Intersection point of 2 lines defined by 2 points each












2












$begingroup$


I'm implementing this in code, but I'll rewrite it so that it is easier understood (like pseudocode):



# a = pt 1 on line 1

# b = pt 2 on line 1

# c = pt 1 on line 2

# d = pt 2 on line 2

def intersect(a,b,c,d):



    # stuff for line 1

    a1 = b.y-a.y

    b1 = a.x-b.x

    c1 = a1*a.x + b1*a.y



    # stuff for line 2

    a2 = d.y-c.y

    b2 = c.x-d.x

    c2 = a2*c.x + b2*c.y



    determinant = a1*b2 - a2*b1



    if (determinant == 0):

        # Return (infinity, infinity) if they never intersect

        # By "never intersect", I mean that the lines are parallel to each other

        return math.inf, math,inf

    else:

        x = (b2*c1 - b1*c2)/determinant

        y = (a1*c2 - a2*c1)/determinant

        return x,y


All the above works, ... but only does by assuming that the lines extend infinitely in each direction, like a linear equation. I'll show what I mean here.



There are the 2 lines, red and green, and the gold dot is what is returned when I test this code ... but the lines don't actually intersect. What can be used to test whether the lines truly intersect?



Heres the actual Python code if needed.






linear-algebra matrices python







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    I'm implementing this in code, but I'll rewrite it so that it is easier understood (like pseudocode):



    # a = pt 1 on line 1
    
# b = pt 2 on line 1
    
# c = pt 1 on line 2
    
# d = pt 2 on line 2
    
def intersect(a,b,c,d):
    

    
    # stuff for line 1
    
    a1 = b.y-a.y
    
    b1 = a.x-b.x
    
    c1 = a1*a.x + b1*a.y
    

    
    # stuff for line 2
    
    a2 = d.y-c.y
    
    b2 = c.x-d.x
    
    c2 = a2*c.x + b2*c.y
    

    
    determinant = a1*b2 - a2*b1
    

    
    if (determinant == 0):
    
        # Return (infinity, infinity) if they never intersect
    
        # By "never intersect", I mean that the lines are parallel to each other
    
        return math.inf, math,inf
    
    else:
    
        x = (b2*c1 - b1*c2)/determinant
    
        y = (a1*c2 - a2*c1)/determinant
    
        return x,y


    All the above works, ... but only does by assuming that the lines extend infinitely in each direction, like a linear equation. I'll show what I mean here.



    There are the 2 lines, red and green, and the gold dot is what is returned when I test this code ... but the lines don't actually intersect. What can be used to test whether the lines truly intersect?



    Heres the actual Python code if needed.






    linear-algebra matrices python







    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I'm implementing this in code, but I'll rewrite it so that it is easier understood (like pseudocode):



      # a = pt 1 on line 1
      
# b = pt 2 on line 1
      
# c = pt 1 on line 2
      
# d = pt 2 on line 2
      
def intersect(a,b,c,d):
      

      
    # stuff for line 1
      
    a1 = b.y-a.y
      
    b1 = a.x-b.x
      
    c1 = a1*a.x + b1*a.y
      

      
    # stuff for line 2
      
    a2 = d.y-c.y
      
    b2 = c.x-d.x
      
    c2 = a2*c.x + b2*c.y
      

      
    determinant = a1*b2 - a2*b1
      

      
    if (determinant == 0):
      
        # Return (infinity, infinity) if they never intersect
      
        # By "never intersect", I mean that the lines are parallel to each other
      
        return math.inf, math,inf
      
    else:
      
        x = (b2*c1 - b1*c2)/determinant
      
        y = (a1*c2 - a2*c1)/determinant
      
        return x,y


      All the above works, ... but only does by assuming that the lines extend infinitely in each direction, like a linear equation. I'll show what I mean here.



      There are the 2 lines, red and green, and the gold dot is what is returned when I test this code ... but the lines don't actually intersect. What can be used to test whether the lines truly intersect?



