Find intersection point of two lines when I have their coordinates ?

I have four COORDINATES of two lines - [x1,y1], [x2,y2], [x3,y3], [x4,y4]. Now, how to get the coordinate of their intersecting point [x,y] ? Any help ??

[John D'Errico]

Oh, come on. You need to make an effort.

https://en.wikipedia.org/wiki/Line–line_intersection

[Roger Stafford]

Suman, I would suggest you break your problem into three steps.

Step 1: Letting the intersection point have the unknown coordinates x0 and y0, write an equation that expresses the equality of the slope of a line connecting (x1,y1) and (x2,y2) to the slope of the line connecting (x1,y1) and (x0,y0). Similarly, write a second equation equating the slopes of lines involving (x3,y3), (x4,y4), and (x0,y0). This gives you two linear equations in two unknowns.

Step 2: Perform appropriate algebraic manipulation on these equations and then translate these into a single matrix equation - that is, for example, the unknown 1-by-2 vector, [x0,y0], multiplied by a 2-by-2 matrix of known coefficients that is to equal a known 1-by-2 vector.

Step 3: Use matlab’s slash “/” operator to solve this matrix equation for the unknown vector, [x0,y0]. (This is equivalent to finding the matrix inverse of the above 2-by-2 matrix.)

[Suman Debnath]

John, I did it. But I think the equation is too long. So, I just wanted to know if there is any short form of that or not.

[Jan]

@Suman: Yes, there is a short form. Did you read the link posted by John? This is an exhaustive solution already. Making an effort means also, to ask an internet search engine.

Answer 1

[Roger Stafford]

The result of the three steps I mentioned would be:

xy = [x1*y2-x2*y1,x3*y4-x4*y3]/[y2-y1,y4-y3;-(x2-x1),-(x4-x3)];

I don’t consider that too long or complicated an expression. The xy variable here is a 1-by-2 vector consisting of the x and y coordinates of the intersection.

[Dimitar Slavchev]

For the edge cases:

Where the two lines are the same line you will get NaNs:

>> LineIntersect(0,0,1,1,0,0,1,1)

Warning: Matrix is singular to working precision.

> In LineIntersect (line 11)

ans =

NaN NaN

And for parallel lines, Infinity:

>> LineIntersect(0,0,1,1,0,1,1,2)

Warning: Matrix is singular to working precision.

> In LineIntersect (line 11)

ans =

-Inf -Inf

LineIntersect is just the above formula from Roger Stafford inside a function. The line is 11, because of comments.