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[freetype2] GSoC-2020-anuj c39b5dd: [sdf] Added shortest distance findin
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From: |
Anuj Verma |
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Subject: |
[freetype2] GSoC-2020-anuj c39b5dd: [sdf] Added shortest distance finding functions. |
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Date: |
Tue, 18 Aug 2020 08:20:05 -0400 (EDT) |
branch: GSoC-2020-anuj
commit c39b5dd8496f7a8b0d530a497682ea8413d70c6a
Author: Anuj Verma <anujv@iitbhilai.ac.in>
Commit: Anuj Verma <anujv@iitbhilai.ac.in>
[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).
---
src/sdf/ftsdf.c | 937 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 937 insertions(+)
diff --git a/src/sdf/ftsdf.c b/src/sdf/ftsdf.c
index 6404805..e38aa74 100644
--- a/src/sdf/ftsdf.c
+++ b/src/sdf/ftsdf.c
@@ -1569,4 +1569,941 @@
return FT_ABS( sdf1.cross ) > FT_ABS( sdf2.cross ) ? sdf1 : sdf2;
}
+ /**************************************************************************
+ *
+ * @Function:
+ * get_min_distance_line
+ *
+ * @Description:
+ * This function find the shortest distance from the `line' to
+ * a given `point' and assigns it to `out'. Only use it for line
+ * segments.
+ *
+ * @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.
+ *
+ * @Return:
+ * out ::
+ * Signed distance from the `point' to the `line'.
+ *
+ * FT_Error ::
+ * FreeType error, 0 means success.
+ *
+ * @Note:
+ * The `line' parameter must have a `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. */
+ /* */
+ /* a = start point of the line segment */
+ /* b = end point of the line segment */
+ /* p = point from which shortest distance is to be calculated */
+ /* ----------------------------------------------------------- */
+ /* => we first write the parametric equation of the line */
+ /* point_on_line = a + ( b - a ) * t ( t is the factor ) */
+ /* */
+ /* => next we find the projection of point p on the line. the */
+ /* projection will be perpendicular to the line, that is */
+ /* why we can find it by making the dot product zero. */
+ /* ( point_on_line - a ) . ( p - point_on_line ) = 0 */
+ /* */
+ /* ( point_on_line ) */
+ /* ( a ) x-------o----------------x ( b ) */
+ /* |_| */
+ /* | */
+ /* | */
+ /* ( p ) */
+ /* */
+ /* => by simplifying the above equation we get the factor of */
+ /* point_on_line such that */
+ /* t = ( ( p - a ) . ( b - a ) ) / ( |b - a| ^ 2 ) */
+ /* */
+ /* => we clamp the factor t between [0.0f, 1.0f], because the */
+ /* point_on_line can be outside the line segment. */
+ /* */
+ /* ( point_on_line ) */
+ /* ( a ) x------------------------x ( b ) -----o--- */
+ /* |_| */
+ /* | */
+ /* | */
+ /* ( p ) */
+ /* */
+ /* => finally the distance becomes | 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 the 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;
+ }
+
+#if !USE_NEWTON_FOR_CONIC
+
+ /**************************************************************************
+ *
+ * @Function:
+ * get_min_distance_conic
+ *
+ * @Description:
+ * This function find the shortest distance from the `conic' bezier
+ * curve to a given `point' and assigns it to `out'. Only use it for
+ * conic/quadratic curves.
+ *
+ * @Input:
+ * conic ::
+ * The conic 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 `conic'.
+ *
+ * FT_Error ::
+ * FreeType error, 0 means success.
+ *
+ * @Note:
+ * The function uses analytical method to find shortest distance
+ * which is faster than the Newton-Raphson's method, but has
+ * underflows at the moment. Use Newton's method if you can
+ * see artifacts in the SDF.
+ *
+ * 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 )
+ {
+ /* The procedure to find the shortest distance from a point to */
+ /* a quadratic bezier curve is similar to a line segment. the */
+ /* shortest distance will be 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 endpo- */
+ /* -ints, because the perpendicular may not be the shortest. */
+ /* */
+ /* p0 = first endpoint */
+ /* p1 = control point */
+ /* p2 = 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 = ( p1 - p0 ) */
+ /* B( t ) = t^2( A ) + 2t( B ) + p0 */
+ /* */
+ /* => the derivative of the above equation is written as */
+ /* B'( t ) = 2( tA + B ) */
+ /* */
+ /* => now 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 )) makes 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 as */
+ /* at^3 + bt^2 + ct + d = 0 */
+ /* a = ( A.A ), b = ( 3A.B ), c = ( 2B.B + A.p0 - A.p ) */
+ /* d = ( p0.B - p.B ) */
+ /* */
+ /* => now the roots of the equation can be computed using the */
+ /* `Cardano's Cubic formula' we clamp the roots in 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; /* 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;
+ }
+
+ /* 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 = 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, but */
+ /* Behdad gave me a lemma: */
+ /* Lemma: */
+ /* * 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. */
+ /* * 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 proof contact: behdad@behdad.org */
+ /* For more details check message: */
+ /*
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 );
+
+ /* 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;
+ }
+
+#else
+
+ /**************************************************************************
+ *
+ * @Function:
+ * get_min_distance_conic
+ *
+ * @Description:
+ * This function find the shortest distance from the `conic' bezier
+ * curve to a given `point' and assigns it to `out'. Only use it for
+ * conic/quadratic curves.
+ *
+ * @Input:
+ * conic ::
+ * The conic 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 `conic'.
+ *
+ * FT_Error ::
+ * FreeType error, 0 means success.
+ *
+ * @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 */
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- [freetype2] GSoC-2020-anuj c39b5dd: [sdf] Added shortest distance finding functions.,
Anuj Verma <=