Sagitta¶
The sagitta is the perpendicular rise from a circular arc's chord midpoint to the arc, linking chord length to radius.
Core Idea¶
The sagitta is how high a circular arc rises above the middle of a chord joining its endpoints. In a circular segment it links chord length, rise, and circle radius through a right triangle.
Scope of Application¶
Geometers solve circle segments with it. In one thin-coating method, a circular wheel's known radius and the measured width of its cut determine the coating thickness as a sagitta; the source uses the small-rise approximation \(T\approx C^2/(8R)\). The inverse calculation can instead infer radius from an independently measured chord and rise.
Boundary: Arc length runs along the curve; sagitta is a perpendicular height from the chord.
Clarity¶
Measure the height perpendicularly at the chord midpoint and say whether the minor arc is intended.
Manages Complexity¶
A curved shape becomes a right-triangle calculation. The direction of inference matters: a very small measured rise makes an inferred radius sensitive to depth error, whereas the coating method infers a small rise from a known radius and measured width.
Abstract Reasoning¶
For radius \(r\), chord \(c\), and minor sagitta \(s\), \((r-s)^2+(c/2)^2=r^2\). Solve this relation for the unknown quantity.
Knowledge Transfer¶
The relation works for drawn circles and suitably circular physical sections, not automatically for any curved arch. Circular Arc is a strict prerequisite: the measure needs an arc and chord but is not a kind of arc.
For example, a radius-5 circle with a length-6 chord has a midpoint-to-arc rise of 1.
Relationships to Other Abstractions¶
Current abstraction Sagitta Domain-specific
Parents (1) — more general patterns this builds on
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Sagitta presupposes Circular arc Domain-specific
A sagitta is defined relative to a selected circular arc and its chord.
Hierarchy path (1) — routes to 1 parentless root
- Sagitta → Circular arc
Neighborhood in Abstraction Space¶
Sagitta sits in a sparse region of the domain-specific corpus (92nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Coordinate Systems & Spatial Measures (29 abstractions)
Nearest neighbors
- Circular arc — 0.84
- Central angle — 0.80
- Secant Line — 0.80
- Ordered geometry — 0.78
- Centered Polygonal Number — 0.77
Computed from structural-signature embeddings · 2026-10-08