Solving intersection...
Enter two COGO lines to calculate their intersection point.
What Is a Line-Line Intersection in COGO?
A line-line intersection is the coordinate where two planar survey lines cross. In coordinate geometry, each line is described by either two known points or by one occupied point and a direction. The calculator solves the crossing point in easting and northing coordinates, then reports line azimuths, the smaller intersection angle, and whether the crossing falls within each entered segment or on an extension.
This is a planar calculator. It is built for projected survey grids, site coordinate systems, CAD drawings, cadastral plans, and GIS layers using metres, feet, or another consistent linear unit. Latitude and longitude degrees are angular values, so they should be projected to an appropriate coordinate reference system before using COGO intersection math.
How the Line Intersection Formula Works
The calculator represents line 1 as P + t*r and line 2 as Q + u*s. P and Q are the start coordinates. r and s are direction vectors. The intersection is found with the two-dimensional cross product:
t = cross(Q - P, s) / cross(r, s) u = cross(Q - P, r) / cross(r, s) intersection = P + t*rThe denominator cross(r, s) measures whether the directions actually cross. When it is near zero, the lines are parallel or lie on the same path, so there is no single clean intersection coordinate to return.
Two-Point Lines vs Bearing Lines
A two-point line uses the vector from the start coordinate to the end coordinate. A bearing line uses a start coordinate and a survey azimuth measured clockwise from north. The bearing vector is east = sin(azimuth) and north = cos(azimuth). This keeps 0 degrees pointing north, 90 degrees east, 180 degrees south, and 270 degrees west.
The calculator supports quadrant bearings such as N 35 E, S 35 E, S 35 W, and N 35 W. These are converted to whole-circle azimuths internally before the intersection is solved.
Parallel and Coincident Lines
Parallel lines have the same or opposite direction and never meet. Coincident lines lie on the same path, which means there are infinitely many shared points rather than one intersection. The calculator separates these cases and returns a clear message instead of dividing by a near-zero denominator.
Why Planar Coordinates Matter
COGO line intersections assume a flat coordinate plane. That assumption is correct on a projected CRS or local construction grid, but it is not correct on raw latitude/longitude degrees. If your source points are GPS coordinates, first convert them to an appropriate projected coordinate system, such as a local State Plane, UTM, national grid, or site control grid.
Common Surveying Uses
Surveyors and GIS users use line intersections to resolve boundary calls, check plat geometry, set construction control, extend alignments, find road centerline crossings, and rebuild coordinates from old bearing descriptions. The segment status is important: a mathematically valid intersection can fall beyond the endpoint of a deed line, CAD segment, or alignment chord.
Line Intersection Calculator FAQ
Can I use latitude and longitude directly?
Use projected coordinates instead. Latitude and longitude are angles, not grid distances, and raw degree math will distort a COGO intersection.
Why does the result say the point is on a line extension?
The calculator solves the infinite line first. If the crossing is outside the two entered endpoints, it reports that the point is on the extension so you can decide whether that is acceptable in your survey or CAD workflow.
What does a shallow-angle warning mean?
A shallow crossing angle makes the result sensitive to small coordinate or bearing errors. In boundary work, check the field evidence and source precision before relying on that point.