(t) . B'(t) /
    *            (P'(t) . B'(t) + P(t) . B''(t))
    *     ```
    *
    *     Note that `P'(t)` is the same as `B'(t)` because the constant is
    *     gone due to the derivative.
    *
    * (8) Finally we get the equation to improve the factor as
    *
    *     ```
    *     t := P(t) . B'(t) /
    *            (B'(t) . B'( t ) + P(t) . B''(t))
    *     ```
    *
    * [note]: `B` and `B(t)` are different in the above equations.
    */

   FT_Error  error = FT_Err_Ok;

   FT_26D6_Vec   aA, bB, cC, dD; /* A, B, C, D in the above comment       */
   FT_16D16_Vec  nearest_point = { 0, 0 };
                                 /* point on curve nearest to `point`     */
   FT_16D16_Vec  direction;      /* direction of curve at `nearest_point` */

   FT_26D6_Vec  p0, p1, p2, p3;  /* control points of a cubic curve       */
   FT_26D6_Vec  p;               /* `point` to which shortest distance    */

   FT_16D16  min_factor    = 0;            /* factor at shortest distance */
   FT_16D16  min_factor_sq = 0;            /* factor at shortest distance */
   FT_16D16  cross;                        /* to determine the sign       */
   FT_16D16  min           = FT_INT_MAX;   /* shortest distance           */

   FT_UShort  iterations;
   FT_UShort  steps;


   if ( !cubic || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( cubic->edge_type != SDF_EDGE_CUBIC )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   p0 = cubic->start_pos;
   p1 = cubic->control_a;
   p2 = cubic->control_b;
   p3 = cubic->end_pos;
   p  = point;

   /* compute substitution coefficients */
   aA.x = -p0.x + 3 * ( p1.x - p2.x ) + p3.x;
   aA.y = -p0.y + 3 * ( p1.y - p2.y ) + p3.y;

   bB.x = 3 * ( p0.x - 2 * p1.x + p2.x );
   bB.y = 3 * ( p0.y - 2 * p1.y + p2.y );

   cC.x = 3 * ( p1.x - p0.x );
   cC.y = 3 * ( p1.y - p0.y );

   dD.x = p0.x;
   dD.y = p0.y;

   for ( iterations = 0; iterations <= MAX_NEWTON_DIVISIONS; iterations++ )
   {
     FT_16D16  factor  = FT_INT_16D16( iterations ) / MAX_NEWTON_DIVISIONS;

     FT_16D16  factor2;         /* factor^2            */
     FT_16D16  factor3;         /* factor^3            */
     FT_16D16  length;

     FT_16D16_Vec  curve_point; /* point on the curve  */
     FT_16D16_Vec  dist_vector; /* `curve_point' - `p' */

     FT_26D6_Vec  d1;           /* first  derivative   */
     FT_26D6_Vec  d2;           /* second derivative   */

     FT_16D16  temp1;
     FT_16D16  temp2;


     for ( steps = 0; steps < MAX_NEWTON_STEPS; steps++ )
     {
       factor2 = FT_MulFix( factor, factor );
       factor3 = FT_MulFix( factor2, factor );

       /* B(t) = t^3 * A + t^2 * B + t * C + D */
       curve_point.x = FT_MulFix( aA.x, factor3 ) +
                       FT_MulFix( bB.x, factor2 ) +
                       FT_MulFix( cC.x, factor ) + dD.x;
       curve_point.y = FT_MulFix( aA.y, factor3 ) +
                       FT_MulFix( bB.y, factor2 ) +
                       FT_MulFix( cC.y, factor ) + dD.y;

       /* convert to 16.16 */
       curve_point.x = FT_26D6_16D16( curve_point.x );
       curve_point.y = FT_26D6_16D16( curve_point.y );

       /* P(t) in the comment */
       dist_vector.x = curve_point.x - FT_26D6_16D16( p.x );
       dist_vector.y = curve_point.y - FT_26D6_16D16( p.y );

       length = VECTOR_LENGTH_16D16( dist_vector );

       if ( length < min )
       {
         min           = length;
         min_factor    = factor;
         min_factor_sq = factor2;
         nearest_point = curve_point;
       }

       /* This the Newton's approximation.         */
       /*                                          */
       /*   t := P(t) . B'(t) /                    */
       /*          (B'(t) . B'(t) + P(t) . B''(t)) */

       /* B'(t) = 3t^2 * A + 2t * B + C */
       d1.x = FT_MulFix( aA.x, 3 * factor2 ) +
              FT_MulFix( bB.x, 2 * factor ) + cC.x;
       d1.y = FT_MulFix( aA.y, 3 * factor2 ) +
              FT_MulFix( bB.y, 2 * factor ) + cC.y;

