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-rw-r--r--src/base/ftbbox.c334
1 files changed, 86 insertions, 248 deletions
diff --git a/src/base/ftbbox.c b/src/base/ftbbox.c
index 6d1c44c..8d3f383 100644
--- a/src/base/ftbbox.c
+++ b/src/base/ftbbox.c
@@ -85,7 +85,7 @@
/* BBox_Conic_Check */
/* */
/* <Description> */
- /* Finds the extrema of a 1-dimensional conic Bezier curve and update */
+ /* Find the extrema of a 1-dimensional conic Bezier curve and update */
/* a bounding range. This version uses direct computation, as it */
/* doesn't need square roots. */
/* */
@@ -108,30 +108,19 @@
FT_Pos* min,
FT_Pos* max )
{
- if ( y1 <= y3 && y2 == y1 ) /* flat arc */
- goto Suite;
-
- if ( y1 < y3 )
- {
- if ( y2 >= y1 && y2 <= y3 ) /* ascending arc */
- goto Suite;
- }
- else
- {
- if ( y2 >= y3 && y2 <= y1 ) /* descending arc */
- {
- y2 = y1;
- y1 = y3;
- y3 = y2;
- goto Suite;
- }
- }
-
- y1 = y3 = y1 - FT_MulDiv( y2 - y1, y2 - y1, y1 - 2*y2 + y3 );
-
- Suite:
- if ( y1 < *min ) *min = y1;
- if ( y3 > *max ) *max = y3;
+ /* This function is only called when a control off-point is outside */
+ /* the bbox that contains all on-points. It finds a local extremum */
+ /* within the segment, equal to (y1*y3 - y2*y2)/(y1 - 2*y2 + y3). */
+ /* Or, offsetting from y2, we get */
+
+ y1 -= y2;
+ y3 -= y2;
+ y2 += FT_MulDiv( y1, y3, y1 + y3 );
+
+ if ( y2 < *min )
+ *min = y2;
+ if ( y2 > *max )
+ *max = y2;
}
@@ -195,9 +184,9 @@
/* BBox_Cubic_Check */
/* */
/* <Description> */
- /* Finds the extrema of a 1-dimensional cubic Bezier curve and */
- /* updates a bounding range. This version uses splitting because we */
- /* don't want to use square roots and extra accuracy. */
+ /* Find the extrema of a 1-dimensional cubic Bezier curve and */
+ /* update a bounding range. This version uses iterative splitting */
+ /* because it is faster than the exact solution with square roots. */
/* */
/* <Input> */
/* p1 :: The start coordinate. */
@@ -213,28 +202,16 @@
/* */
/* max :: The address of the current maximum. */
/* */
-
-#if 0
-
- static void
- BBox_Cubic_Check( FT_Pos p1,
- FT_Pos p2,
- FT_Pos p3,
- FT_Pos p4,
- FT_Pos* min,
- FT_Pos* max )
+ static FT_Pos
+ update_cubic_max( FT_Pos q1,
+ FT_Pos q2,
+ FT_Pos q3,
+ FT_Pos q4,
+ FT_Pos max )
{
- FT_Pos q1, q2, q3, q4;
-
-
- q1 = p1;
- q2 = p2;
- q3 = p3;
- q4 = p4;
-
- /* for a conic segment to possibly reach new maximum */
- /* one of its off-points must be above the current value */
- while ( q2 > *max || q3 > *max )
+ /* for a cubic segment to possibly reach new maximum, at least */
+ /* one of its off-points must stay above the current value */
+ while ( q2 > max || q3 > max )
{
/* determine which half contains the maximum and split */
if ( q1 + q2 > q3 + q4 ) /* first half */
@@ -260,231 +237,91 @@
q3 = q3 / 2;
}
- /* check if either end reached the maximum */
+ /* check whether either end reached the maximum */
if ( q1 == q2 && q1 >= q3 )
{
- *max = q1;
+ max = q1;
break;
}
if ( q3 == q4 && q2 <= q4 )
{
- *max = q4;
+ max = q4;
break;
}
}
- q1 = p1;
- q2 = p2;
- q3 = p3;
- q4 = p4;
-
- /* for a conic segment to possibly reach new minimum */
- /* one of its off-points must be below the current value */
