Tienstra Resection: Understanding the Danger Circle and When to Reposition

The Tienstra resection is a method for computing the position of an unknown station by observing the angles between three known control points. It is elegant, fast to compute, and well-suited to free-station surveying — but it carries a geometric condition that makes it fail silently in certain configurations.

What Is the Danger Circle?

The danger circle is the circumscribed circle of the three control points A, B, and C. When the unknown station P lies on this circle, the observed angles α and β become determinate equations for the same family of solutions — the problem becomes geometrically degenerate and the Tienstra formula yields a singularity.

In practice, what this means: as P approaches the circumcircle of ABC, the computed coordinates become increasingly sensitive to small errors in the observed angles. A 1-arcsecond error that would normally affect the position by a few millimetres can, near the danger circle, propagate into centimetres or metres.

The Mathematics

The Tienstra formula computes weights K_A, K_B, K_C based on the relationship between the observed angles and the interior angles of the triangle:

Where α, β are the observed angles at P and A, B are the interior angles of triangle ABC.

The position of P follows:

The danger circle condition is expressed by the denominators approaching zero simultaneously — when K_A + K_B + K_C ≈ 0.

Detecting the Condition Before Observing

You can detect a potential danger circle condition before setting up the instrument:

1. Plot your three control points 2. Compute the circumcircle of ABC (any geometry package can do this, or use our calculator) 3. Estimate your likely station position from site plans 4. If your estimated position is within one circumradius of the circumcircle boundary, choose different control points or add a fourth observation

The surveyiiing.com Tienstra calculator computes and displays the circumcircle in real time as you enter coordinates. The danger circle warning triggers when P is within 5% of the circumradius from the circle boundary.

When to Reposition

If the warning triggers, your options are:

- **Use a different control station** — even moving 10 metres away from the circumcircle can dramatically improve the geometry - **Add a fourth observation** — a three-point Bowditch resection with four control points provides redundancy and flags geometric weakness via the closure - **Re-observe from a new free station** — set up in a location where your estimated position is clearly inside or clearly outside the circumcircle, not near it

The geometry of your control network determines the geometry of your danger zone. Sites with control points concentrated in a small arc are particularly susceptible.