Model Rocket Dual-Station Altitude Calculator
Enter two stations’ synchronized azimuth and elevation readings to estimate altitude with the vertical-midpoint method. The result also shows each independent altitude, closure error, and the ground point beneath the rocket.
Estimated rocket altitude (m)350.15
- Station 1 altitude (m)
- 322.3
- Station 2 altitude (m)
- 378
- Closure error
- 7.95%
- Measurement agreement
- Within tolerance: the two station estimates have acceptable closure.
- Horizontal range from station 1 (m)
- 382.31
- Horizontal range from station 2 (m)
- 464.94
- Apogee ground position
- From station 1: X = 33.32 m along the baseline; Y = 380.86 m perpendicular.
Both sight heights must use the same vertical datum.
How to use this calculator
- Choose meters or feet, then use that unit for the baseline and both sight heights.
- Measure the baseline horizontally from station 1 to station 2 and enter it before tracking.
- At the same marked instant, enter each station’s interior azimuth and elevation angle.
- Add sight heights referenced to the same ground datum, then compare closure error with your selected tolerance.
How dual-station altitude tracking works
Two observers make a ground triangle with the point directly below the rocket. Put station 1 at (0, 0) and station 2 at (L, 0), where L is the measured baseline. Each observer measures an interior azimuth from the line toward the other station, then marks an elevation angle at the same instant. Synchronized marks matter: two correct angles from different moments do not describe one rocket position.
Let A1 and A2 be the azimuth angles. The law of sines gives the horizontal sight ranges:
R1 = L × sin(A2) / sin(A1 + A2) and R2 = L × sin(A1) / sin(A1 + A2).
With elevation angles E1 and E2, and sight heights I1 and I2, the two independent altitude estimates are H1 = I1 + R1 × tan(E1) and H2 = I2 + R2 × tan(E2). This calculator reports their vertical midpoint, H = (H1 + H2) / 2. It also plots the point under the rocket as X = R1 × cos(A1) along the baseline and Y = R1 × sin(A1) perpendicular to it.
Checking tracking closure
The two altitude estimates should be close but rarely match exactly. Closure error is C = 100 × |H1 − H2| / (H1 + H2). A result at or below the tolerance is marked within tolerance. A larger result is an open track: the midpoint is still displayed, but it should be treated as a diagnostic rather than a dependable apogee value.
Baseline length, azimuth geometry, and timing move the result most. A longer, well-measured baseline can improve angle resolution, but a ground triangle whose azimuth sum is near zero or 180 degrees is poorly conditioned. Small azimuth reading errors then create large range errors. Very low elevation angles similarly magnify angular uncertainty.
What this calculator leaves out
This is a geometric optical estimate, not a flight prediction. It assumes a level common datum for both stations and sight heights; sloping terrain needs surveyed elevation corrections. It does not correct for observer reaction time, tracker calibration, parallax, haze, wind drift between sightings, or failure to mark the same instant. An onboard altimeter remains the better source when precise apogee data is required.
Worked example
Suppose the stations are 300 m apart. Station 1 records an azimuth of 85°, an elevation of 40°, and a sight height of 1.5 m. Station 2 records 55°, 39°, and 1.5 m. The azimuth sum is 140°.
The law of sines gives horizontal ranges of 382.31 m from station 1 and 464.94 m from station 2. The corrected altitude estimates are 322.30 m and 378.00 m, so the vertical midpoint is 350.15 m. Closure is 7.95%, which is within a 10% tolerance. The point beneath the rocket is about 33.32 m along the baseline and 380.86 m perpendicular to it from station 1.
Common questions
How do two tracking stations calculate model-rocket altitude?
The two stations and the ground point below the rocket form a triangle. The law of sines finds each horizontal range from the measured baseline and azimuth angles, then each range is multiplied by the tangent of its elevation angle. The usual vertical-midpoint result averages the two corrected altitude estimates.
How should the two azimuth angles be measured?
At station 1, measure the interior angle from the baseline line pointing toward station 2 to the rocket. At station 2, measure the interior angle from its baseline line pointing toward station 1 to the rocket. The two angles must add to less than 180 degrees to form the required ground triangle.
What does tracking closure error mean?
Closure error is the difference between the two station altitude estimates divided by their sum, expressed as a percentage. It is a quality check on the observations, not an uncertainty interval. High closure commonly points to timing, angle-reading, baseline, datum, or geometry problems.
Why do the two trackers produce different altitudes?
No two visual measurements are perfectly simultaneous or perfectly precise. Small errors in azimuth or elevation, a baseline measured on slope, different sight-height datums, and pointing at a different instant all produce different calculated altitudes. Closure makes that disagreement visible instead of hiding it in an average.
How far apart should model-rocket tracking stations be?
There is no single correct distance because the best baseline depends on expected altitude, the field, and the accuracy of the trackers. Use a safely separated, carefully measured baseline that gives a well-shaped ground triangle rather than azimuth angles close to zero or a sum close to 180 degrees. A field test with known landmarks can help establish a practical setup.
Does tracker eye height need to be added?
Yes, when the desired answer is altitude above the shared ground datum. Add each tracker or theodolite sight height to that station’s range-times-tangent result. The two heights must reference the same datum; on uneven ground, a surveyed station-elevation difference is also needed.