Triangulation Calculator - Intersection From Two Bearings

A triangulation calculator takes the Cartesian coordinates of two known observation points and the compass bearings (azimuths) measured at each one toward an unknown landmark, then returns its (x, y) coordinates where the two bearing lines cross.

• • Locate a landmark from two observation points: Stand at two known positions, sight the unknown object with a compass or theodolite, and the panel resolves the two bearing lines into a single coordinate pair.
• • Reconstruct the layout of a survey network: Given a recorded base of two coordinates and a pair of measured bearings from each, recover the location of a control point that is no longer reachable.
• • Check homework and textbook problems: Type in the two observation points and the two bearings, and the panel shows the intersection point and the observation distances for a quick sanity check.

The native use case is the intersection case: two known observation points A(x1, y1) and B(x2, y2) and two compass bearings alpha and beta that point at the same unknown landmark. The landmark C(x3, y3) sits where the two bearing lines cross, and the panel also reports the distance from each observation point to C for a sanity check on the geometry.

Once the three vertices A, B, and C are known, a general Triangle Calculator can finish the same triangle by computing its perimeter, area, and the remaining angles from the same coordinates.

Behind the panel sit two line equations in Cartesian coordinates, a 2x2 linear system, and Cramer's rule. The form takes two points and two bearings, then returns the unknown landmark (x, y) and the distance from each observation point.

• x1, y1: Cartesian coordinates of the first observation point A, in any consistent length unit (m, ft, km, ...).
• x2, y2: Cartesian coordinates of the second observation point B, in the same length unit as A.
• alpha: Compass bearing (azimuth) measured at A toward the unknown landmark, in degrees clockwise from north. 0 = north, 90 = east, 180 = south, 270 = west.
• beta: Compass bearing (azimuth) measured at B toward the unknown landmark, in the same convention as alpha. Must not be parallel to alpha.
• x3, y3: Computed Cartesian coordinates of the unknown landmark, in the same length unit as the inputs.

Each bearing defines a straight line that passes through one of the observation points. The landmark sits at the intersection of the two lines. The closed-form solution uses the sine and cosine of both bearings and Cramer's rule.

According to Wikipedia: Triangulation (surveying), triangulation in surveying is the method of determining the location of an unknown point by forming triangles to it from known points, with intersection and resection as the two general types.

Each bearing becomes a line with a known slope, and a Slope Calculator gives the same slope from two coordinates if the user wants to verify the geometry of any one of those lines before reading the intersection.

Four ideas carry the whole method. Once you can name them, the intersection case this form expects feels mechanical.

The same identity also explains why the panel takes six inputs instead of fewer. Two coordinates fix one observation point, one bearing fixes one line, and two such pairs give a 2x2 system with a unique solution as long as the two bearings are not parallel.

The same x and y differences that drive the intersection also drive a 2D Distance Calculator, so the two panels can be cross-checked at any time.

Five steps from typing the first coordinate to reading the distance from each observation point. The form expects two observation points and two compass bearings measured at each one toward the same unknown landmark.

1. 1 Pick two observation points and record their coordinates: Choose two positions A and B whose x and y coordinates you know in the same length unit (m, ft, km, ...).
2. 2 Measure the compass bearing at each point: Stand at A, point a compass at the landmark, and read the bearing in degrees clockwise from north. Repeat at B.
3. 3 Type the four coordinates and two bearings: Enter x1, y1, x2, y2, alpha, and beta. The two pairs share the same length unit.
4. 4 Read the (x, y) coordinates of the landmark: The first two rows of the results panel show x3 and y3 in the same unit as the inputs.
5. 5 Read the distance from each observation point: The next two rows show distance A to landmark and distance B to landmark. Use them to sanity-check the geometry.

Try A=(0,0) with bearing 30 degrees and B=(5,0) with bearing 330 degrees. The panel shows x3=2.5, y3=4.33, matching the Omni triangulation example case.

If the two observation points sit on the same line as the landmark, a Midpoint Calculator gives the half-way point along that line.

Four practical reasons to let the panel solve the intersection instead of hand-substituting into the 2x2 system.

• • Closed-form answer from six inputs: Two observation points, two bearings, and a single Cramer's-rule pass return the (x, y) coordinates of the landmark and the distance from each observation point.
• • Works for any pair of bearings: The matrix form handles bearings at 0, 90, 180, and 270 degrees cleanly because cos and sin stay finite.
• • Same length unit for every output: The (x, y) coordinates, the distance from A, and the distance from B all inherit the unit of the input coordinates.
• • Catches the parallel-bearings degeneracy: When the two bearings are equal or differ by 180 degrees, the panel returns a validation error instead of returning Infinity or NaN.

Once A, B, and the landmark are pinned down, an Area Triangle Coordinates Calculator closes the same triangle with the shoelace formula for area, signed sum, and perimeter.

Four things that move the (x, y) coordinates and the observation distances, plus two honest caveats about the method.

• • The form targets the intersection case. Resection (find the observer's own position from two known landmarks) needs a separate panel and is not auto-branched here.
• • The panel cannot detect a typo. Enter 30 in the alpha box when you meant 3 and the panel happily shows a result for a 30-degree bearing.

Cross-check the two observation distances against the real-world geometry before treating the result as final.

As published by Britannica: Triangulation, triangulation in surveying relies on measuring the angles of a network of triangles to determine distances and relative positions of locations spread over the survey area using trigonometry.

According to Wikipedia: Trilateration, trilateration uses distance measurements from known points to determine an unknown location, while triangulation uses angle measurements; GPS uses trilateration, not triangulation.

If the two observation distances look off, recompute one of them by hand with a Length of a Line Segment Calculator from the same x and y values to see whether the coordinates or the bearings are the problem.