[sdf] Added shortest distance finding functions.
* src/sdf/ftsdf.c (get_min_distance_): Added function to find the closest distance from a point to a curves of all three different types (i.e. line segment, conic bezier and cubic bezier).
This commit is contained in:
Anuj Verma
parent
68032a77e3
commit
c39b5dd849
937
src/sdf/ftsdf.c
937
src/sdf/ftsdf.c
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@ -1569,4 +1569,941 @@
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return FT_ABS( sdf1.cross ) > FT_ABS( sdf2.cross ) ? sdf1 : sdf2;
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}
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/**************************************************************************
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*
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* @Function:
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* get_min_distance_line
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*
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* @Description:
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* This function find the shortest distance from the `line' to
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* a given `point' and assigns it to `out'. Only use it for line
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* segments.
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*
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* @Input:
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* line ::
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* The line segment to which the shortest distance is to be
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* computed.
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*
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* point ::
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* Point from which the shortest distance is to be computed.
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*
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* @Return:
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* out ::
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* Signed distance from the `point' to the `line'.
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*
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* FT_Error ::
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* FreeType error, 0 means success.
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*
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* @Note:
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* The `line' parameter must have a `edge_type' of `SDF_EDGE_LINE'.
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*
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*/
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static FT_Error
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get_min_distance_line( SDF_Edge* line,
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FT_26D6_Vec point,
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SDF_Signed_Distance* out )
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{
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/* in order to calculate the shortest distance from a point to */
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/* a line segment. */
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/* */
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/* a = start point of the line segment */
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/* b = end point of the line segment */
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/* p = point from which shortest distance is to be calculated */
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/* ----------------------------------------------------------- */
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/* => we first write the parametric equation of the line */
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/* point_on_line = a + ( b - a ) * t ( t is the factor ) */
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/* */
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/* => next we find the projection of point p on the line. the */
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/* projection will be perpendicular to the line, that is */
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/* why we can find it by making the dot product zero. */
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/* ( point_on_line - a ) . ( p - point_on_line ) = 0 */
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/* */
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/* ( point_on_line ) */
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/* ( a ) x-------o----------------x ( b ) */
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/* |_| */
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/* | */
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/* | */
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/* ( p ) */
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/* */
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/* => by simplifying the above equation we get the factor of */
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/* point_on_line such that */
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/* t = ( ( p - a ) . ( b - a ) ) / ( |b - a| ^ 2 ) */
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/* */
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/* => we clamp the factor t between [0.0f, 1.0f], because the */
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/* point_on_line can be outside the line segment. */
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/* */
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/* ( point_on_line ) */
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/* ( a ) x------------------------x ( b ) -----o--- */
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/* |_| */
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/* | */
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/* | */
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/* ( p ) */
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/* */
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/* => finally the distance becomes | point_on_line - p | */
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FT_Error error = FT_Err_Ok;
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FT_Vector a; /* start position */
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FT_Vector b; /* end position */
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FT_Vector p; /* current point */
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FT_26D6_Vec line_segment; /* `b' - `a'*/
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FT_26D6_Vec p_sub_a; /* `p' - `a' */
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FT_26D6 sq_line_length; /* squared length of `line_segment' */
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FT_16D16 factor; /* factor of the nearest point */
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FT_26D6 cross; /* used to determine sign */
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FT_16D16_Vec nearest_point; /* `point_on_line' */
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FT_16D16_Vec nearest_vector; /* `p' - `nearest_point' */
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if ( !line || !out )
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{