      Heres the actual Python code if needed.






      linear-algebra matrices python







      share|cite|improve this question











      $endgroup$




      I'm implementing this in code, but I'll rewrite it so that it is easier understood (like pseudocode):



      # a = pt 1 on line 1
      
# b = pt 2 on line 1
      
# c = pt 1 on line 2
      
# d = pt 2 on line 2
      
def intersect(a,b,c,d):
      

      
    # stuff for line 1
      
    a1 = b.y-a.y
      
    b1 = a.x-b.x
      
    c1 = a1*a.x + b1*a.y
      

      
    # stuff for line 2
      
    a2 = d.y-c.y
      
    b2 = c.x-d.x
      
    c2 = a2*c.x + b2*c.y
      

      
    determinant = a1*b2 - a2*b1
      

      
    if (determinant == 0):
      
        # Return (infinity, infinity) if they never intersect
      
        # By "never intersect", I mean that the lines are parallel to each other
      
        return math.inf, math,inf
      
    else:
      
        x = (b2*c1 - b1*c2)/determinant
      
        y = (a1*c2 - a2*c1)/determinant
      
        return x,y


      All the above works, ... but only does by assuming that the lines extend infinitely in each direction, like a linear equation. I'll show what I mean here.



      There are the 2 lines, red and green, and the gold dot is what is returned when I test this code ... but the lines don't actually intersect. What can be used to test whether the lines truly intersect?



      Heres the actual Python code if needed.






      linear-algebra matrices python




      linear-algebra matrices python






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago









      Ethan Bolker

      45.6k553120




      45.6k553120










      asked 2 hours ago









      crazicrafter1crazicrafter1

      197




      197






















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          I think you are asking for the intersection point (if any) of two line segments, not two lines.



          Once you find the intersection point $P$ as you have, you can check that it is between the endpoints $A$ and $B$ of a segment by solving the equation
          $$
          tA + (1-t)B = P
          $$
          for $t$ and checking that $t$ is between $0$ and $1$. That equation will have a solution because you know $P$ is on the line. Do that for each of the two segments.



          Warning: you may have numerical instability if the determinant is close to $0$. That will happen when the lines are nearly parallel.



          (There may be a shorter way to do this from scratch, but this will work.)






          share|cite|improve this answer











          $endgroup$





















            2












            $begingroup$

            You have the point $x$ where the infinite lines intersect. You need to check whether that point is on both finite line segments.



            Line segment 1 has endpoints $a$ and $b$. Use these to make a vector $vec{ab}=b-a$. If the dot product $vec{ab}cdotvec{ax}$ is positive, then $x$ is forward of $a$; if it's negative, then $x$ is behind $a$. Likewise, if $vec{ab}cdotvec{bx}$ is positive, then $x$ is forward of $b$. The point $x$ is on the segment if it's between $a$ and $b$.



            Do the same test for the other line segment.






            share|cite|improve this answer









            $endgroup$














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              2 Answers
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              active

              oldest

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              2 Answers
              2






              active

              oldest

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              active

              oldest

              votes






              active

              oldest

              votes









              2












              $begingroup$

              I think you are asking for the intersection point (if any) of two line segments, not two lines.



              Once you find the intersection point $P$ as you have, you can check that it is between the endpoints $A$ and $B$ of a segment by solving the equation
              $$
              tA + (1-t)B = P
              $$
              for $t$ and checking that $t$ is between $0$ and $1$. That equation will have a solution because you know $P$ is on the line. Do that for each of the two segments.



              Warning: you may have numerical instability if the determinant is close to $0$. That will happen when the lines are nearly parallel.



              (There may be a shorter way to do this from scratch, but this will work.)






              share|cite|improve this answer











              $endgroup$


















                2












                $begingroup$

                I think you are asking for the intersection point (if any) of two line segments, not two lines.



                Once you find the intersection point $P$ as you have, you can check that it is between the endpoints $A$ and $B$ of a segment by solving the equation
                $$
                tA + (1-t)B = P
                $$
                for $t$ and checking that $t$ is between $0$ and $1$. That equation will have a solution because you know $P$ is on the line. Do that for each of the two segments.



                Warning: you may have numerical instability if the determinant is close to $0$. That will happen when the lines are nearly parallel.



