GRG Trapezoidal

GRG Trapezoidal

Sep 6, 2025

Earlier Formulation

This page preserves a historical GRG method. Its notation and formulas belong to that version. The current binary radial formulation and verification are available here. Historical computational claims require separate validation.

Δx Trapezoidal

This GRG block computes the circle’s area from the inside using uniform $\Delta x=r/N$ on the quarter. On the graph $y(x)=\sqrt{r^{2}-x^{2}}$, adjacent samples form trapezoids strictly under the arc; multiply by $4$ for the full circle. No angles are required and no external constants appear in the primary formulas.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$
Quarter circle with Δx trapezoids (N=5) Axes, quarter arc of radius 100, and five Δx strips forming interior trapezoids strictly under the arc. x y
Fig. 1 — Δx trapezoids strictly inside the quarter arc (N=5).

Quarter-area sum (→ ×4 for the circle)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl( y_k + y_{k+1} \bigr), \qquad A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N). $$
Single Δx trapezoid between x_k and x_{k+1} One strip showing the trapezoid under the quarter arc; bracket indicates Δx. x_k x_{k+1} y_k y_{k+1} Δx
Fig. 2 — One Δx trapezoid; the dashed arc shows the exact curve above it.

Why use this block

  • Monotone under-approximation (trapezoids are strictly inside the quarter arc).
  • Smooth convergence $\sim O(N^{-2})$ and numerically stable (simple sums and square roots).
  • No angle handling; easy to implement with fixed-point or vectorized code.

How to use

  • Choose $N$ (e.g., $200$–$1000$) to balance speed and accuracy.
  • Step-doubling check: compare results at $N$ and $2N$ to gauge the remaining gap.
  • Optional diagnostic: plot the trapezoids on the quarter to see the under-tiling.

Notes

  • The last strip degenerates to a triangle since $y_N=0$ at $x_N=r$.
  • Perimeter pairs naturally with equal-angle chords; see the Core Turns block for the chord formula.

See also: Core Turns · Original Projection · Applications / Circle

Δx Trapezoidal

Area from the inside: sample the quarter with uniform $\Delta x = r/N$, use trapezoids under the arc on $y(x)=\sqrt{r^{2}-x^{2}}$, then multiply by $4$ for the full circle. No angles required.

Formula box — Quarter setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \ \ (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$

Formula box — Quarter area (→ ×4)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl[\,y_k + y_{k+1}\,\bigr] $$
$$ A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N) $$

Properties

  • Monotone under-approximation: each trapezoid lies strictly inside the quarter arc.
  • Convergence $\sim O(N^{-2})$. Numerically stable: sums and square roots only.
  • The last strip degenerates to a triangle because $y_N=0$ at $x_N=r$.

How to use

  • Choose $N$ (e.g., $200$–$1000$) based on speed/accuracy needs.
  • Step-doubling check: compare results at $N$ and $2N$ to estimate the remaining gap.
  • Optional: plot the trapezoids on the quarter to visualize the under-tiling.
GRG Core (Turns)

GRG Core (Turns)

Aug 24, 2025

Earlier Formulation

This page preserves a historical GRG method. Its notation and formulas belong to that version. The current binary radial formulation and verification are available here. Historical computational claims require separate validation.

Core Turns

Angles are measured in turns (fractions of a full rotation). Primary 2D formulas use products only and avoid sexagesimal units and external constants. We write c(u), s(u) for cosine/sine with period 1 in turns (so $c(0)=1$, $s(0)=0$; $c(\tfrac{1}{4})=0$, $s(\tfrac{1}{4})=1$). Equivalently, $c(u)=\cos(2\pi u)$ and $s(u)=\sin(2\pi u)$.

