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[SOLVED] Finding the coodinates of two lines.

animaguy

New member
Jan 20, 2013
7
I have four points on the same plane (a, b, c, d).

Points a and b form a line ab.

Points c and d form a line cd.

Line ab and cd intersect at point e.

Does anyone have a formula that can find the coordinates of point e?
 

kanderson

Member
Jul 7, 2012
13
You would take the midpoint of opposite vertexes. Take the coordinates of A and C, the midpoint of that, then the midpoint of D to B. That is if the plane is say a square with ABCD, listed in order all around. This is if only it is a square. If a parallelogram, I think you take the midpoint A to D, and B to C. I dont really know if my answer is correct, I have geometry teacher that seems sketchy lol.
 

animaguy

New member
Jan 20, 2013
7
The problem with this scenario is polygon adcd is not a square or a parrallelogram.

I would be happy if I could find a formula that can determine point e, even if the coordinates were not whole and rounded of.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,506

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,909
Hi animaguy! :)

From wikipedia (slightly revised to fit the purpose):
The point where multiple lines meet closest in any number of dimensions is:
[TEX]x= \left(\sum_i I-v_i v_i^T\right)^{-1} \left(\sum_i (I-v_i v_i^T) p_i\right)[/TEX]​
where
$v_i$ is a unit vector along the ith line,
$p_i$ is a point vector on the ith line,
$v_i^T$ is the transpose of $v_i$.​


In your case you have 2 lines and the point where those lines meet closest is the intersection point.
That means:

$v_1 = {b - a \over ||b-a||}$

$p_1 = a$

$v_2 = {d - c \over ||d-c||}$

$p_2 = c$

$x = e$​
 

animaguy

New member
Jan 20, 2013
7
Very quickly, I want to thank the help I have received. I am still working out the calculations regarding how I am applying it and so far I am having some problems but it may just be a simple error.

Regardless, thanks for the help and as soon as I have something more concrete to post I will definitely follow up as a courtesy for your help.

(Happy)
 

animaguy

New member
Jan 20, 2013
7


``````````````````````````````````````````````````````````````````````````
1)
I used the formula on the wikipedia link provided by ILikeSerena.

Assuming this is the formula that I am genuinely looking for, the two coordinates that I produce by using this formula is not the intersection of the two lines.

2)
And I am still unsure of how to test the revised formula ILikeSerena provided.

3)
I am unsure if the formula provided by the Dr. Math link applies because the formula is based on a three dimensional line.

The two lines are on the same plane (x,y) so a z-coordinate at this point is unnecessary.

```````````````````````````````````````````

Question:

Can anyone provide:

a)
a line with the xy-coordinates of two points on that line

b)
a second line with the xy-coordinates of two points on that line and on the same plane as the first line

c)
and demonstrate the use of a formula using those two lines to produce the intersection of the xy-coordinates of the point at which the two lines intersect?

``````````````````````````````````````````````

I would humbly appreciate it.

``````````````````````````````````````````````

In the meantime, I will keep trying.

Thank you,
Animaguy
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,909
I have just created an excel sheet with 2 lines, using the 2-dimensional formula (from wiki):



As you can see, the result matches the intersection point.

The formula I gave is the most generic, which is for m dimensions and for 2 or more lines.
Based on your opening post, I thought you were asking for that.
After your current comment it appears that you don't need it.
 

Attachments

animaguy

New member
Jan 20, 2013
7
I LIKE SERENA,

thanks for the help. i am still having problems solving the formula but I know what the problem is....



(x1 - x2)
(x3 - x4)
(y1 - y2)
(y3 - y4)

are self explanatory for me.....

however the values....

x1y2
y1x2
y3y4
y3x4

confuse me.....

how are these values defined?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,909
however the values....

x1y2
y1x2
y3y4
y3x4

confuse me.....

how are these values defined?
The expression $x_1 y_2$ means $x_1 \times y_2$.
 

animaguy

New member
Jan 20, 2013
7
Problem solved!