Scattered sunlight is linearly polarized. For a scattering angle γ between the sun direction and the line of sight, single-scattering (Rayleigh) theory gives:
Degree of polarization:
p(γ) = p_max · sin²(γ) / (1 + cos²(γ))
E-vector orientation (tangent to the sky, at point V):
e = normalize( S × V )
S = unit vector to the sun, V = unit view direction
The e-vector is always perpendicular to the plane containing the sun, the observer and the observed sky patch, and polarization peaks at γ=90° (a band arcing overhead, perpendicular to the solar meridian). This scene renders that field as small tick marks over the dome.
Dung beetles have a dedicated strip of polarization-sensitive ommatidia in the dorsal rim area of each eye. Before rolling a dung ball away from a competitive pile, a beetle climbs on top and performs a brief spinning "orientation dance," sampling this sky pattern (or, on moonless nights, the Milky Way's light gradient) to pick an arbitrary straight-line bearing -- not a route to any fixed goal. It then keeps rolling along that bearing by continuously comparing the sky pattern to its stored reference: this constant-bearing strategy is called menotaxis. If it is knocked off course, it re-dances and re-locks onto the same bearing.
- Sun elevation -- changes γ at the zenith, so it changes how strongly polarized the overhead band is (strongest when the sun is low).
- Cloud cover -- scales p_max down directly, hiding the pattern the way real overcast skies do.
- New bearing -- triggers a fresh dance and a new random target heading.
- Disturb -- randomly kicks the current heading, then the beetle re-dances and steers back using the compass; heading noise scales as 1 / (DoP + ε), so a hazy sky yields a wobblier, slower recovery and a curved path.