Inside-Out GRG

Aug 24, 2025

GRG — Looking from the Inside: a discrete, π-free language for shapes

Generative Radial Geometry (GRG) — Looking from the Inside

GRG conventions. Angles are measured in turns \(u\in[0,1)\). Radial computations (perimeter and radial area) run on the full turn; the Cartesian area uses the quarter (and then ×4). Trigonometric forms are optional cross-checks, not inputs to the pipeline.

Work-in-progress. GRG is a discrete, π-free way to measure and compare shapes by radial generation. The turning point was this: I looked at the circle from its inside—a point that sends rays outward and measures what they meet. That inside→out switch is small to say and large to practice. The door is open; we will walk through it.

Why GRG matters

  • Discrete by nature: everything runs in finite steps (N)—heights, chords, bands.
  • π-free computation: the pipeline never needs π to run; π appears only when we compare to classical values.
  • Immediate comparability: two shapes at the same resolution are directly comparable with the same grammar.
  • Multi-scale: increase N; the very same procedure refines itself (coarse → fine).
  • GPU-friendly: short loops and sums, ideal for digital geometry, CAD/graphics, and simulation.

Looking from the inside

Not from a grid of squares, not from external coordinates—from a center that emits rays. A shape becomes a radial profile (a finite vector of values) rather than a formula to solve. From there, measurement and comparison happen discretely and immediately.

The joined method (circle)

Perimeter (radial, full turn): split the full turn \(u\in[0,1)\) into \(4N\) equal steps \(\Delta u=1/(4N)\). Join successive GRG border nodes by chords; the quarter view in the figure is for clarity.

Area (Cartesian, quarter): split one radius into \(N\) parts; at each split, drop the perpendicular to the circle (the “height”). Adjacent heights form a trapezoid; summing them tiles the quarter of a disk (the last one degenerates to a triangle). Multiply by 4 for the full disk.

Figure — joined method (circle)

Blue = angular sectors (perimeter, shown on the quarter). Red = single-radius trapezoids (area on the quarter). Dashed red rays start from the height intersections and generally do not coincide with the blue rays.

Figure 1. Quarter circle built from angular sectors (blue) and single-radius trapezoids (red); dashed rays start at height intersections.

Discrete comparability (the practical power)

At the same resolution N, any two shapes share the same grammar (heights, chords). This makes them comparable immediately—π-free inside the computation. We use simple, discrete indices relative to a baseline circle at the same N:

  • Area Index \(a_N\)
  • Perimeter Index \(c_N\)
  • Isoperimetric-like ratio \(q_N=a_N/c_N^2\) (the circle maps to 1 at the same N)
Minimal math — turns-first (π-free, node-based)

Quarter nodes (turns):

\[ \Delta u \;=\; \frac{1}{4N} \] \[ u_k \;=\; k\,\Delta u \quad (k=0,\dots,N) \]

Quarter inscribed perimeter (node distances):

\[ C_{\mathrm{in}}^{(Q)}(N) \;=\; \sum_{k=0}^{N-1} \bigl\|\,P_{k+1}-P_k\,\bigr\| \]

Full inscribed perimeter:

\[ C_{\mathrm{in}}(N) \;=\; 4\,C_{\mathrm{in}}^{(Q)}(N) \]

Radial area (full turn, shoelace on 4N nodes):

\[ A_{\mathrm{in}}(N) \;=\; \frac{1}{2} \sum_{j=0}^{4N-1} \bigl(x_j\,y_{j+1}-x_{j+1}\,y_j\bigr) \]

Cartesian area (quarter, Δx) → ×4:

\[ \Delta x \;=\; \frac{R}{N} \] \[ x_i \;=\; i\,\Delta x \] \[ y_i \;=\; \sqrt{\,R^2 - x_i^{\,2}\,} \] \[ A_{\mathrm{trap}}^{(Q)}(N) \;=\; \sum_{i=1}^{N} \frac{y_{i-1}+y_i}{2}\;\Delta x \] \[ A_{\mathrm{trap}}(N) \;=\; 4\,A_{\mathrm{trap}}^{(Q)}(N) \]

Discrete indices (baseline = circle at the same N):

\[ a_N \;=\; \frac{A_{\text{shape}}(N)}{A_{\text{circle}}(N)} \] \[ c_N \;=\; \frac{C_{\text{shape}}(N)}{C_{\text{circle}}(N)} \] \[ q_N \;=\; \frac{a_N}{c_N^{\,2}} \]
Classical cross-check (optional)

Inscribed perimeter (quarter step in turns):

\[ \Delta u \;=\; \frac{1}{4N} \] \[ s_N \;=\; 2R\,\sin\!\bigl(\pi\,\Delta u\bigr) \] \[ C_{\mathrm{in}}(N) \;=\; 4\,N\,s_N \;=\; 8\,N\,R\,\sin\!\Bigl(\frac{\pi}{4N}\Bigr) \]

Circumscribed perimeter (outer bound):

\[ C_{\mathrm{out}}(N) \;=\; 8\,N\,R\,\tan\!\Bigl(\frac{\pi}{4N}\Bigr) \]

Radial area via triangles (full turn):

\[ A_{\mathrm{tri}}(N) \;=\; 2\,N\,R^{2}\, \sin\!\Bigl(\frac{\pi}{2N}\Bigr) \]

Cartesian area (quarter, Δx) — same as above, then ×4.

Figure — 3D comparator (hemisphere bands)

Clean comparator for surfaces: baseline hemisphere on the left, target surface on the right. Blue = GRG parallels on each latitude; red = bands (average of neighboring circumferences). Below, π-free indices (schematic bars).

Figure 2. Hemisphere bands comparator: left = baseline, right = target. Blue parallels indicate perimeter along latitude; red band shows area via average neighboring circumferences times slant. Bars below sketch π-free indices.

From the circle to the sphere (concept)

We stack latitude bands along the vertical axis and apply the same grammar on each parallel (GRG circumference on the band’s ends, averaged across the slant). Errors shrink fast with balanced vertical and angular splits. The comparator above is designed to show differences band-by-band without clutter.

Applications

  • Graphics/CAD: π-free arc lengths and areas that run fast and parallel.
  • Sampling/tiling on spheres: controllable bands, equal-area approximations.
  • Gear-like and elliptical profiles via directional radial generation.
  • Education: a concrete bridge from discrete geometry to classical values.

Status & invitation

GRG is young and grows with this site. The aim is simple: a practical, comparable, discrete language for shapes and surfaces. If the idea helps, let’s refine proofs, diagrams, and applications together.