       /* B''(t) = 6t * A + 2B */
       d2.x = FT_MulFix( aA.x, 6 * factor ) + 2 * bB.x;
       d2.y = FT_MulFix( aA.y, 6 * factor ) + 2 * bB.y;

       dist_vector.x /= 1024;
       dist_vector.y /= 1024;

       /* temp1 = P(t) . B'(t) */
       temp1 = VEC_26D6_DOT( dist_vector, d1 );

       /* temp2 = B'(t) . B'(t) + P(t) . B''(t) */
       temp2 = VEC_26D6_DOT( d1, d1 ) +
               VEC_26D6_DOT( dist_vector, d2 );

       factor -= FT_DivFix( temp1, temp2 );

       if ( factor < 0 || factor > FT_INT_16D16( 1 ) )
         break;
     }
   }

   /* B'(t) = 3t^2 * A + 2t * B + C */
   direction.x = FT_MulFix( aA.x, 3 * min_factor_sq ) +
                 FT_MulFix( bB.x, 2 * min_factor ) + cC.x;
   direction.y = FT_MulFix( aA.y, 3 * min_factor_sq ) +
                 FT_MulFix( bB.y, 2 * min_factor ) + cC.y;

   /* determine the sign */
   cross = FT_MulFix( nearest_point.x - FT_26D6_16D16( p.x ),
                      direction.y )                           -
           FT_MulFix( nearest_point.y - FT_26D6_16D16( p.y ),
                      direction.x );

   /* assign the values */
   out->distance = min;
   out->sign     = cross < 0 ? 1 : -1;

   if ( min_factor != 0 && min_factor != FT_INT_16D16( 1 ) )
     out->cross = FT_INT_16D16( 1 );   /* the two are perpendicular */
   else
   {
     /* convert to nearest vector */
     nearest_point.x -= FT_26D6_16D16( p.x );
     nearest_point.y -= FT_26D6_16D16( p.y );

     /* compute `cross` if not perpendicular */
     FT_Vector_NormLen( &direction );
     FT_Vector_NormLen( &nearest_point );

     out->cross = FT_MulFix( direction.x, nearest_point.y ) -
                  FT_MulFix( direction.y, nearest_point.x );
   }

 Exit:
   return error;
 }


 /**************************************************************************
  *
  * @Function:
  *   sdf_edge_get_min_distance
  *
  * @Description:
  *   Find shortest distance from `point` to any type of `edge`.  It checks
  *   the edge type and then calls the relevant `get_min_distance_*`
  *   function.
  *
  * @Input:
  *   edge ::
  *     An edge to which the shortest distance is to be computed.
  *
  *   point ::
  *     Point from which the shortest distance is to be computed.
  *
  * @Output:
  *   out ::
  *     Signed distance from `point` to `edge`.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  */
 static FT_Error
 sdf_edge_get_min_distance( SDF_Edge*             edge,
                            FT_26D6_Vec           point,
                            SDF_Signed_Distance*  out )
 {
   FT_Error  error = FT_Err_Ok;


   if ( !edge || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   /* edge-specific distance calculation */
   switch ( edge->edge_type )
   {
   case SDF_EDGE_LINE:
     get_min_distance_line( edge, point, out );
     break;

   case SDF_EDGE_CONIC:
     get_min_distance_conic( edge, point, out );
     break;

   case SDF_EDGE_CUBIC:
     get_min_distance_cubic( edge, point, out );
     break;

   default:
     error = FT_THROW( Invalid_Argument );
   }

 Exit:
   return error;
 }


 /* `sdf_generate' is not used at the moment */
#if 0

 #error "DO NOT USE THIS!"
 #error "The function still outputs 16-bit data, which might cause memory"
 #error "corruption.  If required I will add this later."

 /**************************************************************************
  *
  * @Function:
  *   sdf_contour_get_min_distance
  *
  * @Description:
  *   Iterate over all edges that make up the contour, find the shortest
  *   distance from a point to this contour, and assigns result to `out`.
  *
  * @Input:
  *   contour ::
  *     A contour to which the shortest distance is to be computed.
  *
  *   point ::
  *     Point from which the shortest distance is to be computed.
  *
  * @Output:
  *   out ::
  *     Signed distance from the `point' to the `contour'.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  * @Note:
  *   The function does not return a signed distance for each edge which
  *   makes up the contour, it simply returns the shortest of all the
  *   edges.
  *
  */
 static FT_Error
 sdf_contour_get_min_distance( SDF_Contour*          contour,
                               FT_26D6_Vec           point,
                               SDF_Signed_Distance*  out )
 {
   FT_Error             error    = FT_Err_Ok;
   SDF_Signed_Distance  min_dist = max_sdf;
   SDF_Edge*            edge_list;


   if ( !contour || !out )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   edge_list = contour->edges;

