A hatch's area was not the area of anything. Its boundary was flattened
into a ring by pushing an arc's *centre* into it, and a spline's control
points — neither of which is on the boundary — and the shoelace formula
was run over the result. Boundary edges now go through the kernel, which
measures each for what it encloses: an arc contributes its sector, a
spline is integrated.
AREA had three measurements between them. The bulged-polyline one
carried its own arc corrections; the planar entities it did not cover
fell through to mass properties. Both are now one call on the entity's
curve, which gets the bulges, the ellipse and the spline right in the
same place. What is left is the genuinely spatial — a 3D polyline, a
spline whose points wander off any plane — measured as rings of points
by Newell's method rather than by projecting them flat.
Regions were doing the same Newell sum by hand. They share it now.
Co-Authored-By: Claude Opus 5 <[email protected]>
