Tienstra Resection: Understanding the Danger Circle and When to Reposition

Overview

Resection is the process of determining the coordinates of an unknown observer station ($P$) by measuring the horizontal angles subtended by lines of sight to three known control points ($A, B, C$) 1, 2. Among the hundreds of historical procedures to solve this "three-point problem," the Tienstra method is widely recognized by practitioners as the most compact and elegant analytical solution 1, 3.

Why This Matters

Unlike traditional resection methods that require solving intermediate bearings, distances, or triangle lengths, the Tienstra formula provides coordinates directly using a center-of-gravity type calculation 1, 4. However, it is governed by a critical geometric constraint: the Danger Circle. If your station falls on the circumcircle of the three control points, the math fails, and your position becomes indeterminate 2, 5.

Background

The formula is named after J.M. Tienstra (1895–1951), a professor at the Delft University of Technology who popularized the use of barycentric coordinates for resection 3, 6. While the mathematical roots of the problem date back to Willebrord Snellius in 1615 and later Laurent Pothenot, Tienstra's contribution was a simplified, direct weighting system that is ideally suited for modern field computation 1, 6.

Theory

The Tienstra method treats the unknown point $P$ as the "weighted mean" of the three control points $A, B,$ and $C$ 6. By assigning a specific mathematical weight ($K$) to each control point based on the geometry of the setup, the final coordinates are derived without the need for iterative trials or complex trig sequences 1.

Mathematical Principles

The coordinates of the unknown point $P(E_p, N_p)$ are calculated as follows 1, 7:

$E_p = \frac{K_1 E_a + K_2 E_b + K_3 E_c}{K_1 + K_2 + K_3}$$N_p = \frac{K_1 N_a + K_2 N_b + K_3 N_c}{K_1 + K_2 + K_3}$

Defining the Weights ($K$)

The weights are derived from the cotangents of the interior angles of the control triangle ($A, B, C$) and the observed angles measured at station $P$ ($\alpha, \beta, \gamma$) 1, 7:

• $K_1 = \frac{1}{\cot A - \cot \alpha}$
• $K_2 = \frac{1}{\cot B - \cot \beta}$
• $K_3 = \frac{1}{\cot C - \cot \gamma}$

(Note: $\alpha$ is the angle $\angle BPC$, $\beta$ is $\angle CPA$, and $\gamma$ is $\angle APB$) 7, 8.

Field Workflow

Step-by-Step Example

Scenario: Control triangle points are $A(1000, 2000)$, $B(3000, 2000)$, and $C(2000, 3732)$. The measured angles at $P$ are $\alpha = 110^\circ$, $\beta = 120^\circ$, and $\gamma = 130^\circ$ 10.

1. Triangle Geometry: Assume internal angles $A, B, C$ are all $60^\circ$ (equilateral) 10.
2. Calculate $K_1$:$K_1 = 1 / (\cot 60^\circ - \cot 110^\circ) = 1 / (0.57735 + 0.36397) = 1.0623$
3. Calculate $K_2$ & $K_3$: (Follow the same process using $\beta$ and $\gamma$).
4. Final Easting ($E_p$):$E_p = (1.0623 \times 1000 + K_2 \times 3000 + K_3 \times 2000) / \sum K$.
5. Result: This direct calculation yields the $P$ coordinates without any distance measurements required 4, 11.

Formula Breakdown: The Danger Circle

The Danger Circle occurs when the unknown point $P$ lies on the circumcircle passing through $A, B,$ and $C$ 2, 12.

Practical Tips

• Inside is Safest: To physically guarantee you avoid the Danger Circle, always site your station $P$ inside the triangle formed by $A, B,$ and $C$ 2.
• 4th Point Redundancy: Always observe a fourth control point if available. This allows you to compute a second resection as a check and provides a statistical measure of error 2, 6.
• Check Your Sums: Ensure your measured angles $\alpha + \beta + \gamma = 360^\circ$. Any deviation indicates a gross observation error or a misidentified target 9.

Common Mistakes

• Sign Confusion: When $P$ is outside the control triangle, one or more $K$ values may become negative. Ensure you maintain the correct algebraic signs throughout the summation 1.
• Near-Circle Geometry: Even if you aren't exactly on the Danger Circle, being near it results in very small denominators, which massively magnifies small angular measurement errors 2.

Best Practices

If your reconnaissance shows that you must set up outside the control triangle, perform a quick field check: if the angle at $P$ to the two "outside" points is roughly equal to the internal angle of the triangle at the "middle" point, you are near the danger circle and must reposition 2.

FAQ

Conclusion

The Tienstra method remains a masterclass in geodetic mathematics, offering a direct path from angular field data to coordinate mapping. By understanding the barycentric principles and respecting the "trap" of the Danger Circle, surveyors can utilize this method to establish high-precision site control in even the most challenging environments.

References

Schofield, W. (2001). Engineering Surveying. 5th ed. Butterworth-Heinemann. Allan, A. L. (1969). The Tienstra Method of Resection. Porta, J. M., & Thomas, F. (2009). Concise Proof of Tienstra’s Formula. Journal of Surveying Engineering.