[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[freetype2] anuj-distance-field c433158: [sdf] Added function to find sh
|
From: |
Anuj Verma |
|
Subject: |
[freetype2] anuj-distance-field c433158: [sdf] Added function to find shortest distance from a point to a cubic. |
|
Date: |
Fri, 3 Jul 2020 06:56:33 -0400 (EDT) |
branch: anuj-distance-field
commit c4331583b6078233abf3587468c99bd91442ffff
Author: Anuj Verma <anujv@iitbhilai.ac.in>
Commit: Anuj Verma <anujv@iitbhilai.ac.in>
[sdf] Added function to find shortest distance from a point to a cubic.
---
[GSoC]ChangeLog | 13 +++
src/sdf/ftsdf.c | 247 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2 files changed, 260 insertions(+)
diff --git a/[GSoC]ChangeLog b/[GSoC]ChangeLog
index 15115c9..5b271fc 100644
--- a/[GSoC]ChangeLog
+++ b/[GSoC]ChangeLog
@@ -1,5 +1,18 @@
2020-07-03 Anuj Verma <anujv@iitbhilai.ac.in>
+ [sdf] Added function to find shortest distance from a
+ point to a cubic bezier. Now the sdf module can render
+ all types of fonts, but still has some issues.
+
+ * src/sdf/ftsdf.c (get_min_distance_cubic): The function
+ calculates shortest distance from a point to a cubic
+ bezier curve.
+
+ * src/sdf/ftsdf.c (sdf_edge_get_min_distance): Add the
+ `get_min_distance_cubic' function call.
+
+2020-07-03 Anuj Verma <anujv@iitbhilai.ac.in>
+
* src/sdf/ftsdf.c (resolve_corner): [Bug] Remove the
nearest_point check. Two distances can be same and
can give opposite sign, but they may not have a
diff --git a/src/sdf/ftsdf.c b/src/sdf/ftsdf.c
index 8d8ba8a..9ddcae1 100644
--- a/src/sdf/ftsdf.c
+++ b/src/sdf/ftsdf.c
@@ -1494,6 +1494,7 @@
FT_UShort iterations;
FT_UShort steps;
+
if ( !conic || !out )
{
error = FT_THROW( Invalid_Argument );
@@ -1620,6 +1621,250 @@
/**************************************************************************
*
* @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.
+ * [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.
+ *
+ * @Input:
+ * [TODO]
+ *
+ * @Return:
+ * [TODO]
+ */
+ 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; /* factor at shortest distance */
+ FT_16D16 min_factor_sq; /* 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->nearest_point = nearest_point;
+ out->sign = cross < 0 ? 1 : -1;
+
+ FT_Vector_NormLen( &direction );
+
+ out->direction = direction;
+
+ Exit:
+ return error;
+ }
+
+ /**************************************************************************
+ *
+ * @Function:
* sdf_edge_get_min_distance
*
* @Description:
@@ -1655,6 +1900,8 @@
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 );
}
| [Prev in Thread] |
Current Thread |
[Next in Thread] |
- [freetype2] anuj-distance-field c433158: [sdf] Added function to find shortest distance from a point to a cubic.,
Anuj Verma <=