Arch Geometry API

Circular-segment arch geometry as an API, computed locally and deterministically — the radius, arc-length and set-out numbers a mason, joiner, stonemason or CAD user lays a segmental arch out with. A segmental arch is an arc of a circle struck through the two springings and the crown: the from-span-rise endpoint takes the span and the rise (the height of the crown above the springing line) and returns the radius = (span²/4 + rise²) ÷ (2·rise), the central angle it subtends, the arc length along the curve, and the segment area of the void below it — flatter arches with a small rise have surprisingly huge radii. The from-radius-angle endpoint inverts it, returning the chord (span), the rise (sagitta), the arc length and the area from a known radius and central angle, the way a curve struck with a trammel or a router on a pivot is described. The setout-ordinates endpoint gives the practical numbers to mark a template: the rise of the arc above a straight base line at equally spaced stations across the span (y = √(R² − x²) − (R − rise)), so you can plot the heights, connect them and cut a plywood former or bend a batten without a giant compass — the ends come out zero at the springings and the middle equals the rise at the crown. Everything is computed locally and deterministically, so it is instant and private. Ideal for masonry and joinery layout tools, stair and window-head design, and CAD and woodworking calculators. Pure local computation — no key, no third-party service, instant. Segmental (up to a semicircle) arcs. 3 compute endpoints. For road curves use a horizontal- or vertical-curve API; for plain shape areas a geometry API.