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error = FT_THROW( Invalid_Argument );
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goto Exit;
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}
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if ( line->edge_type != SDF_EDGE_LINE )
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{
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error = FT_THROW( Invalid_Argument );
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goto Exit;
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}
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a = line->start_pos;
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b = line->end_pos;
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p = point;
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line_segment.x = b.x - a.x;
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line_segment.y = b.y - a.y;
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p_sub_a.x = p.x - a.x;
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p_sub_a.y = p.y - a.y;
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sq_line_length = ( line_segment.x * line_segment.x ) / 64 +
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( line_segment.y * line_segment.y ) / 64;
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/* currently factor is 26.6 */
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factor = ( p_sub_a.x * line_segment.x ) / 64 +
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( p_sub_a.y * line_segment.y ) / 64;
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/* now factor is 16.16 */
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factor = FT_DivFix( factor, sq_line_length );
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/* clamp the factor between 0.0 and 1.0 in fixed point */
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if ( factor > FT_INT_16D16( 1 ) )
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factor = FT_INT_16D16( 1 );
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if ( factor < 0 )
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factor = 0;
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nearest_point.x = FT_MulFix( FT_26D6_16D16(line_segment.x),
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factor );
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nearest_point.y = FT_MulFix( FT_26D6_16D16(line_segment.y),
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factor );
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nearest_point.x = FT_26D6_16D16( a.x ) + nearest_point.x;
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nearest_point.y = FT_26D6_16D16( a.y ) + nearest_point.y;
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nearest_vector.x = nearest_point.x - FT_26D6_16D16( p.x );
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nearest_vector.y = nearest_point.y - FT_26D6_16D16( p.y );
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cross = FT_MulFix( nearest_vector.x, line_segment.y ) -
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FT_MulFix( nearest_vector.y, line_segment.x );
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/* assign the output */
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out->sign = cross < 0 ? 1 : -1;
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out->distance = VECTOR_LENGTH_16D16( nearest_vector );
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/* Instead of finding cross for checking corner we */
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/* directly set it here. This is more efficient */
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/* because if the distance is perpendicular we can */
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/* directly set it to 1. */
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if ( factor != 0 && factor != FT_INT_16D16( 1 ) )
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out->cross = FT_INT_16D16( 1 );
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else
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{
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/* [OPTIMIZATION]: Pre-compute this direction. */
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/* if not perpendicular then compute the cross */
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FT_Vector_NormLen( &line_segment );
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FT_Vector_NormLen( &nearest_vector );
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out->cross = FT_MulFix( line_segment.x, nearest_vector.y ) -
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FT_MulFix( line_segment.y, nearest_vector.x );
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}
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Exit:
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return error;
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}
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#if !USE_NEWTON_FOR_CONIC
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/**************************************************************************
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*
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* @Function:
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* get_min_distance_conic
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*
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* @Description:
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* This function find the shortest distance from the `conic' bezier
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* curve to a given `point' and assigns it to `out'. Only use it for
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* conic/quadratic curves.
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*
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* @Input:
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* conic ::
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* The conic bezier to which the shortest distance is to be
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* computed.
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*
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* point ::
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* Point from which the shortest distance is to be computed.
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*
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* @Return:
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* out ::
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* Signed distance from the `point' to the `conic'.
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*
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* FT_Error ::
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* FreeType error, 0 means success.
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*
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* @Note:
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* The function uses analytical method to find shortest distance
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* which is faster than the Newton-Raphson's method, but has
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* underflows at the moment. Use Newton's method if you can
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* see artifacts in the SDF.
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*
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* The `conic' parameter must have a `edge_type' of `SDF_EDGE_CONIC'.
