The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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


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 
Associated Topics:
College Linear Algebra

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.