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

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