Circular Triangle¶
A simple plane figure bounded by three cyclic circular arcs, with corner angles measured between the arcs' tangents.
Core Idea¶
A circular triangle is a simple plane figure enclosed by three circular arcs that meet in order at three vertices. Its corner angles are measured between the tangents of neighboring arcs, rather than between straight chords.[^ref-ef4d00660514] The arcs may have different radii and can bend inward or outward. Their joined boundary must remain simple: sides meet at their intended endpoints without crossing elsewhere.
A mathematical treatment calls this an arc-triangle and also allows straight segments as degenerate arcs.[^ref-ef4d00660514] This entry uses the ordinary curved sense. An all-straight triangle belongs only to the paper's wider limiting convention, which must be stated when using its theorems.
Scope of Application¶
The identity has literal uses within geometry:
- Plane arc geometry. Recognize simple closed figures with three circular-arc sides.[^ref-ef4d00660514]
- Constant-width geometry. The Reuleaux triangle is a special three-arc case made from three equal disks; constant width follows from that added construction.[^ref-b1d3158abec3]
- Curved graph drawing. A suitable three-edge face in a planar Lombardi drawing has arc sides, possibly including straight segments under the source's explicit limiting convention.[^ref-ef4d00660514]
- Angle feasibility. Test whether a proposed triple of tangent-defined corner angles can occur in a simple arc-triangle.[^ref-ef4d00660514]
Across these settings, the three-side incidence and tangent-angle test stays the same; equal radii and constant width remain optional.
Clarity¶
Check the boundary in order: three vertices, three circle-arc sides, no unwanted crossing, and tangent-defined interior angles. A single circular arc is one constituent, not the whole figure. A polygon ordinarily has straight sides. Constant width is an additional property, not part of the circular-triangle definition.
Manages Complexity¶
The figure is recognized by a short set of constraints instead of by one preferred picture. Arc radii, centers, convexity, and corner angles can vary while the three-side structure remains. The distinction between the curved identity and the degenerate-segment convention prevents a straight triangle from being silently counted as a curved example.
Abstract Reasoning¶
To test whether a proposed tangent-angle triple can form a simple arc-triangle under the source's convention, let the angles be θ₁, θ₂, θ₃ in [0, 2π]. Compute ψ = (π − θ₁ − θ₂ − θ₃)/2, then φᵢ = ψ + θᵢ. The paper proves that a simple arc-triangle exists exactly when each φᵢ is strictly between −π and π.[^ref-ef4d00660514] This is an existence test for proposed angles; it is not a requirement to calculate these values every time a circular triangle is recognized.
Knowledge Transfer¶
The test travels literally between geometric constructions and curved-edge graph drawings because both contain plane circle arcs and tangent-defined corners. A three-part cycle in another field may suggest a comparison, but it is not a circular triangle without the geometric carrier. A general rule about local constraints and global closure would need separate evidence before becoming a broader abstraction.
Relationships to Other Abstractions¶
Current abstraction Circular Triangle Domain-specific
Parents (1) — more general patterns this builds on
-
Circular Triangle is part of Circular arc Domain-specific
A circular triangle contains a circular arc as an identity-bearing side of its simple three-sided boundary.
Hierarchy path (1) — routes to 1 parentless root
- Circular Triangle → Circular arc
Neighborhood in Abstraction Space¶
Circular Triangle sits in a sparse region of the domain-specific corpus (88th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Polygon — 0.84
- RAC drawing — 0.81
- Sphericon — 0.79
- Arrangement of Pseudolines — 0.79
- Desargues's Theorem — 0.79
Computed from structural-signature embeddings · 2026-10-17
References¶
[^ref-ef4d00660514]: David Eppstein, Daniel Frishberg, and Martha C. Osegueda, “Angles of Arc-Polygons and Lombardi Drawings of Cacti,” Computational Geometry 112 (2023), 101982; author manuscript, especially Definition 1 and Definition 2 on PDF p. 3, Corollary 4 and Theorem 6 on PDF pp. 5–6, and Theorems 17–18 on PDF pp. 9–10. https://arxiv.org/pdf/2107.03615. [^ref-b1d3158abec3]: Ryan Hynd, “The Blaschke–Lebesgue Theorem Revisited,” arXiv v2 (2024), PDF pp. 1–2, intersection-of-three-equal-disks construction, Figure 2, and constant-width support-line argument. https://arxiv.org/pdf/2303.04359.