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[sdf] Added function to find shortest distance from a point to a conic.

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Anuj Verma 2020-07-01 09:38:24 +05:30 committed by anujverma
parent 5cbcab183a
commit f9b4f37433
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13
[GSoC]ChangeLog
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@ -1,3 +1,16 @@
2020-07-01 Anuj Verma <anujv@iitbhilai.ac.in>
[sdf] Added function to find shortest distance from a
point to a conic bezier. Now the sdf module can render
ttf fonts or fonts with line and conic segments.
* src/sdf/ftsdf.c (get_min_distance_conic): The function
calculates shortest distance from a point to a conic
bezier curve.
* src/sdf/ftsdf.c (sdf_edge_get_min_distance): Add the
`get_min_distance_conic' function call.
2020-07-01 Anuj Verma <anujv@iitbhilai.ac.in>
* src/sdf/ftsdf.c (get_min_distance_line): First check

205
src/sdf/ftsdf.c
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@ -27,6 +27,10 @@
goto Exit; \
} while ( 0 )
#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 ) )
/**************************************************************************
*
* typedefs
@ -1028,6 +1032,7 @@
FT_16D16_Vec nearest_point; /* `point_on_line' */
FT_16D16_Vec nearest_vector; /* `p' - `nearest_point' */
if ( !line || !out )
{
error = FT_THROW( Invalid_Argument );
@ -1094,6 +1099,204 @@
return error;
}
/**************************************************************************
*
* @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:
* [TODO]
*
* @Return:
* [TODO]
*/
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 determin 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 );
/* 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;
/* check this: https://lists.nongnu.org/archive/html/freetype-devel/2020-06/msg00147.html */
/* to see why we clamp the values and not check the endpoints */
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 = FT_MulFix( dist_vector.x, dist_vector.x ) +
FT_MulFix( dist_vector.y, dist_vector.y );
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->squared_distance = min;
out->nearest_point = nearest_point;
out->sign = cross < 0 ? 1 : -1;
FT_Vector_NormLen( &direction );
out->direction = direction;
Exit:
return error;
}
/**************************************************************************
*
* @Function:
@ -1129,6 +1332,8 @@
get_min_distance_line( edge, point, out );
break;
case SDF_EDGE_CONIC:
get_min_distance_conic( edge, point, out );
break;
case SDF_EDGE_CUBIC:
default:
error = FT_THROW( Invalid_Argument );