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*
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*/
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static FT_Error
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get_min_distance_conic( SDF_Edge* conic,
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FT_26D6_Vec point,
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SDF_Signed_Distance* out )
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{
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/* The procedure to find the shortest distance from a point to */
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/* a quadratic bezier curve is similar to a line segment. the */
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/* shortest distance will be perpendicular to the bezier curve */
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/* The only difference from line is that there can be more */
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/* than one perpendicular and we also have to check the endpo- */
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/* -ints, because the perpendicular may not be the shortest. */
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/* */
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/* p0 = first endpoint */
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/* p1 = control point */
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/* p2 = second endpoint */
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/* p = point from which shortest distance is to be calculated */
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/* ----------------------------------------------------------- */
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/* => the equation of a quadratic bezier curve can be written */
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/* B( t ) = ( ( 1 - t )^2 )p0 + 2( 1 - t )tp1 + t^2p2 */
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/* here t is the factor with range [0.0f, 1.0f] */
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/* the above equation can be rewritten as */
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/* B( t ) = t^2( p0 - 2p1 + p2 ) + 2t( p1 - p0 ) + p0 */
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/* */
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/* now let A = ( p0 - 2p1 + p2), B = ( p1 - p0 ) */
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/* B( t ) = t^2( A ) + 2t( B ) + p0 */
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/* */
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/* => the derivative of the above equation is written as */
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/* B'( t ) = 2( tA + B ) */
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/* */
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/* => now to find the shortest distance from p to B( t ), we */
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/* find the point on the curve at which the shortest */
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/* distance vector ( i.e. B( t ) - p ) and the direction */
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/* ( i.e. B'( t )) makes 90 degrees. in other words we make */
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/* the dot product zero. */
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/* ( B( t ) - p ).( B'( t ) ) = 0 */
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/* ( t^2( A ) + 2t( B ) + p0 - p ).( 2( tA + B ) ) = 0 */
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/* */
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/* after simplifying we get a cubic equation as */
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/* at^3 + bt^2 + ct + d = 0 */
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/* a = ( A.A ), b = ( 3A.B ), c = ( 2B.B + A.p0 - A.p ) */
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/* d = ( p0.B - p.B ) */
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/* */
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/* => now the roots of the equation can be computed using the */
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/* `Cardano's Cubic formula' we clamp the roots in range */
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/* [0.0f, 1.0f]. */
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/* */
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/* [note]: B and B( t ) are different in the above equations */
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FT_Error error = FT_Err_Ok;
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FT_26D6_Vec aA, bB; /* A, B in the above comment */
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FT_26D6_Vec nearest_point; /* point on curve nearest to `point' */
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FT_26D6_Vec direction; /* direction of curve at `nearest_point' */
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FT_26D6_Vec p0, p1, p2; /* control points of a conic curve */
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FT_26D6_Vec p; /* `point' to which shortest distance */
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FT_26D6 a, b, c, d; /* cubic coefficients */
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FT_16D16 roots[3] = { 0, 0, 0 }; /* real roots of the cubic eq */
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FT_16D16 min_factor; /* factor at `nearest_point' */
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FT_16D16 cross; /* to determine the sign */
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FT_16D16 min = FT_INT_MAX; /* shortest squared distance */
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FT_UShort num_roots; /* number of real roots of cubic */
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FT_UShort i;
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if ( !conic || !out )
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{
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error = FT_THROW( Invalid_Argument );
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goto Exit;
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}
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if ( conic->edge_type != SDF_EDGE_CONIC )
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{
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error = FT_THROW( Invalid_Argument );
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goto Exit;
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}
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/* assign the values after checking pointer */
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p0 = conic->start_pos;
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p1 = conic->control_a;
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p2 = conic->end_pos;
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p = point;
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/* compute substitution coefficients */
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aA.x = p0.x - 2 * p1.x + p2.x;
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aA.y = p0.y - 2 * p1.y + p2.y;
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bB.x = p1.x - p0.x;
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bB.y = p1.y - p0.y;
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/* compute cubic coefficients */
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a = VEC_26D6_DOT( aA, aA );
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b = 3 * VEC_26D6_DOT( aA, bB );
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c = 2 * VEC_26D6_DOT( bB, bB ) +
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VEC_26D6_DOT( aA, p0 ) -
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VEC_26D6_DOT( aA, p );
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d = VEC_26D6_DOT( p0, bB ) -
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VEC_26D6_DOT( p, bB );
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/* find the roots */
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num_roots = solve_cubic_equation( a, b, c, d, roots );
