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What formula can get a first common point(horizontal) for two parallel lines(segments)? A couple of examples:

First example Second example Third example Fourth example

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Update: Interactive visualization using IntervalSliders:

DynamicModule[{i1 = {5, 10}, i2 = {6, 11}}, 
 Panel@Column[{Row[{Style["interval 1    ", 16, ColorData[97]@4], 
      Dynamic@Pane[i1[[1]], Alignment -> Right, ImageSize -> {40, 20},
         BaselinePosition -> Center], 
      IntervalSlider[Dynamic[i1], {0, 20, 1}, Method -> "Stop", 
       Appearance -> {"Paired"}, ImageSize -> 400], 
      Dynamic@Pane[i1[[2]], Alignment -> Left, ImageSize -> {40, 20}, 
        BaselinePosition -> Center]}, Spacer[10]], 
    Row[{Style["intersection", 16, ColorData[97]@2], 
      Dynamic@Pane[If[Max[Min /@ {i1, i2}] > Min[Max /@ {i1, i2}], " ", 
         Max[Min /@ {i1, i2}]], Alignment -> Right, 
        ImageSize -> {40, 20}, BaselinePosition -> Center], 
      Dynamic@Graphics[{If[Max[Min /@ {i1, i2}] > Min[Max /@ {i1, i2}], 
       {Thick, Red, Line[{{0, 0}, {20, 0}}]},
       {Orange, CapForm["Round"], AbsoluteThickness[7],
         Line[{{Max[Min /@ {i1, i2}], 0}, {Min[Max /@ {i1, i2}], 0}}],  
        Thick, Gray, Line[{{0, 0}, {20, 0}}],
        Black, PointSize[Medium], 
            Point@{{Max[Min /@ {i1, i2}], 0}, {Min[Max /@ {i1, i2}], 
              0}}}]}, 
      ImageSize -> 400], 
      Dynamic@Pane[If[Max[Min /@ {i1, i2}] > Min[Max /@ {i1, i2}], " ", 
         Min[Max /@ {i1, i2}]], Alignment -> Left, 
        ImageSize -> {40, 20}, BaselinePosition -> Center]}, 
     Spacer[10]], 
    Row[{Style["interval 2    ", 16, ColorData[97]@1], 
      Dynamic@Pane[i2[[1]], Alignment -> Right, ImageSize -> {40, 20},
         BaselinePosition -> Center],
      IntervalSlider[Dynamic[i2], {0, 20, 1}, Method -> "Stop", 
       Appearance -> {"Paired"}, ImageSize -> 400], 
      Dynamic@Pane[i2[[2]], Alignment -> Left, ImageSize -> {40, 20}, 
        BaselinePosition -> Center]}, Spacer[10]]}, 
   Spacings -> {0, -.5}, Alignment -> Center]]

enter image description here

enter image description here

enter image description here

Using NumberLinePlot and IntervalSlider:

nlP = NumberLinePlot[{Interval@First@#, 
     IntervalIntersection @@ Interval /@ #, 
     min = Min[IntervalIntersection @@ Interval /@ #], 
     Interval@Last@#}, 
    Epilog -> If[min === DirectedInfinity[1], {}, 
      Text[Style[min, 14], {min, 2}, {3, 0}]], 
    Spacings -> {1, 1, 0, 1}, 
    Ticks -> {{#, Style[#, 14]} & /@ Flatten[#], Automatic}, ##2, 
    ImageSize -> 1 -> 30, PlotRange -> {{0, 20}, {0, 4}}] &;

DynamicModule[{i1 = {5, 10}, i2 = {6, 11}}, 
 Panel @ Column[{Row[{Style["interval 1    ", 16, ColorData[97]@4], 
      Dynamic@Pane[i1[[1]], Alignment -> Right, ImageSize -> {40, 20},
         BaselinePosition -> Center], 
      IntervalSlider[Dynamic[i1], {0, 20, 1}, Method -> "Stop", 
       Appearance -> {"Paired"}, ImageSize -> 300], 
      Dynamic@Pane[i1[[2]], Alignment -> Left, ImageSize -> {40, 20}, 
        BaselinePosition -> Center]}, Spacer[10]], 
    Row[{Style["interval 2    ", 16, ColorData[97]@1], 
      Dynamic@Pane[i2[[1]], Alignment -> Right, ImageSize -> {40, 20},
         BaselinePosition -> Center],
      IntervalSlider[Dynamic[i2], {0, 20, 1}, Method -> "Stop", 
       Appearance -> {"Paired"}, ImageSize -> 300], 
      Dynamic@Pane[i2[[2]], Alignment -> Left, ImageSize -> {40, 20}, 
        BaselinePosition -> Center]}, Spacer[10]], 
    Dynamic[Panel[#, ImageMargins -> 10] &@
      nlP[{i2, i1}, ImageSize -> 1 -> {30, 25}, 
       PlotRange -> {{0, 20}, {0, 4}}, 
       PlotStyle -> (Directive[CapForm["Round"], #,                 
            AbsolutePointSize[5], AbsoluteThickness[7]] & /@ 
         {ColorData[97]@1, ColorData[97]@2, Black, ColorData[97]@4})]]}, 
   Spacings -> {0, -.5, 1.5}, Alignment -> Center]]

enter image description here

Original answer:

ClearAll[f]
f[i__Interval] := Min @ IntervalIntersection[i] /. Infinity -> {}

f[l__Line] := Module[{int = Interval /@ Sort[{l}[[All, 1, All, 1]]]}, f @@ int]

Examples:

f[Interval[{5, 10}], Interval[{6, 11}]]
6
f[Line[{{5, 1}, {10, 1}}], Line[{{6, 2}, {11, 2}}]]
6
f[Interval[{5, 100}], Interval[{7, 9}]]
7
f[Interval[{5, 10}], Interval[{11, 15}]]
 {}
| improve this answer | |
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  • 1
    $\begingroup$ Similar approach for Intervals: ArgMin[{x, And @@ (Element[{x}, #] & /@ {##})}, x] &. $\endgroup$ – kirma Apr 6 at 9:26

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