static void
compute_inner_joint (cairo_point_t *p1, cairo_int64_t d_p1,
const cairo_point_t *p2, cairo_int64_t d_p2,
cairo_int64_t half_line_width)
{
int32_t dx = p2->x - p1->x;
int32_t dy = p2->y - p1->y;
half_line_width = _cairo_int64_sub (half_line_width, d_p1);
d_p2 = _cairo_int64_sub (d_p2, d_p1);
p1->x += _cairo_int_96by64_32x64_divrem (_cairo_int64x32_128_mul (half_line_width, dx),
d_p2).quo;
p1->y += _cairo_int_96by64_32x64_divrem (_cairo_int64x32_128_mul (half_line_width, dy),
d_p2).quo;
}
/* Compute the intersection of two lines as defined by two edges. The
* result is provided as a coordinate pair of 128-bit integers.
*
* Returns %CAIRO_BO_STATUS_INTERSECTION if there is an intersection or
* %CAIRO_BO_STATUS_PARALLEL if the two lines are exactly parallel.
*/
static cairo_bool_t
intersect_lines (cairo_bo_edge_t *a,
cairo_bo_edge_t *b,
cairo_bo_intersect_point_t *intersection)
{
cairo_int64_t a_det, b_det;
/* XXX: We're assuming here that dx and dy will still fit in 32
* bits. That's not true in general as there could be overflow. We
* should prevent that before the tessellation algorithm begins.
* What we're doing to mitigate this is to perform clamping in
* cairo_bo_tessellate_polygon().
*/
int32_t dx1 = a->edge.line.p1.x - a->edge.line.p2.x;
int32_t dy1 = a->edge.line.p1.y - a->edge.line.p2.y;
int32_t dx2 = b->edge.line.p1.x - b->edge.line.p2.x;
int32_t dy2 = b->edge.line.p1.y - b->edge.line.p2.y;
cairo_int64_t den_det;
cairo_int64_t R;
cairo_quorem64_t qr;
den_det = det32_64 (dx1, dy1, dx2, dy2);
/* Q: Can we determine that the lines do not intersect (within range)
* much more cheaply than computing the intersection point i.e. by
* avoiding the division?
*
* X = ax + t * adx = bx + s * bdx;
* Y = ay + t * ady = by + s * bdy;
* ∴ t * (ady*bdx - bdy*adx) = bdx * (by - ay) + bdy * (ax - bx)
* => t * L = R
*
* Therefore we can reject any intersection (under the criteria for
* valid intersection events) if:
* L^R < 0 => t < 0, or
* L<R => t > 1
*
* (where top/bottom must at least extend to the line endpoints).
*
* A similar substitution can be performed for s, yielding:
* s * (ady*bdx - bdy*adx) = ady * (ax - bx) - adx * (ay - by)
*/
R = det32_64 (dx2, dy2,
b->edge.line.p1.x - a->edge.line.p1.x,
b->edge.line.p1.y - a->edge.line.p1.y);
if (_cairo_int64_le (den_det, R))
return FALSE;
R = det32_64 (dy1, dx1,
a->edge.line.p1.y - b->edge.line.p1.y,
a->edge.line.p1.x - b->edge.line.p1.x);
if (_cairo_int64_le (den_det, R))
return FALSE;
/* We now know that the two lines should intersect within range. */
a_det = det32_64 (a->edge.line.p1.x, a->edge.line.p1.y,
a->edge.line.p2.x, a->edge.line.p2.y);
b_det = det32_64 (b->edge.line.p1.x, b->edge.line.p1.y,
b->edge.line.p2.x, b->edge.line.p2.y);
/* x = det (a_det, dx1, b_det, dx2) / den_det */
qr = _cairo_int_96by64_32x64_divrem (det64x32_128 (a_det, dx1,
b_det, dx2),
den_det);
if (_cairo_int64_eq (qr.rem, den_det))
return FALSE;
intersection->x = round_to_nearest (qr, den_det);
/* y = det (a_det, dy1, b_det, dy2) / den_det */
qr = _cairo_int_96by64_32x64_divrem (det64x32_128 (a_det, dy1,
b_det, dy2),
den_det);
if (_cairo_int64_eq (qr.rem, den_det))
return FALSE;
intersection->y = round_to_nearest (qr, den_det);
return TRUE;
}
