Three-site shadow model comparison
Enter three northward site distances and signed zenith angles on the same date. Fit one spherical curvature and one shared local-Sun geometry to the same observations.
The sphere fits angles linear in meridian distance; the local-Sun model fits arctan((distance − Sun position)/height). Each fits two parameters to three readings. With independent angle uncertainty 0.1°, χ² compares the remaining mismatches. Calibration or shared timing error needs a separate correction.
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Sphere: z = a + x / R; local Sun: z = atan((x − s) / H)
Take the measurements at each location’s local solar noon on the same date, using sites close to one meridian. Set distance zero at one chosen origin and measure northward separation along the surface. The signed angle is positive when the Sun lies south of the zenith and negative when it lies north. A shadow-to-upright ratio gives the angle’s magnitude with atan.
The sphere predicts equal angular increments for equal meridian distances when the distant Sun’s rays are parallel. Its intercept places the subsolar latitude; its slope is inverse radius. The local-Sun model uses the same measured baselines as straight distances on a level plane and fits one height and one northward position.
Two readings allow both models to fit their two parameters. A third reading tests their different angle-versus-distance curves. Symmetric sites can still produce indistinguishable fits; add more non-symmetric sites and retain the raw measurements. χ² uses independent equal standard uncertainties, with one remaining degree of freedom. These sample defaults are constructed angles, not a reported field survey.