   /* iterate over all the edges manually */
   while ( edge_list )
   {
     SDF_Signed_Distance  current_dist = max_sdf;
     FT_16D16             diff;


     FT_CALL( sdf_edge_get_min_distance( edge_list,
                                         point,
                                         &current_dist ) );

     if ( current_dist.distance >= 0 )
     {
       diff = current_dist.distance - min_dist.distance;


       if ( FT_ABS( diff ) < CORNER_CHECK_EPSILON )
         min_dist = resolve_corner( min_dist, current_dist );
       else if ( diff < 0 )
         min_dist = current_dist;
     }
     else
       FT_TRACE0(( "sdf_contour_get_min_distance: Overflow.\n" ));

     edge_list = edge_list->next;
   }

   *out = min_dist;

 Exit:
   return error;
 }


 /**************************************************************************
  *
  * @Function:
  *   sdf_generate
  *
  * @Description:
  *   This is the main function that is responsible for generating signed
  *   distance fields.  The function does not align or compute the size of
  *   `bitmap`; therefore the calling application must set up `bitmap`
  *   properly and transform the `shape' appropriately in advance.
  *
  *   Currently we check all pixels against all contours and all edges.
  *
  * @Input:
  *   internal_params ::
  *     Internal parameters and properties required by the rasterizer.  See
  *     @SDF_Params for more.
  *
  *   shape ::
  *     A complete shape which is used to generate SDF.
  *
  *   spread ::
  *     Maximum distances to be allowed in the output bitmap.
  *
  * @Output:
  *   bitmap ::
  *     The output bitmap which will contain the SDF information.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  */
 static FT_Error
 sdf_generate( const SDF_Params  internal_params,
               const SDF_Shape*  shape,
               FT_UInt           spread,
               const FT_Bitmap*  bitmap )
 {
   FT_Error  error = FT_Err_Ok;

   FT_UInt  width = 0;
   FT_UInt  rows  = 0;
   FT_UInt  x     = 0;   /* used to loop in x direction, i.e., width     */
   FT_UInt  y     = 0;   /* used to loop in y direction, i.e., rows      */
   FT_UInt  sp_sq = 0;   /* `spread` [* `spread`] as a 16.16 fixed value */

   FT_Short*  buffer;


   if ( !shape || !bitmap )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( spread < MIN_SPREAD || spread > MAX_SPREAD )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   width  = bitmap->width;
   rows   = bitmap->rows;
   buffer = (FT_Short*)bitmap->buffer;

   if ( USE_SQUARED_DISTANCES )
     sp_sq = FT_INT_16D16( spread * spread );
   else
     sp_sq = FT_INT_16D16( spread );

   if ( width == 0 || rows == 0 )
   {
     FT_TRACE0(( "sdf_generate:"
                 " Cannot render glyph with width/height == 0\n" ));
     FT_TRACE0(( "             "
                 " (width, height provided [%d, %d])\n",
                 width, rows ));

     error = FT_THROW( Cannot_Render_Glyph );
     goto Exit;
   }

   /* loop over all rows */
   for ( y = 0; y < rows; y++ )
   {
     /* loop over all pixels of a row */
     for ( x = 0; x < width; x++ )
     {
       /* `grid_point` is the current pixel position; */
       /* our task is to find the shortest distance   */
       /* from this point to the entire shape.        */
       FT_26D6_Vec          grid_point = zero_vector;
       SDF_Signed_Distance  min_dist   = max_sdf;
       SDF_Contour*         contour_list;

       FT_UInt   index;
       FT_Short  value;


       grid_point.x = FT_INT_26D6( x );
       grid_point.y = FT_INT_26D6( y );

       /* This `grid_point' is at the corner, but we */
       /* use the center of the pixel.               */
       grid_point.x += FT_INT_26D6( 1 ) / 2;
       grid_point.y += FT_INT_26D6( 1 ) / 2;

       contour_list = shape->contours;

       /* iterate over all contours manually */
       while ( contour_list )
       {
         SDF_Signed_Distance  current_dist = max_sdf;


         FT_CALL( sdf_contour_get_min_distance( contour_list,
                                                grid_point,
                                                &current_dist ) );

         if ( current_dist.distance < min_dist.distance )
           min_dist = current_dist;

         contour_list = contour_list->next;
       }

       /* [OPTIMIZATION]: if (min_dist > sp_sq) then simply clamp  */
       /*                 the value to spread to avoid square_root */

       /* clamp the values to spread */
       if ( min_dist.distance > sp_sq )
         min_dist.distance = sp_sq;

