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[smooth] Simplify cubic Bézier flattening.

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The previous implementation is correct but it is too complex.
The revised algorithm is based on the fact that each split moves
the control points closer to the trisection points on the chord.
The corresponding distances are good surrogates for the curve
deviation from the straight line.

This cubic flattening algorithm is somewhat similar to the conic
algorithm based the distance from the control point to the middle of
the chord.  The cubic distances, however, decrease less predictably
but are easy enough to calculate on each step.

* src/smooth/ftgrays.c (gray_render_cubic): Replace the split
condition.
This commit is contained in:
Alexei Podtelezhnikov 2019-04-29 22:49:15 -04:00
parent 80475edead
commit 2ea511eed8
2 changed files with 25 additions and 42 deletions
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18
ChangeLog
View File

@ -1,3 +1,21 @@
2019-04-29 Alexei Podtelezhnikov <apodtele@gmail.com>
[smooth] Simplify cubic Bézier flattening.
The previous implementation is correct but it is too complex.
The revised algorithm is based on the fact that each split moves
the control points closer to the trisection points on the chord.
The corresponding distances are good surrogates for the curve
deviation from the straight line.
This cubic flattening algorithm is somewhat similar to the conic
algorithm based the distance from the control point to the middle of
the chord. The cubic distances, however, decrease less predictably
but are easy enough to calculate on each step.
* src/smooth/ftgrays.c (gray_render_cubic): Replace the split
condition.
2019-04-26 Alexei Podtelezhnikov <apodtele@gmail.com>
[smooth] Bithacks and cosmetics.

49
src/smooth/ftgrays.c
View File

@ -1093,9 +1093,6 @@ typedef ptrdiff_t FT_PtrDist;
{
FT_Vector bez_stack[16 * 3 + 1]; /* enough to accommodate bisections */
FT_Vector* arc = bez_stack;
TPos dx, dy, dx_, dy_;
TPos dx1, dy1, dx2, dy2;
TPos L, s, s_limit;
arc[0].x = UPSCALE( to->x );
@ -1124,45 +1121,13 @@ typedef ptrdiff_t FT_PtrDist;
for (;;)
{
/* Decide whether to split or draw. See `Rapid Termination */
/* Evaluation for Recursive Subdivision of Bezier Curves' by Thomas */
/* F. Hain, at */
/* http://www.cis.southalabama.edu/~hain/general/Publications/Bezier/Camera-ready%20CISST02%202.pdf */
/* dx and dy are x and y components of the P0-P3 chord vector. */
dx = dx_ = arc[3].x - arc[0].x;
dy = dy_ = arc[3].y - arc[0].y;
L = FT_HYPOT( dx_, dy_ );
/* Avoid possible arithmetic overflow below by splitting. */
if ( L > 32767 )
goto Split;
/* Max deviation may be as much as (s/L) * 3/4 (if Hain's v = 1). */
s_limit = L * (TPos)( ONE_PIXEL / 6 );
/* s is L * the perpendicular distance from P1 to the line P0-P3. */
dx1 = arc[1].x - arc[0].x;
dy1 = arc[1].y - arc[0].y;
s = FT_ABS( SUB_LONG( MUL_LONG( dy, dx1 ), MUL_LONG( dx, dy1 ) ) );
if ( s > s_limit )
goto Split;
/* s is L * the perpendicular distance from P2 to the line P0-P3. */
dx2 = arc[2].x - arc[0].x;
dy2 = arc[2].y - arc[0].y;
s = FT_ABS( SUB_LONG( MUL_LONG( dy, dx2 ), MUL_LONG( dx, dy2 ) ) );
if ( s > s_limit )
goto Split;
/* Split super curvy segments where the off points are so far
from the chord that the angles P0-P1-P3 or P0-P2-P3 become
acute as detected by appropriate dot products. */
if ( dx1 * ( dx1 - dx ) + dy1 * ( dy1 - dy ) > 0 ||
dx2 * ( dx2 - dx ) + dy2 * ( dy2 - dy ) > 0 )
/* with each split, control points quickly converge towards */
/* chord trisection points and the vanishing distances below */
/* indicate when the segment is flat enough to draw */
if ( FT_ABS( 2 * arc[0].x - 3 * arc[1].x + arc[3].x ) > ONE_PIXEL / 2 ||
FT_ABS( 2 * arc[0].y - 3 * arc[1].y + arc[3].y ) > ONE_PIXEL / 2 ||
FT_ABS( arc[0].x - 3 * arc[2].x + 2 * arc[3].x ) > ONE_PIXEL / 2 ||
FT_ABS( arc[0].y - 3 * arc[2].y + 2 * arc[3].y ) > ONE_PIXEL / 2 )
goto Split;
gray_render_line( RAS_VAR_ arc[0].x, arc[0].y );