In this case, we are failing at the first test (bounding box intersection between the two curves), so we go home early and don't do any subdivision test:

seg1 = [(0.0, 0.0), (0.0, 0.25), (0.0, 0.75), (0.0, 1.0)]
seg2 = [(0.0, 0.5), (0.25, 0.5), (0.75, 0.5), (1.0, 0.5)]

bounds1 = _curve_bounds(seg1)
bounds2 = _curve_bounds(seg2)

print(bounds1, bounds2)
# (0.0, 0.0, 0.0, 1.0) (0.0, 0.5, 1.0, 0.5)

print(sectRect(bounds1, bounds2))
# (False, (0, 0, 0, 0))

good catch! would it be enough to special-case this edge case when both curves's bounds are flattened to perperdicular lines?

What I would suggest is that if a curve is sufficiently line-like (for some definition...), it gets turned into a line, and then either curveLineIntersections or lineLineIntersections handles it. This'll be both faster and more correct.

maybe it's our sectRect that's at fault.

In the classic "Graphics Gems" IV book, the chapter about curve intersections, they have the code below in the routine that checks two beziers' bounds intersection (in particular, see the comment // Not >= : need boundary case)

https://web.archive.org/web/20130613230901/http://users.soe.ucsc.edu/~pang/160/f98/Gems/GemsIV/curve_isect/Bezier.cc

    // Test bounding box of b against bounding box of a
    if( ( minax > maxbx ) || ( minay > maxby )  // Not >= : need boundary case
	|| ( minbx > maxax ) || ( minby > maxay ) )
	return 0; // they don't intersect
    else
	return 1; // they intersect

Possibly true. But if you can get away with treating curves as lines, the maths gets a lot simpler and less liable to go wrong, so we should do that.

the fix you proposed in #3353 only addresses the issue when both curves (or either one) can be reduced to a line; what if two curves cannot be reduced to line-like and they (as well as their bounds) actually touch one another?

E.g. take these two curves:

    seg1 = [(0, 0), (0, 100), (100, 100), (100, 0)]
    seg2 = [(0, 150), (0, 50), (100, 50), (100, 150)]

Both evaluate (50, 75) at t=0.5, however they don't cross one another, they simply "touch" at that point. Do we expect that our algorithm also covers these cases?

My proposed #3354 would detect these.

On the other end, I'm not 100% sure if sectRect was designed to deliberately not catch similar boundary cases for some reasons.

/cc @justvanrossum @behdad

Let's do both!

ok