diff options
| -rw-r--r-- | src/cairo-path-stroke.c | 76 |
1 files changed, 75 insertions, 1 deletions
diff --git a/src/cairo-path-stroke.c b/src/cairo-path-stroke.c index 6502a94c7..853582899 100644 --- a/src/cairo-path-stroke.c +++ b/src/cairo-path-stroke.c @@ -280,6 +280,9 @@ _cairo_stroker_join (cairo_stroker_t *stroker, cairo_stroke_face_t *in, cairo_st double in_dot_out = ((-in->usr_vector.x * out->usr_vector.x)+ (-in->usr_vector.y * out->usr_vector.y)); double ml = stroker->style->miter_limit; + double tolerance_squared = stroker->tolerance * stroker->tolerance; + double line_width_squared = (stroker->style->line_width * + stroker->style->line_width); /* * Check the miter limit -- lines meeting at an acute angle @@ -308,8 +311,79 @@ _cairo_stroker_join (cairo_stroker_t *stroker, cairo_stroke_face_t *in, cairo_st * * 2 <= ml² (1 - in · out) * + * + * That gives us the condition to avoid generating miters that + * are too large from angles that are too large. But we also + * need to avoid generating miters when the angle is very small. + * + * The miter formed from a tiny angle is also tiny, so the + * miter limit is not a concern. But with a tiny angle we will + * be computing the intersection of two lines that are very + * near parallel. Also, the limits of the fixed-point grid on + * the input face coordinates mean that the resulting + * intersection could be wildly wrong. (See the + * get-path-extents test case for a call to cairo_arc that + * results in two problematic faces.) + * + * Fortunately we can also derive an expression for when using + * a bevel join instead of a miter will introduce an error no + * larger than the tolerance. Consider the same join from + * before but with the miter now chopped off and replaced with + * a bevel join. The drawing is zoomed in a bit again, the + * point marked as '*' is the center of the stroke---the point + * where the two line segments of interest intersect: + * + * ----- . + * ^ .. + * | . . + * | . . + * 1/2 . . + * miter . . | + * length . . | + * | .______. ___v___ + * | | . \ 1/2 bevel + * v | . \ width + * ---- * \ ------- + * | \ ^ + * + * + * The length of interest here is the vertical length of the + * miter that is eliminated. It's length can be obtained by + * starting with 1/2 the miter length and the subtracting off + * the vertical length that is included by the bevel join, + * (here termed 1/2 bevel width). To determine this new bevel + * width, we have a small right triangle shown, the hypotenuse + * of which has a length of 1/2 the line width, and the small + * angle at the upper right of the figure is psi/2. + * + * So we have: + * + * sin (psi/2) = (bevel_width / 2) / (line_width / 2) + * + * And we can determine when the miter is required by + * calculating when the eliminated portion of the miter is + * greater than the tolerance: + * + * (miter_length / 2) - (bevel_width / 2) > tolerance + * + * Substituting in the above expressions for miter_length and + * bevel_width: + * + * (line_width/2) / sin (psi/2) - (line_width/2) * sin (psi/2) > tolerance + * 1 / sin(psi/2) - sin (psi/2) > 2 * tolerance / line_width + * 1 / sin²(psi/2) -2 + sin²(psi/2) > 4 * (tolerance/line_width)² + * + * Use identity: sin²(psi/2) = (1-cos(psi))/2 + + * 2/(1 - cos(psi)) - 2 + (1-cos(psi))/2 > 4 * (tolerance/line_width)² + * 4/(1 - cos(psi)) - 4 + (1-cos(psi)) > 8 * (tolerance/line_width)² + * 4/(1 - cos(psi)) + (1-cos(psi)) > 8 * ((tolerance/line_width)² + 0.5) */ - if (2 <= ml * ml * (1 - in_dot_out)) { + if ((2 <= ml * ml * (1 - in_dot_out)) && + ((8 * (tolerance_squared / line_width_squared + 0.5)) < + 4 / (1 - in_dot_out) + (1 - in_dot_out)) + ) + { double x1, y1, x2, y2; double mx, my; double dx1, dx2, dy1, dy2; |