Sextant Horizontal Angles

What is modelled

The angle subtended by a chord at a point of a circle is the same for every point on the same arc (inscribed-angle theorem). So a measured angle between two landmarks fixes a circle of position through them, and the second circle through the middle landmark gives the ship. A danger angle is the angle read on the circle that just touches a hazard.

Equations

• Circle radius R = |AB| / (2 sin alpha), centre at distance (|AB| / 2) cot alpha from the chord midpoint.
• The fix is the second intersection of two circles that share the landmark B: the mirror image of B in the line of centres.
• Error sigma_pos = sigma_angle sqrt(|dF/dalpha|^2 + |dF/dbeta|^2) from the Jacobian.
• Safe while the observed angle stays below the danger angle angle AHB.

Checks run in the page

• Inscribed angle on 200 random positions.
• Exact fixes recovered.
• 4000 noisy fixes against the Jacobian prediction.
• Danger circle sampled all round, inside and outside.
• The circle through all three marks, where no fix exists.

Honest simplifications

A flat chart with exact landmark positions, independent Gaussian errors on the two angles and no index error, dip or refraction; the real sextant optics are in the 3D page and the 2D double-reflection sim.

• Press Close to the coast: the fix tightens but the rock comes close to the track.
• Press Far offshore, poor angles to see error growth.
• Watch the weakness readout rise near the circle through all three marks.