Generative Radial Geometry (GRG)
A discrete–exact grammar for curves and surfaces using turns, chords, and radial projections. π is not imported as a magic constant; it emerges from dyadic steps and converges in the classical limit.
Turns-first definitions
We discretize the unit turn; dyadic refinement keeps operations exact on the mesh and drives accuracy.
For a circle of radius \(R\), the inscribed polygon at \(N\) turns has:
Interactive Lab
Switch presets and adjust the dyadic exponent \(b\). For the star case \(r(\theta)=1+a\cos(k\theta)\), the exact area is \(A=\pi\!\left(1+\frac{a^{2}}{2}\right)\).
Notes. Circle & atlas use exact finite-step formulas (chords) and converge monotonically. Star-shape uses inscribed polygons; C-shape uses odd–even radial sweep for area (perimeter shown exact for reference).
Critical FAQ (condensed)
Q: Is GRG just numerical integration with a trick?
A: For circles with dyadic bisections steps are exact at finite \(N\) (chords & projections). The limit recovers classical values; nothing is imported.
Q: Where does \(\pi\) come from?
A: From the discrete perimeter/area growth of dyadic polygons. \(\pi\) appears as a rotational-symmetry shadow as \(N\to\infty\).
Q: Non-convex shapes “jump across the outside”.
A: We prevent that with visibility-aware schemes: (i) odd–even radial sweep; (ii) radial atlas (locally star-shaped patches with ± signs for outer components/holes).
Q: Convergence?
A: \(O(N^{-2})\) for \(C^{2}\) boundaries; \(O(N^{-1})\) near corners (mitigated via adaptive dyadics). Inscribed perimeter/area are monotone.
Q: Why is GRG pedagogically useful?
A: You can teach Euclidean geometry from bisections and chords; radians & trig tables emerge rather than being prerequisites.
Technical Appendix — Non-Convex & Complex Parametrizations
- Star-Check. If a center sees the whole boundary (star-shaped), sample \(\theta\) in dyadic turns and use \(r(\theta)\). Inscribed area/perimeter are monotone \(\uparrow\) with \(O(N^{-2})\).
- Odd–Even radial sweep. For each \(\theta_{k}\), intersect the ray with the boundary, sort radii \(r_{k,j}\), and add inside segments \([r_{k,1},\, r_{k,2}],\ [r_{k,3},\, r_{k,4}], \dots\).
- Radial atlas (± signs). Decompose into locally star-shaped patches; outer components \(+\), holes \(−\). Sum areas with signs and perimeters by absolute length. Same \(O(N^{-2})\) patch-wise.
- Robustness. \(C^{2} \Rightarrow O(N^{-2})\); corners \(\Rightarrow O(N^{-1})\) unless adaptively refined. Self-intersections: use winding rule instead of parity.
All computations use turns, chords, projections; operations are exact at finite dyadic steps for circular primitives and exact on the mesh for polygons.
Team
Roles are presented by responsibility area. Individual bios can be added later.
Leadership & Research
- Sets the research agenda and publishes specifications with reproducible examples.
- Maintains the “turns-first” glossary and consistency across demos and notes.
- Approves public statements and abstracts before release.
Methods & Algorithms
- Designs visibility-aware procedures (odd–even sweep, radial atlas with ± signs).
- Owns convergence studies and baselines for circles, ellipses, and polygons.
- Runs complexity/accuracy benchmarks; documents limits and mitigations.
Systems & Frontend
- Implements interactive canvases; keeps demos accessible and performant.
- Maintains build tooling and test fixtures for numerical correctness.
- Coordinates localization and analytics (non-invasive, anonymized).
Editorial Review & QA
- Checks clarity, tone, and scope; enforces the evidence-led style.
- Validates formulas and captions (MathJax + SVG semantics and accessibility).
- Controls versioning of figures; ensures non-repetitive hero imagery.
Collaborations
We welcome collaborations with educators and researchers. A public contact form will be added once finalized.