Radial point

$$ P(u) \;=\; \bigl( r\cdot c(u),\; r\cdot s(u) \bigr). $$

Perimeter (equal-angle chords)

$$ C_N \;=\; 2\cdot N \cdot r \cdot s\!\Bigl(\tfrac{1}{2N}\Bigr). $$

Area (equal-angle sectors)

$$ A_N \;=\; \tfrac{1}{2}\cdot N \cdot r^{2} \cdot s\!\Bigl(\tfrac{1}{N}\Bigr). $$

How to use

  • Pick a precision $N$ (e.g., $256$). If you store angles as $k/2^{B}$ turns, choose $N$ that divides $2^{B}$.
  • Dyadic refinement: with $b\in\mathbb{N}_{\ge 0}$, set $N=2^{b}$ so the turn step is $\Delta u=2^{-b}$. Quarter sampling uses $\Delta u=\tfrac{1}{4N}=2^{-b-2}$.
  • Optional check (step-doubling): compare $N$ vs. $2N$ to gauge the remaining gap.

Notes

  • Both $C_N$ and $A_N$ increase with $N$ and converge to the circle values $2\pi r$ and $\pi r^2$; the error typically scales like $O\!\bigl((\Delta u)^2\bigr)=O(N^{-2})$.
  • This block pairs with the Trapezoidal and Original blocks on the Applications pages.

Figure: turns, chord, and sector

Unit circle with radial point P(u), equal-angle step Δu=1/N, chord and sector Circle of radius r with angle u in turns; two rays separated by Δu illustrate the chord s(1/(2N)) and the associated sector s(1/N). u Δu = 1/N P(u) chord O
Radial point $P(u)$, equal-angle step $\Delta u=1/N$, chord $s(\tfrac{1}{2N})$, and sector $s(\tfrac{1}{N})$.

See also

Δx Trapezoidal · Original Projection · Applications / Circle · Applications / Sphere

Formalisation (2025 snapshot). GRG is fully angular: in 2D circumference via equal-angle Δu chords, and area via equal-angle Δu sectors (products only); in 3D, surface/volume are angular sums over meridian bands. No sexagesimal units, no π.

Generative Radial Geometry (GRG) — Formalisation (Turns)

GRG is a constructive, discrete framework based on full-turn angular partitions (turns). All angles are fractions of a full rotation $u\in[0,1)$. We use period-1 trigonometric functions $c(u), s(u)$ with $c(0)=1$, $s(0)=0$, $(c(\tfrac14), s(\tfrac14))=(0,1)$, and standard identities. Primary formulas contain no sexagesimal units and no external constants.

1) Setup and Notation (full turn)

  • Spaces. Plane $\mathbb{R}^2$ and space $\mathbb{R}^3$ with axes $x,y,z$.

  • Circle/Sphere. Center $O$, radius $r>0$. Circle: $x^2+y^2=r^2$. Sphere: $x^2+y^2+z^2=r^2$.

  • Angles. $u\in[0,1)$ turns. Turns-trigonometry: $c(u), s(u)$ with unit identities.

Fig. A — Radial construction over the full turn Axes, circle, and two radii separated by angle u in turns; point P(u) on the circle. u R
Fig. A — Angle $u$ (turns). Radial point $P(u)=(r\,c(u),\,r\,s(u))$.

2) Exact radial nodes (full turn)

For any $u\in[0,1)$:

Radial node

$$ P(u)\;=\;\bigl(r\,c(u),\; r\,s(u)\bigr). $$
Evenly-spaced angles: $u_k=k/N$ over the full turn.

3) 2D operators (products, no sums)

2D Circumference (inscribed regular N-gon)

$$ \boxed{\, C_N \;=\; 2\,N\,r\,s\!\Bigl(\tfrac{1}{2N}\Bigr) \,} $$
Sum of $N$ identical chords → product.

2D Area (N equal-angle sectors)

$$ \boxed{\, A_N \;=\; \tfrac{N}{2}\,r^{2}\,s\!\Bigl(\tfrac{1}{N}\Bigr) \,} $$
Sum of $N$ identical sectors → product.

