Intersecting Lines

Date: 05/08/97 at 18:41:18
From: Alain CUYPERS
Subject: Intersection of two lines in space

Hi Dr. Math,

Could you please help me with the following problem?

I have to calculate the coordinates of the intersection point of two 
lines in space (if there is an intersection). The two given lines are 
both determinated by two points, so I don't have their equations.

Line A contains the points (x1,y1,z1) and (x2,y2,z2)
Line B contains the points (x3,y3,z3) and (x4,y4,z4)

How do I solve this problem?

Thanks,
Alain Cuypers

Date: 05/09/97 at 09:40:43
From: Doctor Jerry
Subject: Re: Intersection of two line in space

Hi Alain,

Line A can be described by the three parametric equations:

x = x1+t*(x2-x1)
y = y1+t*(y2-y1)
z = z1+t*(z2-z1)

Line B can be described by the three parametric equations: 

x = x3+s*(x4-x3)
y = y3+s*(y4-y3)
z = z3+s*(z4-z3)

To determine whether these two lines intersect, solve simultaneously 
the equations:

x1+t*(x2-x1) = x3+s*(x4-x3)
y1+t*(y2-y1) = y3+s*(y4-y3)

Are there values of t and s that satisfy these two equations?  If 
there are, then the lines intersect. If so, check to see if the two 
z-values are also equal. If not, then the lines do not intersect.

Note: if the same parameter, for example, t, is used to describe both 
lines, then you may run into trouble. The equations that result from 
this set-up model another, related question.  

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/