- while ( q2 < *min || q3 < *min )
- {
- /* determine which half contains the minimum and split */
- if ( q1 + q2 < q3 + q4 ) /* first half */
- {
- q4 = q4 + q3;
- q3 = q3 + q2;
- q2 = q2 + q1;
- q4 = q4 + q3;
- q3 = q3 + q2;
- q4 = ( q4 + q3 ) / 8;
- q3 = q3 / 4;
- q2 = q2 / 2;
- }
- else /* second half */
- {
- q1 = q1 + q2;
- q2 = q2 + q3;
- q3 = q3 + q4;
- q1 = q1 + q2;
- q2 = q2 + q3;
- q1 = ( q1 + q2 ) / 8;
- q2 = q2 / 4;
- q3 = q3 / 2;
- }
-
- /* check if either end reached the minimum */
- if ( q1 == q2 && q1 <= q3 )
- {
- *min = q1;
- break;
- }
- if ( q3 == q4 && q2 >= q4 )
- {
- *min = q4;
- break;
- }
- }
- }
-
-#else
-
- static void
- test_cubic_extrema( FT_Pos y1,
- FT_Pos y2,
- FT_Pos y3,
- FT_Pos y4,
- FT_Fixed u,
- FT_Pos* min,
- FT_Pos* max )
- {
- /* FT_Pos a = y4 - 3*y3 + 3*y2 - y1; */
- FT_Pos b = y3 - 2*y2 + y1;
- FT_Pos c = y2 - y1;
- FT_Pos d = y1;
- FT_Pos y;
- FT_Fixed uu;
-
- FT_UNUSED ( y4 );
-
-
- /* The polynomial is */
- /* */
- /* P(x) = a*x^3 + 3b*x^2 + 3c*x + d , */
- /* */
- /* dP/dx = 3a*x^2 + 6b*x + 3c . */
- /* */
- /* However, we also have */
- /* */
- /* dP/dx(u) = 0 , */
- /* */
- /* which implies by subtraction that */
- /* */
- /* P(u) = b*u^2 + 2c*u + d . */
-
- if ( u > 0 && u < 0x10000L )
- {
- uu = FT_MulFix( u, u );
- y = d + FT_MulFix( c, 2*u ) + FT_MulFix( b, uu );
-
- if ( y < *min ) *min = y;
- if ( y > *max ) *max = y;
- }
+ return max;
}
static void
- BBox_Cubic_Check( FT_Pos y1,
- FT_Pos y2,
- FT_Pos y3,
- FT_Pos y4,
+ BBox_Cubic_Check( FT_Pos p1,
+ FT_Pos p2,
+ FT_Pos p3,
+ FT_Pos p4,
FT_Pos* min,
FT_Pos* max )
{
- /* always compare first and last points */
- if ( y1 < *min ) *min = y1;
- else if ( y1 > *max ) *max = y1;
+ FT_Pos nmin, nmax;
+ FT_Int shift;
- if ( y4 < *min ) *min = y4;
- else if ( y4 > *max ) *max = y4;
- /* now, try to see if there are split points here */
- if ( y1 <= y4 )
+ /* This function is only called when a control off-point is outside */
+ /* the bbox that contains all on-points. It finds a local extremum */
+ /* within the segment using iterative bisection of the segment. */
+ /* The fixed-point arithmetic of bisection is inherently stable */
+ /* but may loose accuracy in the two lowest bits. To compensate, */
+ /* we upscale the segment if there is room. Large values may need */
+ /* to be downscaled to avoid overflows during bisection. */
+ /* The control off-point outside the bbox is likely to have the top */
+ /* absolute value among arguments. */
+
+ shift = 27 - FT_MSB( FT_ABS( p2 ) | FT_ABS( p3 ) );
+
+ if ( shift > 0 )
{
- /* flat or ascending arc test */
- if ( y1 <= y2 && y2 <= y4 && y1 <= y3 && y3 <= y4 )
- return;
+ /* upscaling too much just wastes time */
+ if ( shift > 2 )
+ shift = 2;
+
+ p1 <<= shift;
+ p2 <<= shift;
+ p3 <<= shift;
+ p4 <<= shift;
+ nmin = *min << shift;
+ nmax = *max << shift;
}
- else /* y1 > y4 */
+ else
{
- /* descending arc test */
- if ( y1 >= y2 && y2 >= y4 && y1 >= y3 && y3 >= y4 )
- return;
+ p1 >>= -shift;
+ p2 >>= -shift;
+ p3 >>= -shift;
+ p4 >>= -shift;
+ nmin = *min >> -shift;
+ nmax = *max >> -shift;
}
- /* There are some split points. Find them. */
- /* We already made sure that a, b, and c below cannot be all zero. */
