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Mathematica: Disjoint Segments

How can we say that Mathematica gives us a set of disjoint lines? In this case, two lines intersect if they have a point (and not an end point). Consider this simple case:

l1 = {{-1, 0}, {1, 0}};
l2 = {{0, -1}, {0, 1}};
lines = {l1, l2};

The idea is to create a function that, given a set of strings, returns a set of disjoint strings. If such a function exists, say split, then the conclusion

split[lines]

will be

{
 {{-1, 0}, {0,0}},
 {{ 0, 0}, {1,0}}, 
 {{ 0,-1}, {0,0}}, 
 {{ 0, 0}, {0,1}}
}

, {0,0} , , , . , . - , Mathematica ? , .

Mathematica . .

+1
2

, , ;

ClearAll[permsnodups];
permsnodups[lp_] := DeleteDuplicates[Permutations[lp, {2}],
   ((#1[[1]] == #2[[1]]) &&(#1[[2]] == #2[[2]]) || 
   (#1[[1]] == #2[[2]]) && (#1[[2]] == #2[[1]])) &]

: permsnodups[{a, b, c, d}] {{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}}, split (.. , , {a,b} , {b,a} - $\ sum_ {i, j > i} $ $\ sum_ {i, j} $).

EDIT: split ( - , , , , ):

ClearAll[split2]
split2[{{ai_, bi_}, {ci_, di_}}] := Module[
{g1, g2, a, b, c, d, x0, y0, alpha, beta},
(*make sure that a is to the left of b*)

If[ai[[1]] > bi[[1]], {a, b} = {bi, ai}, {a, b} = {ai, bi}];
If[ci[[1]] > di[[1]], {c, d} = {di, ci}, {c, d} = {ci, di}];
g1 = (b[[2]] - a[[2]])/(b[[1]] - a[[1]]);
g2 = (d[[2]] - c[[2]])/(d[[1]] - c[[1]]);
If[g2 \[Equal] g1,
    {{a, b}, {c, d}},(*they're parallel*)

alpha = a[[2]] - g1*a[[1]];
    beta = c[[2]] - g2*c[[1]];
    x0 = (alpha - beta)/(g2 - g1);(*intersection x*)

If[(a[[1]] < x0 < b[[1]]) && (c[[1]] < x0 < 
   d[[1]]),(*they do intersect*)
            y0 = alpha + g1*x0;
            {{a, #}, {#, b}, {c, #}, {#, d}} &@{x0, y0},
            {{a, b}, {c, d}}(*they don't intersect after all*)]]]

( ). , , :

Manipulate[
Grid[{{Graphics[{Line[{p1, p2}, VertexColors \[Rule] {Red, Green}], 
  Line[{p3, p4}]},
        PlotRange \[Rule] 3, Axes \[Rule] True],
        (*Reap@split2[{{p1,p2},{p3,p4}}]//Last,*)
        If[
            Length@split2[{{p1, p2}, {p3, p4}}] \[Equal] 2,
            "not intersecting",
            "intersecting"]}}],
{{p1, {0, 1}}, Locator}, {{p2, {1, 1}}, Locator},
{{p3, {2.3, -.1}}, Locator}, {{p4, {2, 1}}, Locator}]

,

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( ). , split2 , ( , ).

. , ( , ), ( ). .

, , - (, , ). , , , , , , . , :)

+3

, , (.. ...):

f[t_, l_List] := l[[1]] + t (l[[2]] - l[[1]])
split[l1_, l2_] := Module[{s},
  If[(s = ToRules@
       Reduce[f[t1, l1]==f[t2, l2] && 0 <t2< 1 && 0 <t1< 1, {t1,t2},Reals]) =={},
   Return[{l1, l2}],
   Return[{{f[0, l1], f[t1, l1] /. s}, {f[1, l1], f[t1, l1] /. s},
           {f[0, l2], f[t2, l2] /. s}, {f[1, l2], f[t2, l2] /. s}}]
   ]]
+2

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