# line intersection equation

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Okay im trying to make out the equalation(sp?, word?) for line intersection. As described here: http://astronomy.swin.edu.au/~pbourke/geometry/lineline2d/ It starts with Pa = P1 + ua ( P2 - P1 ) Pb = P3 + ub ( P4 - P3 ) Where ua and ub are in range [0..1] This I understand, as it is just interpolation between 2 poins to generate a line. Now the next thing is to solve for ua and ub, since the line intersects if: ( P1 + ua( P2 - P1 ) ) = ( P3 + ub ( P4 - P3 ) ) I also have no problem understanding this: It means that if a point on P1P2 and P3P4 have the same location, but ua and ub dont need to be equal, just the curret point on the line the article now solves for ua and ub, this is where I get trubble. How can I solve for these, I have tried and tried, just cant get it working. So if anyone could be so kind to solve these step by step, and say what you do each step, I'did be happy. - CoMaNdore

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Now you want to have either ua or ub in the following system :

So, let us recap what you know :
- P1, P2, P3, P4 are known (and all the x1..4 and y1..4)
- There should be a point P = P1 + ua(P2-P1) = P3 + ub(P3-P4) such that both equations :
x1 + ua(x2 - x1) = x3 + ub(x4 - x3) (E1) AND
y1 + ua(y2 - y1) = y3 + ub(y4 - y3) (E2)
are true.

You want to know P, so you ONLY need to know ua or ub, expressed with something you know.

(E1) && (E2) (y4-y3)*(E1) && (x4-x3)*(E2) (1)
x1*(y4-y3) + ua(x2 - x1)*(y4-y3) = x3*(y4-y3) + ub(x4 - x3)*(y4-y3) (E1') &&
y1*(x4-x3) + ua(y2 - y1)*(x4-x3) = y3*(x4-x3) + ub(y4 - y3)*(x4-x3) (E2') (2)

(1) I am just multiplying stuff. Well if a=b, 2a=2b, isn'it ?

(2) Know, if you take have a=b and c=d, than a+c = b+d, no ? an a-c = b-d, after all ? Then you
practicaly write (E1' - E2') to get rid of ub, and voila :

(E1' && E2')

[x1*(y4-y3) + ua(x2 - x1)*(y4-y3)] - [y1*(x4-x3) + ua(y2 - y1)*(x4-x3)] =
[x3*(y4-y3) + ub(x4 - x3)*(y4-y3)] - [y3*(x4-x3) + ub(y4 - y3)*(x4-x3)]

[x1*(y4-y3) + ua(x2 - x1)*(y4-y3)] - [y1*(x4-x3) + ua(y2 - y1)*(x4-x3)] =
x3*(y4-y3) + ub(x4 - x3)*(y4-y3) - y3*(x4-x3) - ub(y4 - y3)*(x4-x3)

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

ua[(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)] + x1*(y4-y3) - y1*(x4-x3) = x3*(y4-y3) - y3*(x4-x3)

ua[(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)] = x3*(y4-y3) + y1*(x4-x3) - y3*(x4-x3) - x1*(y4-y3)

ua = [x3*(y4-y3) + y1*(x4-x3) - y3*(x4-x3) - x1*(y4-y3)] / [(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)]

ua = [(x3-x1)*(x4-x3) + (y1-y3)*(x4-x3)] / [(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)]

Which is what you want . (CQFD, as we say in french...)

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Sorry, I messed up with the message (and I forgot my account detail, so a I cant edit it)

I wrote the equivalent sign many times, and it disapear. So, the demo will use the == sign to tell that stuff are logically equivalent.

Now you want to have either ua or ub in the following system :

So, let us recap what you know :
- P1, P2, P3, P4 are known (and all the x1..4 and y1..4)
- There should be a point P = P1 + ua(P2-P1) = P3 + ub(P3-P4) such that both equations :
x1 + ua(x2 - x1) = x3 + ub(x4 - x3) (E1) AND
y1 + ua(y2 - y1) = y3 + ub(y4 - y3) (E2)
are true.

You want to know P, so you ONLY need to know ua or ub, expressed with something you know.

(E1) && (E2) == (y4-y3)*(E1) && (x4-x3)*(E2) (1)
== x1*(y4-y3) + ua(x2 - x1)*(y4-y3) = x3*(y4-y3) + ub(x4 - x3)*(y4-y3) (E1') &&
y1*(x4-x3) + ua(y2 - y1)*(x4-x3) = y3*(x4-x3) + ub(y4 - y3)*(x4-x3) (E2') (2)

(1) I am just multiplying stuff. Well if a=b, 2a=2b, isn'it ?

(2) Know, if you take have a=b and c=d, than a+c = b+d, no ? an a-c = b-d, after all ? Then you
practicaly write (E1' - E2') to get rid of ub, and voila :

(E1' && E2') ==

[x1*(y4-y3) + ua(x2 - x1)*(y4-y3)] - [y1*(x4-x3) + ua(y2 - y1)*(x4-x3)] =
[x3*(y4-y3) + ub(x4 - x3)*(y4-y3)] - [y3*(x4-x3) + ub(y4 - y3)*(x4-x3)]

==

[x1*(y4-y3) + ua(x2 - x1)*(y4-y3)] - [y1*(x4-x3) + ua(y2 - y1)*(x4-x3)] =
x3*(y4-y3) + ub(x4 - x3)*(y4-y3) - y3*(x4-x3) - ub(y4 - y3)*(x4-x3)

==

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

==

ua[(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)] + x1*(y4-y3) - y1*(x4-x3) = x3*(y4-y3) - y3*(x4-x3)

==

ua[(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)] = x3*(y4-y3) + y1*(x4-x3) - y3*(x4-x3) - x1*(y4-y3)

==

ua = [x3*(y4-y3) + y1*(x4-x3) - y3*(x4-x3) - x1*(y4-y3)] / [(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)]

==

ua = [(x3-x1)*(x4-x3) + (y1-y3)*(x4-x3)] / [(x2-x1)*(y4-y3) - (y2-y1)*(x4-x3)]

Which is what you want . (CQFD, as we say in french...)

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