                (There may be a shorter way to do this from scratch, but this will work.)






                share|cite|improve this answer











                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  I think you are asking for the intersection point (if any) of two line segments, not two lines.



                  Once you find the intersection point $P$ as you have, you can check that it is between the endpoints $A$ and $B$ of a segment by solving the equation
                  $$
                  tA + (1-t)B = P
                  $$
                  for $t$ and checking that $t$ is between $0$ and $1$. That equation will have a solution because you know $P$ is on the line. Do that for each of the two segments.



                  Warning: you may have numerical instability if the determinant is close to $0$. That will happen when the lines are nearly parallel.



                  (There may be a shorter way to do this from scratch, but this will work.)






                  share|cite|improve this answer











                  $endgroup$



                  I think you are asking for the intersection point (if any) of two line segments, not two lines.



                  Once you find the intersection point $P$ as you have, you can check that it is between the endpoints $A$ and $B$ of a segment by solving the equation
                  $$
                  tA + (1-t)B = P
                  $$
                  for $t$ and checking that $t$ is between $0$ and $1$. That equation will have a solution because you know $P$ is on the line. Do that for each of the two segments.



                  Warning: you may have numerical instability if the determinant is close to $0$. That will happen when the lines are nearly parallel.



                  (There may be a shorter way to do this from scratch, but this will work.)







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 1 hour ago

























                  answered 1 hour ago









                  Ethan BolkerEthan Bolker

                  45.6k553120




                  45.6k553120























                      2












                      $begingroup$

                      You have the point $x$ where the infinite lines intersect. You need to check whether that point is on both finite line segments.



                      Line segment 1 has endpoints $a$ and $b$. Use these to make a vector $vec{ab}=b-a$. If the dot product $vec{ab}cdotvec{ax}$ is positive, then $x$ is forward of $a$; if it's negative, then $x$ is behind $a$. Likewise, if $vec{ab}cdotvec{bx}$ is positive, then $x$ is forward of $b$. The point $x$ is on the segment if it's between $a$ and $b$.



                      Do the same test for the other line segment.






                      share|cite|improve this answer









                      $endgroup$


















                        2












                        $begingroup$

                        You have the point $x$ where the infinite lines intersect. You need to check whether that point is on both finite line segments.



                        Line segment 1 has endpoints $a$ and $b$. Use these to make a vector $vec{ab}=b-a$. If the dot product $vec{ab}cdotvec{ax}$ is positive, then $x$ is forward of $a$; if it's negative, then $x$ is behind $a$. Likewise, if $vec{ab}cdotvec{bx}$ is positive, then $x$ is forward of $b$. The point $x$ is on the segment if it's between $a$ and $b$.



                        Do the same test for the other line segment.






                        share|cite|improve this answer









                        $endgroup$
















                          2












                          2








                          2





                          $begingroup$

                          You have the point $x$ where the infinite lines intersect. You need to check whether that point is on both finite line segments.



                          Line segment 1 has endpoints $a$ and $b$. Use these to make a vector $vec{ab}=b-a$. If the dot product $vec{ab}cdotvec{ax}$ is positive, then $x$ is forward of $a$; if it's negative, then $x$ is behind $a$. Likewise, if $vec{ab}cdotvec{bx}$ is positive, then $x$ is forward of $b$. The point $x$ is on the segment if it's between $a$ and $b$.



                          Do the same test for the other line segment.






                          share|cite|improve this answer









                          $endgroup$



                          You have the point $x$ where the infinite lines intersect. You need to check whether that point is on both finite line segments.



                          Line segment 1 has endpoints $a$ and $b$. Use these to make a vector $vec{ab}=b-a$. If the dot product $vec{ab}cdotvec{ax}$ is positive, then $x$ is forward of $a$; if it's negative, then $x$ is behind $a$. Likewise, if $vec{ab}cdotvec{bx}$ is positive, then $x$ is forward of $b$. The point $x$ is on the segment if it's between $a$ and $b$.



                          Do the same test for the other line segment.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 1 hour ago









                          mr_e_manmr_e_man

                          1,1401424




                          1,1401424






























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