Clean & efficient bounding box non-intersection test

Question

I would like a quick test to tell me that segments $ab$ and $cd$ cannot intersect in the plane. If the maximum $x$-coordinate of $a$ and $b$ is less than the minimum $x$-coordinate of $c$ and $d$, then they cannot intersect. Etc.:

max = Max[a[[1]], b[[1]]];
min = Min[c[[1]], d[[1]]];
(*ab leftof cd*)
If[max < min, Return[False]];
max = Max[c[[1]], d[[1]]];
min = Min[a[[1]], b[[1]]];
(*cd leftof ab*)
If[max < min, Return[False]];
...

I feel this is rather clumsy coding. What is (a) a more concise coding, and (b) recognizing this is for speed, a more efficient bounding-box rejection.

The same question can be asked for $\mathbb{R}^d$ for $d>2$ (athough then it would likely be higher-dimensional objects determining the bounding box).

Answer 1

Let's assume you have the following points a-f, which form the line segments ab and cd and ef.

a = {0, 0};
b = {3, 3};
c = {1, .5};
d = {2, 1.5};
e = {0, 2};
f = {2, 0};

In order to test whether they intersect you can use RegionIntersection. First form lines from your points, then use RegionIntersection on the two lines. Compare the intersection to an empty region to see if the lines intersect. I built the following functions for 2D points, but you can change that by specifying a different dimension for EmptyRegion

IntersectQ[a_, b_, c_, d_] := Not[SameQ[RegionIntersection[Line[{a, b}],
 Line[{c, d}]], EmptyRegion[2]]]

IntersectQ[a,b,c,d];
(*False*)
IntersectQ[a,b,e,f];
(*True*)

The evaluation was quick when I ran it, around 0.014 seconds.

As AndyRoss mentioned, you can use a variety of shapes within the intersection to suit your needs.

If you need to find the box surrounding the lines, use the following:

 RegionBounds[RegionUnion[Line[{a, b}], Line[{c, d}]]]

Answer 2

Starting with M11.1 you can use RegionDisjoint:

a = {0, 0};
b = {3, 3};
c = {1, .5};
d = {2, 1.5};
e = {0, 2};
f = {2, 0};

Not @ RegionDisjoint[Line[{a, b}], Line[{c, d}]] //RepeatedTiming
Not @ RegionDisjoint[Line[{a, b}], Line[{e, f}]] //RepeatedTiming

{0.0020, False}

{0.00083, True}

Compare to the accepted answer:

IntersectQ[a_, b_, c_, d_] := Not[
    SameQ[RegionIntersection[Line[{a, b}], Line[{c, d}]], EmptyRegion[2]]
]

IntersectQ[a,b,c,d] //RepeatedTiming
IntersectQ[a,b,e,f] //RepeatedTiming

{0.0035, False}

{0.0023, True}