# Common point for parallel lines(segments)

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

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## 1 Answer

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]]


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]]


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}]]

 {}

• Similar approach for Intervals: ArgMin[{x, And @@ (Element[{x}, #] & /@ {##})}, x] &. Apr 6 '20 at 9:26