       /* square_root the values and fit in a 6.10 fixed-point */
       if ( USE_SQUARED_DISTANCES )
         min_dist.distance = square_root( min_dist.distance );

       if ( internal_params.orientation == FT_ORIENTATION_FILL_LEFT )
         min_dist.sign = -min_dist.sign;
       if ( internal_params.flip_sign )
         min_dist.sign = -min_dist.sign;

       min_dist.distance /= 64; /* convert from 16.16 to 22.10 */

       value  = min_dist.distance & 0x0000FFFF; /* truncate to 6.10 */
       value *= min_dist.sign;

       if ( internal_params.flip_y )
         index = y * width + x;
       else
         index = ( rows - y - 1 ) * width + x;

       buffer[index] = value;
     }
   }

 Exit:
   return error;
 }

#endif /* 0 */


 /**************************************************************************
  *
  * @Function:
  *   sdf_generate_bounding_box
  *
  * @Description:
  *   This function does basically the same thing as `sdf_generate` above
  *   but more efficiently.
  *
  *   Instead of checking all pixels against all edges, we loop over all
  *   edges and only check pixels around the control box of the edge; the
  *   control box is increased by the spread in all directions.  Anything
  *   outside of the control box that exceeds `spread` doesn't need to be
  *   computed.
  *
  *   Lastly, to determine the sign of unchecked pixels, we do a single
  *   pass of all rows starting with a '+' sign and flipping when we come
  *   across a '-' sign and continue.  This also eliminates the possibility
  *   of overflow because we only check the proximity of the curve.
  *   Therefore we can use squared distanced safely.
  *
  * @Input:
  *   internal_params ::
  *     Internal parameters and properties required by the rasterizer.
  *     See @SDF_Params for more.
  *
  *   shape ::
  *     A complete shape which is used to generate SDF.
  *
  *   spread ::
  *     Maximum distances to be allowed in the output bitmap.
  *
  * @Output:
  *   bitmap ::
  *     The output bitmap which will contain the SDF information.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  */
 static FT_Error
 sdf_generate_bounding_box( const SDF_Params  internal_params,
                            const SDF_Shape*  shape,
                            FT_UInt           spread,
                            const FT_Bitmap*  bitmap )
 {
   FT_Error   error  = FT_Err_Ok;
   FT_Memory  memory = NULL;

   FT_Int  width, rows, i, j;
   FT_Int  sp_sq;            /* max value to check   */

   SDF_Contour*   contours;  /* list of all contours */
   FT_SDFFormat*  buffer;    /* the bitmap buffer    */

   /* This buffer has the same size in indices as the    */
   /* bitmap buffer.  When we check a pixel position for */
   /* a shortest distance we keep it in this buffer.     */
   /* This way we can find out which pixel is set,       */
   /* and also determine the signs properly.             */
   SDF_Signed_Distance*  dists = NULL;

   const FT_16D16  fixed_spread = (FT_16D16)FT_INT_16D16( spread );


   if ( !shape || !bitmap )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( spread < MIN_SPREAD || spread > MAX_SPREAD )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   memory = shape->memory;
   if ( !memory )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   if ( FT_ALLOC( dists,
                  bitmap->width * bitmap->rows * sizeof ( *dists ) ) )
     goto Exit;

   contours = shape->contours;
   width    = (FT_Int)bitmap->width;
   rows     = (FT_Int)bitmap->rows;
   buffer   = (FT_SDFFormat*)bitmap->buffer;

   if ( USE_SQUARED_DISTANCES )
     sp_sq = FT_INT_16D16( (FT_Int)( spread * spread ) );
   else
     sp_sq = fixed_spread;

   if ( width == 0 || rows == 0 )
   {
     FT_TRACE0(( "sdf_generate:"
                 " Cannot render glyph with width/height == 0\n" ));
     FT_TRACE0(( "             "
                 " (width, height provided [%d, %d])", width, rows ));

     error = FT_THROW( Cannot_Render_Glyph );
     goto Exit;
   }

   /* loop over all contours */
   while ( contours )
   {
     SDF_Edge*  edges = contours->edges;


     /* loop over all edges */
     while ( edges )
     {
       FT_CBox  cbox;
       FT_Int   x, y;


       /* get the control box and increase it by `spread' */
       cbox = get_control_box( *edges );

       cbox.xMin = ( cbox.xMin - 63 ) / 64 - ( FT_Pos )spread;
       cbox.xMax = ( cbox.xMax + 63 ) / 64 + ( FT_Pos )spread;
       cbox.yMin = ( cbox.yMin - 63 ) / 64 - ( FT_Pos )spread;
       cbox.yMax = ( cbox.yMax + 63 ) / 64 + ( FT_Pos )spread;