4) Sphere: meridian bands over a half turn (angular-only)

Meridian parameterization (angles only)

$$ \Delta v=\tfrac{1}{2M}, \qquad v_i=i\,\Delta v, \qquad \rho_i=r\,s(v_i). $$

4.1 Surface (angular sum over bands)

Meridian step (constant) & parallel circumference

$$ h=2\,r\,s\!\Bigl(\tfrac{\Delta v}{2}\Bigr)=2\,r\,s\!\Bigl(\tfrac{1}{4M}\Bigr), \qquad \mathrm{C}_{\mathrm{GRG}}(\rho;L)=2\,L\,\rho\,s\!\Bigl(\tfrac{1}{2L}\Bigr). $$

Surface via trapezoids (bands)

$$ S_{M,L}=h\left[ \tfrac12\,\mathrm{C}_{\mathrm{GRG}}(\rho_0;L) + \sum_{i=1}^{M-1}\mathrm{C}_{\mathrm{GRG}}(\rho_i;L) + \tfrac12\,\mathrm{C}_{\mathrm{GRG}}(\rho_M;L) \right]. $$

Surface — midpoints (often more accurate)

$$ S_{M,L}\;\approx\; h\sum_{i=0}^{M-1}\mathrm{C}_{\mathrm{GRG}}\!\bigl(\rho_{i+\tfrac12};L\bigr), \qquad \rho_{i+\tfrac12}=r\,s\!\bigl(v_i+\tfrac{\Delta v}{2}\bigr). $$
Fig. D — Meridian bands: parallels front/mid/back Three parallels drawn as ellipses to suggest a sphere; axes and reference labels included. v_i v_{i+1}
Fig. D — Meridian bands over a half turn; 2D angular operators on each parallel.

4.2 Volume (angular sum over sections)

Axial thickness & section area (angular)

$$ \Delta z_i = r\,\big|\,c(v_{i+1})-c(v_i)\,\big|, \qquad \mathrm{A}_{\mathrm{GRG}}(\rho;N)=\tfrac{N}{2}\,\rho^2\,s\!\Bigl(\tfrac{1}{N}\Bigr). $$

Volume via trapezoids (sections)

$$ V_{M,N}\;\approx\;\sum_{i=0}^{M-1} \frac{\Delta z_i}{2}\Bigl(\mathrm{A}_{\mathrm{GRG}}(\rho_i;N)+\mathrm{A}_{\mathrm{GRG}}(\rho_{i+1};N)\Bigr). $$

Volume — midpoints

$$ V_{M,N}\;\approx\;\sum_{i=0}^{M-1}\mathrm{A}_{\mathrm{GRG}}\!\bigl(\rho_{i+\tfrac12};N\bigr)\,\Delta z_i. $$

5) Compact “no-π” operators (summary)

  • 2D circumference: $\displaystyle \mathrm{C}_{\mathrm{GRG}}(r; N)=2\,N\,r\,s\!\bigl(\tfrac{1}{2N}\bigr)$.

  • 2D area: $\displaystyle \mathrm{A}_{\mathrm{GRG}}(r; N)=\tfrac{N}{2}\,r^2\,s\!\bigl(\tfrac{1}{N}\bigr)$.

Dyadic refinement (turns-first): $b\in\mathbb{N}_{\ge 0}$, $N=2^{b}$, and $\Delta u=2^{-b}$ (quarter sampling uses $\Delta u = \tfrac{1}{4N}=2^{-b-2}$).

Core Turns

Angles are measured in turns (fractions of a full rotation, u∈[0,1)). We write c(u) and s(u) for cosine/sine with period 1 in turns (so c(0)=1, s(0)=0; c(1/4)=0, s(1/4)=1). This block gives 2D, product-only formulas without sexagesimal units or external constants.

Radial point

$$ P(u) \;=\; \bigl( r\,c(u),\; r\,s(u) \bigr) $$

Perimeter (equal-angle chords)

$$ C_N \;=\; 2\,N\,r\, s\!\Bigl(\tfrac{1}{2N}\Bigr) $$

Area (equal-angle sectors)

$$ A_N \;=\; \tfrac{1}{2}\, N\, r^{2}\, s\!\Bigl(\tfrac{1}{N}\Bigr) $$

How to use

  • Choose a precision N (e.g., 256). If you store angles as k/2^B turns, pick N dividing 2^B.
  • For a quick accuracy check, compare results at N and 2N (step-doubling).

Notes

  • C_N and A_N increase with N and converge to the circle’s values.
  • This is one GRG method block. See also the Δx Trapezoidal and Original Projection blocks.

See also: Δx Trapezoidal · Original Projection · Applications / Circle · Applications / Sphere

Note: 3D (surface/volume) lives under Applications / Sphere. This page focuses on the 2D Core Turns block.