math - Find point of intersection between two vectors in MATLAB

Question

I have a very simple MATLAB question. What is the easiest way to find the point of intersection between two vectors. I am not familiar with the various MATLAB functions -- it seems like there should be one for this.

For example, if I have one vector from (0,0) to (6,6) and another vector from (0,6) to (6,0), I need to determine that they intersect at (3,3).

Answer 1

One solution is to use the equations derived in this tutorial for finding the intersection point of two lines in 2-D (update: this is an internet archive link since the site no longer exists). You can first create two matrices: one to hold the x coordinates of the line endpoints and one to hold the y coordinates.

x = [0 0; 6 6];  %# Starting points in first row, ending points in second row
y = [0 6; 6 0];

The equations from the above source can then be coded up as follows:

dx = diff(x);  %# Take the differences down each column
dy = diff(y);
den = dx(1)*dy(2)-dy(1)*dx(2);  %# Precompute the denominator
ua = (dx(2)*(y(1)-y(3))-dy(2)*(x(1)-x(3)))/den;
ub = (dx(1)*(y(1)-y(3))-dy(1)*(x(1)-x(3)))/den;

And you can now compute the intersection point of the two lines:

xi = x(1)+ua*dx(1);
yi = y(1)+ua*dy(1);

For the example in the question, the above code gives xi = 3 and yi = 3, as expected. If you want to check that the intersection point lies between the endpoints of the lines (i.e. they are finite line segments), you just have to check that the values ua and ub both lie between 0 and 1:

isInSegment = all(([ua ub] >= 0) & ([ua ub] <= 1));

A couple more points from the tutorial I linked to above:

• If the denominator den is 0 then the two lines are parallel.
• If the denominator and numerator for the equations for ua and ub are 0 then the two lines are coincident.