       /* now loop over the pixels in the control box. */
       for ( y = cbox.yMin; y < cbox.yMax; y++ )
       {
         for ( x = cbox.xMin; x < cbox.xMax; x++ )
         {
           FT_26D6_Vec          grid_point = zero_vector;
           SDF_Signed_Distance  dist       = max_sdf;
           FT_UInt              index      = 0;
           FT_16D16             diff       = 0;


           if ( x < 0 || x >= width )
             continue;
           if ( y < 0 || y >= rows )
             continue;

           grid_point.x = FT_INT_26D6( x );
           grid_point.y = FT_INT_26D6( y );

           /* This `grid_point` is at the corner, but we */
           /* use the center of the pixel.               */
           grid_point.x += FT_INT_26D6( 1 ) / 2;
           grid_point.y += FT_INT_26D6( 1 ) / 2;

           FT_CALL( sdf_edge_get_min_distance( edges,
                                               grid_point,
                                               &dist ) );

           if ( internal_params.orientation == FT_ORIENTATION_FILL_LEFT )
             dist.sign = -dist.sign;

           /* ignore if the distance is greater than spread;       */
           /* otherwise it creates artifacts due to the wrong sign */
           if ( dist.distance > sp_sq )
             continue;

           /* take the square root of the distance if required */
           if ( USE_SQUARED_DISTANCES )
             dist.distance = square_root( dist.distance );

           if ( internal_params.flip_y )
             index = (FT_UInt)( y * width + x );
           else
             index = (FT_UInt)( ( rows - y - 1 ) * width + x );

           /* check whether the pixel is set or not */
           if ( dists[index].sign == 0 )
             dists[index] = dist;
           else
           {
             diff = FT_ABS( dists[index].distance - dist.distance );

             if ( diff <= CORNER_CHECK_EPSILON )
               dists[index] = resolve_corner( dists[index], dist );
             else if ( dists[index].distance > dist.distance )
               dists[index] = dist;
           }
         }
       }

       edges = edges->next;
     }

     contours = contours->next;
   }

   /* final pass */
   for ( j = 0; j < rows; j++ )
   {
     /* We assume the starting pixel of each row is outside. */
     FT_Char  current_sign = -1;
     FT_UInt  index;


     if ( internal_params.overload_sign != 0 )
       current_sign = internal_params.overload_sign < 0 ? -1 : 1;

     for ( i = 0; i < width; i++ )
     {
       index = (FT_UInt)( j * width + i );

       /* if the pixel is not set                     */
       /* its shortest distance is more than `spread` */
       if ( dists[index].sign == 0 )
         dists[index].distance = fixed_spread;
       else
         current_sign = dists[index].sign;

       /* clamp the values */
       if ( dists[index].distance > fixed_spread )
         dists[index].distance = fixed_spread;

       /* flip sign if required */
       dists[index].distance *= internal_params.flip_sign ? -current_sign
                                                          :  current_sign;

       /* concatenate to appropriate format */
       buffer[index] = map_fixed_to_sdf( dists[index].distance,
                                         fixed_spread );
     }
   }

 Exit:
   FT_FREE( dists );
   return error;
 }


 /**************************************************************************
  *
  * @Function:
  *   sdf_generate_subdivision
  *
  * @Description:
  *   Subdivide the shape into a number of straight lines, then use the
  *   above `sdf_generate_bounding_box` function to generate the SDF.
  *
  *   Note: After calling this function `shape` no longer has the original
  *         edges, it only contains lines.
  *
  * @Input:
  *   internal_params ::
  *     Internal parameters and properties required by the rasterizer.
  *     See @SDF_Params for more.
  *
  *   shape ::
  *     A complete shape which is used to generate SDF.
  *
  *   spread ::
  *     Maximum distances to be allowed in the output bitmap.
  *
  * @Output:
  *   bitmap ::
  *     The output bitmap which will contain the SDF information.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  */
 static FT_Error
 sdf_generate_subdivision( const SDF_Params  internal_params,
                           SDF_Shape*        shape,
                           FT_UInt           spread,
                           const FT_Bitmap*  bitmap )
 {
   /*
    * Thanks to Alexei for providing the idea of this optimization.
    *
    * We take advantage of two facts.
    *
    * (1) Computing the shortest distance from a point to a line segment is
    *     very fast.
    * (2) We don't have to compute the shortest distance for the entire
    *     two-dimensional grid.
    *
    * Both ideas lead to the following optimization.
    *
    * (1) Split the outlines into a number of line segments.
    *
    * (2) For each line segment, only process its neighborhood.
    *
    * (3) Compute the closest distance to the line only for neighborhood
    *     grid points.
    *
    * This greatly reduces the number of grid points to check.
    */

