PHALIN THAKKAR
Fitting a direction without searching every angle
How does the double angle identity turn the fit into a linear algebra problem?
Note 9 · Research extension
Analytical result · Hypothetical examples are not physical measurements.
Write:
where:
For known analyzer angles, form one row of a design matrix as [1, cos2a, sin2a]. Estimate A, C and S with linear least squares. With known unequal independent uncertainties, use weighted least squares. Use a stable QR or SVD implementation.
Recover:
Illustrative readings at 0°, 45°, 90° and 135° are 83, 63, 23 and 43 units. They give A = 53, C = 30, S = 10, R ≈ 31.6228 and φ ≈ 9.2175°.
Under the assumed fully linear input plus constant background model, I0 = 2R ≈ 63.2456 and b = A − R ≈ 21.3772.
That last interpretation depends on the model. Unpolarized light and detector background both contribute to the constant term. The fit alone cannot uniquely assign that term to one cause. A dark calibration adds independent information.
If R = 0, no linear axis is identified. Do not use the numerical behaviour of atan2 at the origin as an inference.