- {
- FT_Pos a = y4 - 3*y3 + 3*y2 - y1;
- FT_Pos b = y3 - 2*y2 + y1;
- FT_Pos c = y2 - y1;
- FT_Pos d;
- FT_Fixed t;
- FT_Int shift;
-
-
- /* We need to solve `ax^2+2bx+c' here, without floating points! */
- /* The trick is to normalize to a different representation in order */
- /* to use our 16.16 fixed-point routines. */
- /* */
- /* We compute FT_MulFix(b,b) and FT_MulFix(a,c) after normalization. */
- /* These values must fit into a single 16.16 value. */
- /* */
- /* We normalize a, b, and c to `8.16' fixed-point values to ensure */
- /* that their product is held in a `16.16' value including the sign. */
- /* Necessarily, we need to shift `a', `b', and `c' so that the most */
- /* significant bit of their absolute values is at position 22. */
- /* */
- /* This also means that we are using 23 bits of precision to compute */
- /* the zeros, independently of the range of the original polynomial */
- /* coefficients. */
- /* */
- /* This algorithm should ensure reasonably accurate values for the */
- /* zeros. Note that they are only expressed with 16 bits when */
- /* computing the extrema (the zeros need to be in 0..1 exclusive */
- /* to be considered part of the arc). */
-
- shift = FT_MSB( FT_ABS( a ) | FT_ABS( b ) | FT_ABS( c ) );
-
- if ( shift > 22 )
- {
- shift -= 22;
+ nmax = update_cubic_max( p1, p2, p3, p4, nmax );
- /* this loses some bits of precision, but we use 23 of them */
- /* for the computation anyway */
- a >>= shift;
- b >>= shift;
- c >>= shift;
- }
- else
- {
- shift = 22 - shift;
-
- a <<= shift;
- b <<= shift;
- c <<= shift;
- }
+ /* now flip the signs to update the minimum */
+ nmin = -update_cubic_max( -p1, -p2, -p3, -p4, -nmin );
- /* handle a == 0 */
- if ( a == 0 )
- {
- if ( b != 0 )
- {
- t = - FT_DivFix( c, b ) / 2;
- test_cubic_extrema( y1, y2, y3, y4, t, min, max );
- }
- }
- else
- {
- /* solve the equation now */
- d = FT_MulFix( b, b ) - FT_MulFix( a, c );
- if ( d < 0 )
- return;
-
- if ( d == 0 )
- {
- /* there is a single split point at -b/a */
- t = - FT_DivFix( b, a );
- test_cubic_extrema( y1, y2, y3, y4, t, min, max );
- }
- else
- {
- /* there are two solutions; we need to filter them */
- d = FT_SqrtFixed( (FT_Int32)d );
- t = - FT_DivFix( b - d, a );
- test_cubic_extrema( y1, y2, y3, y4, t, min, max );
-
- t = - FT_DivFix( b + d, a );
- test_cubic_extrema( y1, y2, y3, y4, t, min, max );
- }
- }
+ if ( shift > 0 )
+ {
+ nmin >>= shift;
+ nmax >>= shift;
+ }
+ else
+ {
+ nmin <<= -shift;
+ nmax <<= -shift;
}
- }
-#endif
+ if ( nmin < *min )
+ *min = nmin;
+ if ( nmax > *max )
+ *max = nmax;
+ }
/*************************************************************************/
@@ -521,8 +358,9 @@
FT_Vector* to,
TBBox_Rec* user )
{
- /* we don't need to check `to' since it is always an `on' point, thus */
- /* within the bbox */
+ /* We don't need to check `to' since it is always an on-point, */
+ /* thus within the bbox. Only segments with an off-point outside */
+ /* the bbox can possibly reach new extreme values. */
if ( CHECK_X( control1, user->bbox ) ||
CHECK_X( control2, user->bbox ) )
generated by cgit v1.2.3 (git 2.25.1) at 2024-04-25 06:12:19 +0000