   FT_Error  error = FT_Err_Ok;


   FT_CALL( split_sdf_shape( shape ) );
   FT_CALL( sdf_generate_bounding_box( internal_params,
                                       shape, spread, bitmap ) );

 Exit:
   return error;
 }


 /**************************************************************************
  *
  * @Function:
  *   sdf_generate_with_overlaps
  *
  * @Description:
  *   This function can be used to generate SDF for glyphs with overlapping
  *   contours.  The function generates SDF for contours separately on
  *   separate bitmaps (to generate SDF it uses
  *   `sdf_generate_subdivision`).  At the end it simply combines all the
  *   SDF into the output bitmap; this fixes all the signs and removes
  *   overlaps.
  *
  * @Input:
  *   internal_params ::
  *     Internal parameters and properties required by the rasterizer.  See
  *     @SDF_Params for more.
  *
  *   shape ::
  *     A complete shape which is used to generate SDF.
  *
  *   spread ::
  *     Maximum distances to be allowed in the output bitmap.
  *
  * @Output:
  *   bitmap ::
  *     The output bitmap which will contain the SDF information.
  *
  * @Return:
  *   FreeType error, 0 means success.
  *
  * @Note:
  *   The function cannot generate a proper SDF for glyphs with
  *   self-intersecting contours because we cannot separate them into two
  *   separate bitmaps.  In case of self-intersecting contours it is
  *   necessary to remove the overlaps before generating the SDF.
  *
  */
 static FT_Error
 sdf_generate_with_overlaps( SDF_Params        internal_params,
                             SDF_Shape*        shape,
                             FT_UInt           spread,
                             const FT_Bitmap*  bitmap )
 {
   FT_Error  error = FT_Err_Ok;

   FT_Int      num_contours;        /* total number of contours      */
   FT_Int      i, j;                /* iterators                     */
   FT_Int      width, rows;         /* width and rows of the bitmap  */
   FT_Bitmap*  bitmaps;             /* separate bitmaps for contours */

   SDF_Contour*  contour;           /* temporary variable to iterate */
   SDF_Contour*  temp_contour;      /* temporary contour             */
   SDF_Contour*  head;              /* head of the contour list      */
   SDF_Shape     temp_shape;        /* temporary shape               */

   FT_Memory      memory;           /* to allocate memory            */
   FT_SDFFormat*  t;                /* target bitmap buffer          */
   FT_Bool        flip_sign;        /* flip sign?                    */

   /* orientation of all the separate contours */
   SDF_Contour_Orientation*  orientations;


   bitmaps      = NULL;
   orientations = NULL;
   head         = NULL;

   if ( !shape || !bitmap || !shape->memory )
     return FT_THROW( Invalid_Argument );

   /* Disable `flip_sign` to avoid extra complication */
   /* during the combination phase.                   */
   flip_sign                 = internal_params.flip_sign;
   internal_params.flip_sign = 0;

   contour           = shape->contours;
   memory            = shape->memory;
   temp_shape.memory = memory;
   width             = (FT_Int)bitmap->width;
   rows              = (FT_Int)bitmap->rows;
   num_contours      = 0;

   /* find the number of contours in the shape */
   while ( contour )
   {
     num_contours++;
     contour = contour->next;
   }

   /* allocate the bitmaps to generate SDF for separate contours */
   if ( FT_ALLOC( bitmaps,
                  (FT_UInt)num_contours * sizeof ( *bitmaps ) ) )
     goto Exit;

   /* allocate array to hold orientation for all contours */
   if ( FT_ALLOC( orientations,
                  (FT_UInt)num_contours * sizeof ( *orientations ) ) )
     goto Exit;

   contour = shape->contours;

   /* Iterate over all contours and generate SDF separately. */
   for ( i = 0; i < num_contours; i++ )
   {
     /* initialize the corresponding bitmap */
     FT_Bitmap_Init( &bitmaps[i] );

     bitmaps[i].width      = bitmap->width;
     bitmaps[i].rows       = bitmap->rows;
     bitmaps[i].pitch      = bitmap->pitch;
     bitmaps[i].num_grays  = bitmap->num_grays;
     bitmaps[i].pixel_mode = bitmap->pixel_mode;

     /* allocate memory for the buffer */
     if ( FT_ALLOC( bitmaps[i].buffer,
                    bitmap->rows * (FT_UInt)bitmap->pitch ) )
       goto Exit;

     /* determine the orientation */
     orientations[i] = get_contour_orientation( contour );

