A sundog is geometry doing optics.

Every bright feature has a name.

When you photograph a sundog you don't see one phenomenon; you see a stack of them. The bright spots at the sun's left and right are the parhelia. The white ring around the sun is the 22° halo. The colorful arc near the top of the sky is the circumzenithal arc. The vertical light shaft is the sun pillar. Each is its own geometric primitive, but they do not carry the same evidence weight.

22° halo

Tight ring around the sun. Most commonly seen alone.

46° halo

Larger faint ring, roughly twice the 22° halo's radius.

Sundog / parhelion

Bright spot on the parhelic circle at the sun's altitude, just outside the 22° halo.

Parhelic circle

Horizontal great circle through the sun. Looks roughly horizontal in a photo.

Circumzenithal arc (CZA)

Colorful "smile" near the zenith, tangent to the 46° halo at its top.

Supralateral arc

Less common arc tangent to the 46° halo, just outside the CZA.

Upper tangent arc

"Eyelid" tangent to the 22° halo's top, from column-oriented crystals.

Suncave Parry arc

Parry-orientation cousin of the upper tangent, nested inside it.

Lower tangent arc

Mirror of upper tangent at the 22° halo's bottom edge.

Why do the bright spots sit at 22° from the sun?

Sundogs form when sunlight refracts through hexagonal ice crystals: tiny six-sided columns or plates falling through high cirrus clouds. A light ray entering one face of a hexagonal column and exiting an adjacent face traverses a 60° prism. The minimum-deviation angle of that prism for visible light in ice is almost exactly 22°. So most refracted light bunches up at 22° from the source. To an observer, that's a bright spot 22° away from the sun.

Machine-checked core: the public Lean proof now checks the calculus part of that sentence: the symmetric ray is stationary for the prism deviation function. The proof names, but does not prove, the physical optics wall. Lean method ledger.

When the sun rises, the parhelia slide outward.

The 22° angle is measured along the shortest path on the celestial sphere: the great-circle distance from the sun. But the parhelion sits on the parhelic circle, which is horizontal through the sun. Those two are the same thing only when the sun is on the horizon. When the sun rises, the parhelic-circle azimuthal offset has to grow to keep the actual angle from the sun at 22°. The exact relationship is

parhelion offset = R₂₂ / cos(h)

where R₂₂ is the apparent 22° halo radius and h is the sun's altitude. This is one of those rare cases where you can read the sun's altitude directly from a photograph, with no metadata and no instruments: just measure where the parhelion sits, divide.

This is also how a sundog photograph can become a measuring instrument. For the plain step-by-step inverse, start with the h-of-x math demo. The 7-photo calibration pass shipped with this project (see §Multi-Photo Calibration Pass) recovers the sun's altitude from each photograph with no EXIF or astronomical metadata, within roughly 1 pixel of residual.

Current evidence keeps this narrow. The promoted inverse handle is parhelion offset on eligible photos with bilateral parhelia and an independent 22° halo anchor; the CZA, supralateral, tangent-family, parhelic-belt, and pillar layers are rendered context, vocabulary, or coverage-gated routes unless their own receipts say otherwise.

Every visible arc, mapped to a circle.

Atmospheric optics has spent two centuries naming the arcs in a halo display. The twelve primitives below are the set this atlas renders. Some are scale or context; some are optional vocabulary; one route, parhelion offset, is promoted as the current inverse measurement handle. One paragraph each: what it is, where it sits, how the atlas constructs it, and what status it carries.

For the broader vocabulary ledger — including named-only, not-modeled, and speculative families — see the Sundog Halo Legend. The h-of-x page isolates the first promoted inverse equation; the atlas shows the current rendered geometry; the legend is the definition layer that separates rendered primitives, optional labels, and not-yet-modeled halo names.

Boundary landmarks: tangent arcs merge into the circumscribed halo near 29° sun elevation; the CZA leaves the visible hemisphere at about 32.2°; parhelic-belt y and the stylized pillar generator remain visual QA, not hidden-state evidence.

Named in the literature, beyond the rendered set

The set above is a deliberate subset. The standard atmospheric-optics literature names several more halo families this atlas does not draw as separate primitives — some are rare, some need a crystal-population model the geometry layer does not carry, and some require viewing geometry outside this explainer's audience. They are catalogued here so the canonical vocabulary is complete in one place: formation mechanism in a sentence, where to read more (Tape chapter · HaloSim simulation file · Cowley's Atmospheric Optics), and this project's rendering status.

Run the atlas on your own sundog photograph.

Upload a sundog photo, mark the sun and the 22° halo edge, and the atlas will overlay every visible arc — and tell you the sun's altitude when the photograph was taken.

The atlas render runs entirely in your browser. EXIF metadata is stripped before any opt-in sharing. For the backend pipeline and privacy policy, see PHASE5_BACKEND_PLAN.md · PHOTO_DATA_POLICY.md. Command-line equivalent: scripts/overlay_calibrate.py.

People have been measuring sundogs since 1535.

Vädersoltavlan (Sun Dog Painting) — Stockholm, 1535. One of the earliest detailed depictions of a complex halo display, stored in Storkyrkan, Stockholm.

The phenomenon has been observed for centuries, but rigorous geometric description waited for the 19th and 20th-century atmospheric-optics literature. The atlas math on this page is standard refraction geometry; what's specific to this project is the integrated primitive-atlas presentation and the inverse-inference workflow.

Further reading:

• Greenler, R. Rainbows, Halos, and Glories. Cambridge University Press (1980, reissued 1990). The canonical reference for refraction geometry in hexagonal ice crystals.
• Tape, W. Atmospheric Halos. American Geophysical Union, Antarctic Research Series Vol. 64 (1994). Detailed arc geometry including supralateral/infralateral symmetry and altitude bindings.
• Cowley, L. Atmospheric Optics (atoptics.co.uk). The standard public reference site with HaloSim ray-trace renderings.
• Sun dog on Wikipedia.

Reproducible math, citable references.

The atlas is built so other tools, papers, and Wikipedia entries can cite the math reproducibly. Every formula has a source; the canonical pose is a JSON file; the calibration evidence is a runnable script.