Call function with kind, stock, ambient, mask, normal, and three points.

Test point in ABC after projection into the authored face chart.

Return the nearest chart-projected point of ABC and its squared distance.

When chart projects inside ABC, retain the original authored-normal ray intersection exactly. Otherwise the nearest point of the closed projected triangle lies on one of its three edges; lift that edge parameter back onto the actual three-dimensional primitive.

Test point in the authored face's inclusive one-cell footprint.

Classify one admissible candidate against the current geometric optimum.

Resolve face chart coordinates U/V against finished meshes.

U and V are normalized logical-face coordinates in [-1,1]. The square chart is compacted radially onto |U|+|V|<=1: points already inside that diamond are unchanged, while logical corners map continuously onto the realized junction domain instead of casting through empty space beyond a chamfered corner.

For every facing finished triangle, the resolver finds the closest point in the authored face chart and lifts it onto that actual primitive. Candidates are ranked first by minimum tangential distance and then by outermost normal displacement. Thus any triangle under the old authored-normal ray has zero tangential distance and produces the exact old result; only a genuine ray miss moves tangentially onto the nearest finished face, band, transition triangle, or junction fan.

The finished surface of the authored support cell cannot lie outward of its cubical face or more than the mesh's maximum bevel width inward or tangentially away from the mapped chart point. It must also stay inside the exact one-cell support-face footprint. Enforcing that slab, radius, footprint, and outward facing cone is an ownership condition, not a global nearest-hit heuristic: parallel or neighboring surfaces elsewhere must never steal the attachment.

At a non-smooth point there is no unique differential normal. Rather than falling back to the cubical face, this function returns the normalized sum of the distinct tied primitive normals: a deterministic bisector of the actual surface normal cone.