     /* The `overload_sign` property is specific to  */
     /* `sdf_generate_bounding_box`.  This basically */
     /* overloads the default sign of the outside    */
     /* pixels, which is necessary for               */
     /* counter-clockwise contours.                  */
     if ( orientations[i] == SDF_ORIENTATION_CCW                   &&
          internal_params.orientation == FT_ORIENTATION_FILL_RIGHT )
       internal_params.overload_sign = 1;
     else if ( orientations[i] == SDF_ORIENTATION_CW                   &&
               internal_params.orientation == FT_ORIENTATION_FILL_LEFT )
       internal_params.overload_sign = 1;
     else
       internal_params.overload_sign = 0;

     /* Make `contour->next` NULL so that there is   */
     /* one contour in the list.  Also hold the next */
     /* contour in a temporary variable so as to     */
     /* restore the original value.                  */
     temp_contour  = contour->next;
     contour->next = NULL;

     /* Use `temp_shape` to hold the new contour. */
     /* Now, `temp_shape` has only one contour.   */
     temp_shape.contours = contour;

     /* finally generate the SDF */
     FT_CALL( sdf_generate_subdivision( internal_params,
                                        &temp_shape,
                                        spread,
                                        &bitmaps[i] ) );

     /* Restore the original `next` variable. */
     contour->next = temp_contour;

     /* Since `split_sdf_shape` deallocated the original */
     /* contours list we need to assign the new value to */
     /* the shape's contour.                             */
     temp_shape.contours->next = head;
     head                      = temp_shape.contours;

     /* Simply flip the orientation in case of post-script fonts */
     /* so as to avoid modificatons in the combining phase.      */
     if ( internal_params.orientation == FT_ORIENTATION_FILL_LEFT )
     {
       if ( orientations[i] == SDF_ORIENTATION_CW )
         orientations[i] = SDF_ORIENTATION_CCW;
       else if ( orientations[i] == SDF_ORIENTATION_CCW )
         orientations[i] = SDF_ORIENTATION_CW;
     }

     contour = contour->next;
   }

   /* assign the new contour list to `shape->contours` */
   shape->contours = head;

   /* cast the output bitmap buffer */
   t = (FT_SDFFormat*)bitmap->buffer;

   /* Iterate over all pixels and combine all separate    */
   /* contours.  These are the rules for combining:       */
   /*                                                     */
   /* (1) For all clockwise contours, compute the largest */
   /*     value.  Name this as `val_c`.                   */
   /* (2) For all counter-clockwise contours, compute the */
   /*     smallest value.  Name this as `val_ac`.         */
   /* (3) Now, finally use the smaller value of `val_c'   */
   /*     and `val_ac'.                                   */
   for ( j = 0; j < rows; j++ )
   {
     for ( i = 0; i < width; i++ )
     {
       FT_Int  id = j * width + i;       /* index of current pixel    */
       FT_Int  c;                        /* contour iterator          */

       FT_SDFFormat  val_c  = 0;         /* max clockwise value       */
       FT_SDFFormat  val_ac = UCHAR_MAX; /* min counter-clockwise val */


       /* iterate through all the contours */
       for ( c = 0; c < num_contours; c++ )
       {
         /* current contour value */
         FT_SDFFormat  temp = ( (FT_SDFFormat*)bitmaps[c].buffer )[id];


         if ( orientations[c] == SDF_ORIENTATION_CW )
           val_c = FT_MAX( val_c, temp );   /* clockwise         */
         else
           val_ac = FT_MIN( val_ac, temp ); /* counter-clockwise */
       }

       /* Finally find the smaller of the two and assign to output. */
       /* Also apply `flip_sign` if set.                            */
       t[id] = FT_MIN( val_c, val_ac );

       if ( flip_sign )
         t[id] = invert_sign( t[id] );
     }
   }

 Exit:
   /* deallocate orientations array */
   if ( orientations )
     FT_FREE( orientations );

   /* deallocate temporary bitmaps */
   if ( bitmaps )
   {
     if ( num_contours == 0 )
       error = FT_THROW( Raster_Corrupted );
     else
     {
       for ( i = 0; i < num_contours; i++ )
         FT_FREE( bitmaps[i].buffer );

       FT_FREE( bitmaps );
     }
   }

   /* restore the `flip_sign` property */
   internal_params.flip_sign = flip_sign;

   return error;
 }


 /**************************************************************************
  *
  * interface functions
  *
  */

 static FT_Error
 sdf_raster_new( void*       memory_,   /* FT_Memory    */
                 FT_Raster*  araster_ ) /* SDF_PRaster* */
 {
   FT_Memory     memory  = (FT_Memory)memory_;
   SDF_PRaster*  araster = (SDF_PRaster*)araster_;