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if ( num_roots == 0 )
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{
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roots[0] = 0;
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roots[1] = FT_INT_16D16( 1 );
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num_roots = 2;
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}
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/* [OPTIMIZATION]: Check the roots, clamp them and discard */
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/* duplicate roots. */
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/* convert these values to 16.16 for further computation */
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aA.x = FT_26D6_16D16( aA.x );
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aA.y = FT_26D6_16D16( aA.y );
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bB.x = FT_26D6_16D16( bB.x );
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bB.y = FT_26D6_16D16( bB.y );
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p0.x = FT_26D6_16D16( p0.x );
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p0.y = FT_26D6_16D16( p0.y );
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p.x = FT_26D6_16D16( p.x );
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p.y = FT_26D6_16D16( p.y );
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for ( i = 0; i < num_roots; i++ )
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{
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FT_16D16 t = roots[i];
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FT_16D16 t2 = 0;
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FT_16D16 dist = 0;
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FT_16D16_Vec curve_point;
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FT_16D16_Vec dist_vector;
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/* Ideally we should discard the roots which are outside the */
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/* range [0.0, 1.0] and check the endpoints of the bezier, but */
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/* Behdad gave me a lemma: */
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/* Lemma: */
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/* * If the closest point on the curve [0, 1] is to the endpoint */
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/* at t = 1 and the cubic has no real roots at t = 1 then, the */
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/* cubic must have a real root at some t > 1. */
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/* * Similarly, */
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/* If the closest point on the curve [0, 1] is to the endpoint */
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/* at t = 0 and the cubic has no real roots at t = 0 then, the */
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/* cubic must have a real root at some t < 0. */
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/* */
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/* Now because of this lemma we only need to clamp the roots and */
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/* that will take care of the endpoints. */
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/* */
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/* For proof contact: behdad@behdad.org */
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/* For more details check message: */
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/* https://lists.nongnu.org/archive/html/freetype-devel/2020-06/msg00147.html */
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if ( t < 0 )
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t = 0;
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if ( t > FT_INT_16D16( 1 ) )
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t = FT_INT_16D16( 1 );
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t2 = FT_MulFix( t, t );
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/* B( t ) = t^2( A ) + 2t( B ) + p0 - p */
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curve_point.x = FT_MulFix( aA.x, t2 ) +
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2 * FT_MulFix( bB.x, t ) + p0.x;
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curve_point.y = FT_MulFix( aA.y, t2 ) +
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2 * FT_MulFix( bB.y, t ) + p0.y;
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/* `curve_point' - `p' */
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dist_vector.x = curve_point.x - p.x;
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dist_vector.y = curve_point.y - p.y;
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dist = VECTOR_LENGTH_16D16( dist_vector );
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if ( dist < min )
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{
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min = dist;
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nearest_point = curve_point;
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min_factor = t;
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}
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}
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/* B'( t ) = 2( tA + B ) */
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direction.x = 2 * FT_MulFix( aA.x, min_factor ) + 2 * bB.x;
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direction.y = 2 * FT_MulFix( aA.y, min_factor ) + 2 * bB.y;
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/* determine the sign */
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cross = FT_MulFix( nearest_point.x - p.x, direction.y ) -
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FT_MulFix( nearest_point.y - p.y, direction.x );
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/* assign the values */
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out->distance = min;
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out->sign = cross < 0 ? 1 : -1;
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if ( min_factor != 0 && min_factor != FT_INT_16D16( 1 ) )
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out->cross = FT_INT_16D16( 1 ); /* the two are perpendicular */
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else
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{
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/* convert to nearest vector */
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nearest_point.x -= FT_26D6_16D16( p.x );
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nearest_point.y -= FT_26D6_16D16( p.y );
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/* if not perpendicular then compute the cross */
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FT_Vector_NormLen( &direction );
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FT_Vector_NormLen( &nearest_point );
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out->cross = FT_MulFix( direction.x, nearest_point.y ) -
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FT_MulFix( direction.y, nearest_point.x );
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}
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Exit:
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return error;
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}
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#else
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/**************************************************************************
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*
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* @Function:
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* get_min_distance_conic
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*
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* @Description:
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* This function find the shortest distance from the `conic' bezier
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* curve to a given `point' and assigns it to `out'. Only use it for
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* conic/quadratic curves.
