Resection Calculator

Resection, also called backward intersection or free stationing, is a classic surveying technique used to find the coordinates of an unknown station by measuring horizontal angles to three known control points. It is one of the fastest ways to set up a total station on a job site because you do not need to occupy a known point; you only sight three points whose coordinates are already established.

The method has been used since the era of plane-table mapping and remains a core field procedure today. A total station or theodolite is placed at the unknown point P, the instrument is oriented to any convenient reference, and the angles between the rays to control points A, B, and C are measured. From those two angles and the known coordinates, the position of P can be computed analytically.

How Tienstra's Method Works

This calculator uses Tienstra's method, a closed-form solution to the three-point resection problem. Given the interior angles of the known triangle ABC and the two measured angles at P, the method assigns a weight to each control point based on the difference between the cotangent of the triangle angle and the cotangent of the corresponding angle at P. The unknown coordinates are then the weighted average of the three known coordinates.

Enter the control points in the order they appear around the horizon, measure the angle from A to B and from B to C, and the calculator computes the remaining sector angle automatically. The result includes the station coordinates, distances back to each control point, and a geometry-margin check that warns you when the solution is near the danger circle.

When to Use Backward Intersection in the Field

Resection is the right choice whenever you need a temporary instrument setup and occupying a known point is impractical. Common scenarios include setting up a total station on a busy roadway, establishing a control point inside a building where no monument exists, or extending control into a remote area where only three distant control points are visible. It is also a standard classroom exercise in surveying and geomatics programs because it ties together coordinate geometry, angle measurement, and error analysis.

Resection vs. Intersection vs. Radiation

These three techniques are often confused because they all use angles and known points. Resection measures angles from an unknown point to known points to locate the observer. Intersection measures angles from two known points to an unknown point and computes where the rays cross. Radiation places the instrument on a known point and measures angles and distances to locate new points. Choosing the right method depends on which points you can occupy and which angles you can measure safely.

How to Avoid the Danger Circle in Resection

The danger circle is the circumcircle that passes through the three control points. If the unknown station lies on or near this circle, the resection becomes mathematically unstable and small measurement errors produce large coordinate errors. You can avoid the problem by selecting control points that surround the station rather than placing the station on the same arc as the control triangle. The geometry-margin value reported by this tool is a quick indicator: larger values mean a more stable solution.

Worked Example

The table below shows the equilateral-triangle preset used by the quick-load button. With three symmetric control points and two 120° angles, the station falls exactly at the centroid of the triangle.

Frequently Asked Questions