   FT_Error     error;
   SDF_PRaster  raster = NULL;


   if ( !FT_NEW( raster ) )
     raster->memory = memory;

   *araster = raster;

  return error;
 }


 static void
 sdf_raster_reset( FT_Raster       raster,
                   unsigned char*  pool_base,
                   unsigned long   pool_size )
 {
   FT_UNUSED( raster );
   FT_UNUSED( pool_base );
   FT_UNUSED( pool_size );
 }


 static FT_Error
 sdf_raster_set_mode( FT_Raster      raster,
                      unsigned long  mode,
                      void*          args )
 {
   FT_UNUSED( raster );
   FT_UNUSED( mode );
   FT_UNUSED( args );

   return FT_Err_Ok;
 }


 static FT_Error
 sdf_raster_render( FT_Raster                raster,
                    const FT_Raster_Params*  params )
 {
   FT_Error                  error      = FT_Err_Ok;
   SDF_TRaster*              sdf_raster = (SDF_TRaster*)raster;
   FT_Outline*               outline    = NULL;
   const SDF_Raster_Params*  sdf_params = (const SDF_Raster_Params*)params;

   FT_Memory   memory = NULL;
   SDF_Shape*  shape  = NULL;
   SDF_Params  internal_params;


   /* check for valid arguments */
   if ( !sdf_raster || !sdf_params )
   {
     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   outline = (FT_Outline*)sdf_params->root.source;

   /* check whether outline is valid */
   if ( !outline )
   {
     error = FT_THROW( Invalid_Outline );
     goto Exit;
   }

   /* if the outline is empty, return */
   if ( outline->n_points == 0 || outline->n_contours == 0 )
     goto Exit;

   /* check whether the outline has valid fields */
   if ( !outline->contours || !outline->points )
   {
     error = FT_THROW( Invalid_Outline );
     goto Exit;
   }

   /* check whether spread is set properly */
   if ( sdf_params->spread > MAX_SPREAD ||
        sdf_params->spread < MIN_SPREAD )
   {
     FT_TRACE0(( "sdf_raster_render:"
                 " The `spread' field of `SDF_Raster_Params' is invalid,\n" ));
     FT_TRACE0(( "                  "
                 " the value of this field must be within [%d, %d].\n",
                 MIN_SPREAD, MAX_SPREAD ));
     FT_TRACE0(( "                  "
                 " Also, you must pass `SDF_Raster_Params' instead of\n" ));
     FT_TRACE0(( "                  "
                 " the default `FT_Raster_Params' while calling\n" ));
     FT_TRACE0(( "                  "
                 " this function and set the fields properly.\n" ));

     error = FT_THROW( Invalid_Argument );
     goto Exit;
   }

   memory = sdf_raster->memory;
   if ( !memory )
   {
     FT_TRACE0(( "sdf_raster_render:"
                 " Raster not setup properly,\n" ));
     FT_TRACE0(( "                  "
                 " unable to find memory handle.\n" ));

     error = FT_THROW( Invalid_Handle );
     goto Exit;
   }

   /* set up the parameters */
   internal_params.orientation   = FT_Outline_Get_Orientation( outline );
   internal_params.flip_sign     = sdf_params->flip_sign;
   internal_params.flip_y        = sdf_params->flip_y;
   internal_params.overload_sign = 0;

   FT_CALL( sdf_shape_new( memory, &shape ) );

   FT_CALL( sdf_outline_decompose( outline, shape ) );

   if ( sdf_params->overlaps )
     FT_CALL( sdf_generate_with_overlaps( internal_params,
                                          shape, sdf_params->spread,
                                          sdf_params->root.target ) );
   else
     FT_CALL( sdf_generate_subdivision( internal_params,
                                        shape, sdf_params->spread,
                                        sdf_params->root.target ) );

   if ( shape )
     sdf_shape_done( &shape );

 Exit:
   return error;
 }


 static void
 sdf_raster_done( FT_Raster  raster )
 {
   FT_Memory  memory = (FT_Memory)((SDF_TRaster*)raster)->memory;


   FT_FREE( raster );
 }


 FT_DEFINE_RASTER_FUNCS(
   ft_sdf_raster,

   FT_GLYPH_FORMAT_OUTLINE,

   (FT_Raster_New_Func)     sdf_raster_new,       /* raster_new      */
   (FT_Raster_Reset_Func)   sdf_raster_reset,     /* raster_reset    */
   (FT_Raster_Set_Mode_Func)sdf_raster_set_mode,  /* raster_set_mode */
   (FT_Raster_Render_Func)  sdf_raster_render,    /* raster_render   */
   (FT_Raster_Done_Func)    sdf_raster_done       /* raster_done     */
 )


/* END */