Point of Intersection between Two Lines

Date: 07/08/2003 at 09:28:24
From: Mr. Pintu Saha
Subject: Getting point of intersection between two lines

Sir, 

I am developing a software program (civil work) and am facing a 
problem. I have one line whose coordinates are (x1,y1) and (x2,y2).
There is another line with coordinates (x3,y3) and (x4,y4). The 
problem is how to check if these lines intersect between 
these coordinates, and if yes, find the point of intersection (x,y)

I am doing the software in VB.

Date: 07/08/2003 at 13:23:43
From: Doctor George
Subject: Re: Getting point of intersection between two lines

Hi Pintu,

Thanks for writing to Doctor Math.

Let L12 and L34 be the lines through the two pairs of points, and call 
each point Pi = (xi,yi). We can write vector equations for the two 
lines like this.

              L12 = P1 + a(P2 - P1)

              L34 = P3 + b(P4 - P3)

Where L12 and L34 intersect we must have

              P1 + a(P2 - P1) = P3 + b(P4 - P3)

              a(P2 - P1) = (P3 - P1) + b(P4 - P3)

Now define vector VP to be perpendicular to (P4 - P3) and take the dot 
product of both sides with VP.

              a[(P2 - P1).VP] = (P3 - P1).VP

Note that the 'b' term has dropped out. If (P2 - P1).VP = 0 then L12 
and L34 are parallel, otherwise we can solve for 'a'. In a similar way 
we can also make 'a' drop out and solve for 'b.'

If 0 < a < 1 the intersection is between P1 and P2. If 0 < b < 1 the 
intersection is between P3 and P4.

To get the intersection point, use 'a' in the equation for L12, or 'b' 
in the equation for L34.

Does that make sense? Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/