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Plot line intersections in 3D?

Alex Gausman

Posted 8 years ago

I have spent the last hour trying to figure out how to plot a line in 3D in either WolframAlpha or Mathematica. I am not experienced with either of these products, but I should not be having such a hard time trying to perform such a simple task. Ultimately I would like to check if two lines intersect in 3D, but I do not know how to properly represent the equations to these lines so that they are plotted as vectors in 3D. Any help would be much appreciated,

POSTED BY: Alex Gausman

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Alex Gausman

Posted 8 years ago

Wow! Thanks so much everybody for your suggestions. This is my first time using Mathematica and posting on the discussion boards and I'm super impressed with how many helpful responses I received in just a day. This must mean I've come to the right place (and have found the right software), haha. During my searching I did manage to find the documentation for describing 3D lines in terms of points along the line, however (because I am interested in the topic for calculus homework) I was particularly curious about how to define a line in terms of a point and a vector. This last explanation is exactly what I was looking for. Thanks for the help!

POSTED BY: Alex Gausman

David Keith

Posted 8 years ago

Here is another approach. Lines are defined by vector equations. The vector equation for each line is derived from a pair of points on the line, but they are just two points on the line -- it is not a line segment with endpoints. We then have two lines, each parameterized by its own parameter. FindInstance is used to determine a pair of parameter values for which the lines share a common point. If no intersection exists, it will return an empty list. (* a vector equation for a line from two points ) pa1 = {-2, -3, -1}; pa2 = {3, 2, 2}; lineA[t_] := pa1 + t Normalize[pa2 - pa1] ( a second line ) ( notice they intersect at {3,2,2} ) pb1 = {-4, 1, -3}; pb2 = {3, 2, 2}; lineB[t_] := pb1 + t Normalize[pb2 - pb1] ( find an intersection ) ( if they don't intersect, FindInstance would return { } *) intersection = FindInstance[lineA[t1] == lineB[t2], {t1, t2}] {{t1 -> Sqrt[59], t2 -> 5 Sqrt[3]}} Show[Graphics3D[{PointSize[Large], Red, Point[lineA[t1] /. intersection[[1]]]}], ParametricPlot3D[{lineA[t], lineB[t]}, {t, -5, 15}]]

Here is another approach. Lines are defined by vector equations. The vector equation for each line is derived from a pair of points on the line, but they are just two points on the line -- it is not a line segment with endpoints. We then have two lines, each parameterized by its own parameter. FindInstance is used to determine a pair of parameter values for which the lines share a common point. If no intersection exists, it will return an empty list.

(* a vector equation for a line from two points *)

pa1 = {-2, -3, -1}; pa2 = {3, 2, 2};

lineA[t_] := pa1 + t Normalize[pa2 - pa1]

(* a second line *)

(* notice they intersect at {3,2,2} *)

pb1 = {-4, 1, -3}; pb2 = {3, 2, 2};

lineB[t_] := pb1 + t Normalize[pb2 - pb1]

(* find an intersection *)
(* if they don't intersect, FindInstance would return { } *)

intersection = FindInstance[lineA[t1] == lineB[t2], {t1, t2}]

{{t1 -> Sqrt[59], t2 -> 5 Sqrt[3]}}

Show[Graphics3D[{PointSize[Large], Red, 
   Point[lineA[t1] /. intersection[[1]]]}], 
 ParametricPlot3D[{lineA[t], lineB[t]}, {t, -5, 15}]]

enter image description here

POSTED BY: David Keith

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 8 years ago

Maybe like this: p1 = {-1, 0, 0}; (This defines point no.1) p2 = {1, 1, 1}; (This defines point no.2) p3 = {-1, 1, 1}; (This defines point no.3) p4 = {0, 0, 0}; (This defines point no.4) Show[Plot3D[0, {x, -1, 1}, {y, -1, 1}, PlotStyle -> None, Mesh -> None, RegionFunction -> Function[{x}, 0 < x < 0], AxesLabel -> Automatic], Graphics3D[{{Blue, Thick, Line[{p1, p2}]}, {Green, Thick, Line[{p3, p4}]}}]]

Maybe like this:

p1 = {-1, 0, 0};  (*This defines point no.1*)
p2 = {1, 1, 1}; (*This defines point no.2*)
p3 = {-1, 1, 1};  (*This defines point no.3*)
p4 = {0, 0, 0}; (*This defines point no.4*)

  Show[Plot3D[0, {x, -1, 1}, {y, -1, 1}, PlotStyle -> None, 
  Mesh -> None, RegionFunction -> Function[{x}, 0 < x < 0], 
  AxesLabel -> Automatic], 
  Graphics3D[{{Blue, Thick, Line[{p1, p2}]}, {Green, Thick, Line[{p3, p4}]}}]]

enter image description here

POSTED BY: Mariusz Iwaniuk

Hans Milton

Posted 8 years ago

A slightly more simple variation? p1 = {-1, 0, 0}; (This defines point no.1) p2 = {1, 1, 1}; (This defines point no.2) p3 = {-1, 1, 1}; (This defines point no.3) p4 = {0, 0, 0}; (This defines point no.4) Graphics3D[{ Thick, Blue, Line[{p1, p2}], Green, Line[{p3, p4}] }, Axes -> True ]

A slightly more simple variation?

p1 = {-1, 0, 0}; (*This defines point no.1*)
p2 = {1, 1, 1}; (*This defines point no.2*)
p3 = {-1, 1, 1}; (*This defines point no.3*)
p4 = {0, 0, 0}; (*This defines point no.4*)
Graphics3D[{
    Thick,
    Blue,
    Line[{p1, p2}],
    Green,
    Line[{p3, p4}]
  }, Axes -> True
]

enter image description here