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*
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* @Input:
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* conic ::
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* The conic bezier to which the shortest distance is to be
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* computed.
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*
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* point ::
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* Point from which the shortest distance is to be computed.
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*
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* @Return:
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* out ::
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* Signed distance from the `point' to the `conic'.
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*
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* FT_Error ::
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* FreeType error, 0 means success.
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||||
*
|
||||
* @Note:
|
||||
* 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. Use is upto your needs.
|
||||
*
|
||||
* The `conic' parameter must have a `edge_type' of `SDF_EDGE_CONIC'.
|
||||
*
|
||||
*/
|
||||
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 conic curve which does */
|
||||
/* not involve solving any cubic equation, that is why there */
|
||||
/* is no risk of underflow. The method is as follows: */
|
||||
/* */
|
||||
/* p0 = first endpoint */
|
||||
/* p1 = control point */
|
||||
/* p3 = second endpoint */
|
||||
/* p = point from which shortest distance is to be calculated */
|
||||
/* ----------------------------------------------------------- */
|
||||
/* => the equation of a quadratic bezier curve can be written */
|
||||
/* B( t ) = ( ( 1 - t )^2 )p0 + 2( 1 - t )tp1 + t^2p2 */
|
||||
/* here t is the factor with range [0.0f, 1.0f] */
|
||||
/* the above equation can be rewritten as */
|
||||
/* B( t ) = t^2( p0 - 2p1 + p2 ) + 2t( p1 - p0 ) + p0 */
|
||||
/* */
|
||||
/* now let A = ( p0 - 2p1 + p2), B = 2( p1 - p0 ) */
|
||||
/* B( t ) = t^2( A ) + t( B ) + p0 */
|
||||
/* */
|
||||
/* => the derivative of the above equation is written as */
|
||||
/* B'( t ) = 2t( A ) + B */
|
||||
/* */
|
||||
/* => further derivative of the above equation is written as */
|
||||
/* B''( t ) = 2A */
|
||||
/* */
|
||||
/* => the equation of distance from point `p' to the curve */
|
||||
/* P( t ) can be written as */
|
||||
/* P( t ) = t^2( A ) + t^2( B ) + p0 - p */
|
||||
/* Now let C = ( p0 - p ) */
|
||||
/* P( t ) = t^2( A ) + t( B ) + C */
|
||||
/* */
|
||||
/* => finally the equation of angle between curve B( t ) and */
|
||||
/* point to curve distance P( t ) can be written as */
|
||||
/* Q( t ) = P( t ).B'( t ) */
|
||||
/* */
|
||||
/* => now our task is to find a value of t such that the above */
|
||||
/* equation Q( t ) becomes zero. in other words the point */
|
||||
/* to curve vector makes 90 degree with curve. this is done */
|
||||
/* by Newton-Raphson's method. */
|
||||
/* */
|
||||
/* => we first assume a arbitary value of the factor `t' and */
|
||||
/* then we improve it using Newton's equation such as */
|
||||
/* */
|
||||
/* t -= Q( t ) / Q'( t ) */
|
||||
/* putting 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 ) ) */
|
||||
/* */
|
||||
/* P'( t ) is noting but B'( t ) because the constant are */
|
||||
/* gone due to derivative */
|
||||
/* */
|
||||
/* => 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; /* 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;
|
||||
}
|
||||
|
||||
/* assign the values after checking pointer */
|
||||
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 );
|
||||
|
||||
/* B( t ) = t^2( A ) + t( B ) + p0 - p. 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 the actual 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 ) = 2tA + 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 );
|
||||
|
||||
/* if not perpendicular then compute the cross */
|
||||
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
|
||||
|
||||
/**************************************************************************
|
||||
*
|
||||
* @Function:
|
||||
* get_min_distance_cubic
|
||||
*
|
||||
* @Description:
|
||||
* This function find the shortest distance from the `cubic' bezier
|
||||
* curve to a given `point' and assigns it to `out'. Only use it for
|
||||
* cubic curves.
