What is the formula of the intersection of two lines?

Answer 1

Given coordinates of two points and directions (bearings or azimuths) from those two points, find the coordinates of the point of intersection, assuming that the lines do intersect and are not parallel. Use the Cantuland method to calculate the coordinates of the northing and the easting. This is a simplification of a process that came from the use of simultaneous equations from matrix algebra that employed a trigonomic identity for tangent functions.

Northing of the point of intersection:
1. Convert the azimuth of the first line to degrees and decimal degrees.
2. Find the tangent of the azimuth of the first line.
3. Step-two, times the northing of the point on the first line.
4. Step-three, minus the easting of the point on the first line.

5. Convert the azimuth of the second line to degrees and decimal degrees.
6. Find the tangent of the azimuth of the second line.
7. Step-six, times the northing of the point on the second line.
8. Step-seven, minus the easting of the point on the second line.

9. Step-four, minus step-eight.
10. Step-two, minus step-six.
11. Step-nine, divided by step-ten. That's the northing of the intersection.

Now let's find the easting. Most of the steps are the same, except a little bit is added into the process. See steps 4A, 8A and 9.

Easting of the point of intersection:
1. Convert the azimuth of the first line to degrees and decimal degrees.
2. Find the tangent of the azimuth of the first line.
3. Step-two, times the northing of the point on the first line.
4. Step-three, minus the easting of the point on the first line.
4A. Step-four, times step-six.

5. Convert the azimuth of the second line to degrees and decimal degrees.
6. Find the tangent of the azimuth of the second line.
7. Step-six, times the northing of the point on the second line.
8. Step-seven, minus the easting of the point on the second line.
8A. Step-eight, times step-two.

9. Step-4A, minus step-8A.
10. Step-two, minus step-six.
11. Step-nine, divided by step-ten. That's the easting of the intersection.

This works unless the azimuth of one of the lines is 90 degrees or 270 degrees. Tangent of the azimuth of 90 degrees or 270 degrees will result in "undefined", and the above will not work. In this case, swap all calls for "easting" to "northing"; and swap all calls for "northing" to "easting"; and swap the calls for the Tangent function to replace them with Cotangent functions. This adjustment to the process will work for all intersections except when the azimuth of one of the lines is zero degrees or 180 degrees. In those cases, use the unmodified steps as outlined above to take care of those issues.