|
||||
*
|
||||
* @Input:
|
||||
* cubic ::
|
||||
* The cubic bezier to which the shortest distance is to be
|
||||
* computed.
|
||||
*
|
||||
* point ::
|
||||
* Point from which the shortest distance is to be computed.
|
||||
*
|
||||
* @Return:
|
||||
* out ::
|
||||
* Signed distance from the `point' to the `cubic'.
|
||||
*
|
||||
* FT_Error ::
|
||||
* 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 a `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 a quadratic curve. */
|
||||
/* The only difference is that while calculating the factor */
|
||||
/* `t', instead of a cubic polynomial equation we have to find */
|
||||
/* the roots of a 5th degree polynomial equation. */
|
||||
/* But since solving a 5th degree polynomial equation require */
|
||||
/* significant amount of time and still the results may not be */
|
||||
/* accurate, we are going to directly approximate the value of */
|
||||
/* `t' using Newton-Raphson method */
|
||||
/* */
|
||||
/* p0 = first endpoint */
|
||||
/* p1 = first control point */
|
||||
/* p2 = second control point */
|
||||
/* p3 = second endpoint */
|
||||
/* p = point from which shortest distance is to be calculated */
|
||||
/* ----------------------------------------------------------- */
|
||||
/* => the equation of a cubic bezier curve can be written as: */
|
||||
/* B( t ) = ( ( 1 - t )^3 )p0 + 3( ( 1 - t )^2 )tp1 + */
|
||||
/* 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 ) + */
|
||||
/* 3( t^2 )( p0 - 2p1 + p2 ) + 3t( -p0 + p1 ) + p0 */
|
||||
/* */
|
||||
/* Now let A = ( -p0 + 3p1 - 3p2 + p3 ), */
|
||||
/* B = 3( p0 - 2p1 + p2 ), C = 3( -p0 + p1 ) */
|
||||
/* B( t ) = t^3( A ) + t^2( B ) + tC + p0 */
|
||||
/* */
|
||||
/* => the derivative of the above equation is written as */
|
||||
/* B'( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
/* */
|
||||
/* => further derivative of the above equation is written as */
|
||||
/* B''( t ) = 6t( A ) + 2B */
|
||||
/* */
|
||||
/* => the equation of distance from point `p' to the curve */
|
||||
/* P( t ) can be written as */
|
||||
/* P( t ) = t^3( A ) + t^2( B ) + tC + p0 - p */
|
||||
/* Now let D = ( p0 - p ) */
|
||||
/* P( t ) = t^3( A ) + t^2( B ) + tC + D */
|
||||
/* */
|
||||
/* => finally the equation of angle between curve B( t ) and */
|
||||
/* point to curve distance P( t ) can be written as */
|
||||
/* Q( t ) = P( t ).B'( t ) */
|
||||
/* */
|
||||
/* => now our task is to find a value of t such that the above */
|
||||
/* equation Q( t ) becomes zero. in other words the point */
|
||||
/* to curve vector makes 90 degree with curve. this is done */
|
||||
/* by Newton-Raphson's method. */
|
||||
/* */
|
||||
/* => we first assume a arbitary value of the factor `t' and */
|
||||
/* then we improve it using Newton's equation such as */
|
||||
/* */
|
||||
/* t -= Q( t ) / Q'( t ) */
|
||||
/* putting 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 ) ) */
|
||||
/* */
|
||||
/* P'( t ) is noting but B'( t ) because the constant are */
|
||||
/* gone due to derivative */
|
||||
/* */
|
||||
/* => 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 in the above comment */
|
||||
FT_16D16_Vec nearest_point; /* 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 = FT_INT_MAX; /* 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_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;
|
||||
}
|
||||
|
||||
/* assign the values after checking pointer */
|
||||
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 ) + tC + 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 actual 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 );
|
||||
|
||||
/* if not perpendicular then compute the cross */
|
||||
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;
|
||||
}
|
||||
|
||||
|
||||
/* END */
|
||||
|
|
|
|||
Loading…
Reference in New Issue