GRG Original Projection

GRG Original Projection

Sep 4, 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.

Original Projection — historical method This page presents the original GRG idea: the circle is generated from the inside. The circle area is computed on a quadrant by projecting heights onto the x-axis and using midpoint rectangles (discrete upper); a matching right-endpoint rule gives a discrete lower, forming an under/over corridor that shrinks as $N$ increases (the midpoint rule is second-order). The perimeter is obtained from the sum of equal-angle chords on the quadrant (then $\times 4$). This method remains alongside other GRG blocks and is kept intact for comparison and continued study. See also Core Turns and Δx Trapezoidal.

Original Projection (Quarter → ×4)

Work on the quarter $x\in[0,r]$ with the profile $y(x)=\sqrt{r^{2}-x^{2}}$. Use uniform $\Delta x=r/N$. Midpoint rectangles provide a discrete upper estimate; right-endpoint rectangles provide a discrete lower. Multiply by $4$ to obtain full-circle values.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \ (k=0,\dots,N), \qquad x_{k+\tfrac12} \;=\; \Bigl(k+\tfrac12\Bigr)\Delta x, \qquad y(x) \;=\; \sqrt{\,r^{2}-x^{2}\,}. $$

Quarter area: discrete corridor (upper vs lower)

Midpoint (discrete upper, second-order)

$$ A_{\text{quarter}}^{\text{mid}}(N) \;=\; \sum_{k=0}^{N-1} \Delta x \; y\!\bigl(x_{k+\tfrac12}\bigr), \qquad A_{\text{circle}}^{\text{mid}}(N) \;=\; 4\,A_{\text{quarter}}^{\text{mid}}(N). $$

Right-endpoint (discrete lower)

$$ A_{\text{quarter}}^{\text{right}}(N) \;=\; \sum_{k=0}^{N-1} \Delta x \; y\!\bigl(x_{k+1}\bigr), \qquad A_{\text{circle}}^{\text{right}}(N) \;=\; 4\,A_{\text{quarter}}^{\text{right}}(N). $$

Corridor: $A_{\text{circle}}^{\text{right}}(N) \;\le\; \text{Area(circle)} \;\le\; A_{\text{circle}}^{\text{mid}}(N)$, with gap shrinking like $O(N^{-2})$ under smoothness.

Perimeter via equal-angle chords (quadrant → ×4)

Partition the quarter by equal angular steps in turns. Let the quarter have $Q$ equal-angle chords of step $\Delta u=\tfrac{1}{4Q}$. The sum over the quarter is

$$ C_{\text{quarter}}(Q) \;=\; 2\,Q\,r\,s\!\Bigl(\tfrac{\Delta u}{2}\Bigr) \;=\; 2\,Q\,r\,s\!\Bigl(\tfrac{1}{8Q}\Bigr), \qquad C_{\text{circle}} \;\approx\; 4\,C_{\text{quarter}}(Q) \;=\; 8\,Q\,r\,s\!\Bigl(\tfrac{1}{8Q}\Bigr). $$

Equivalently, with $N=4Q$ steps on the full turn: $C_{N}=2\,N\,r\,s\!\bigl(\tfrac{1}{2N}\bigr)$ (see Core Turns).

Figure — Quarter with midpoint and right rectangles

Quarter: midpoint (upper) vs right-endpoint (lower) Quarter arc of a circle (r=100) with Δx strips. Green: midpoint rectangles; orange: right-endpoint rectangles. Both lie inside the quarter; midpoint sits above right. x y midpoint (upper) right (lower)
Midpoint vs right-endpoint corridor on the quarter.

Why use this block

  • Explicit under/over corridor with midpoint/right rules on the quarter.
  • Second-order convergence for the midpoint rule ($O(N^{-2})$) with simple operations.
  • No angle handling; complements chord-based perimeter via equal-angle steps.

How to use

  • Choose $N$ (e.g., $200$–$1000$) for the quarter; multiply results by $4$ for the circle.
  • Step-doubling check: compare $N$ vs $2N$ to estimate the residual gap between upper and lower.
  • Pair with equal-angle chords on the quarter to report both area and perimeter consistently.

See also

Core Turns · Δx Trapezoidal · Applications / Circle

Original Projection

The original GRG area method: sample the quarter along the x-axis with uniform step $\Delta x=r/N$. At each sample, project the height on the circle graph $y(x)=\sqrt{r^{2}-x^{2}}$ and build discrete rectangles. Two rules create a shrinking corridor: midpoint rectangles (upper) and right-endpoint rectangles (lower). Multiply by $4$ for the full circle. No angles are required in this block.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad x_{k+\tfrac12} \;=\; \Bigl(k+\tfrac12\Bigr)\Delta x, \qquad y(x) \;=\; \sqrt{\,r^{2}-x^{2}\,}. $$

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

$$ \textbf{Midpoint (upper):}\quad A_{\mathrm{mid}}^{\mathrm{q}}(N) \;=\; \sum_{k=0}^{N-1} y\!\bigl(x_{k+\tfrac12}\bigr)\,\Delta x $$
$$ \textbf{Right-end (lower):}\quad A_{\mathrm{right}}^{\mathrm{q}}(N) \;=\; \sum_{k=0}^{N-1} y\!\bigl(x_{k+1}\bigr)\,\Delta x $$
$$ \textbf{Circle totals:}\quad A_{\mathrm{mid}} \;=\; 4\,A_{\mathrm{mid}}^{\mathrm{q}}, \qquad A_{\mathrm{right}} \;=\; 4\,A_{\mathrm{right}}^{\mathrm{q}} $$
$$ \textbf{Corridor:}\quad A_{\mathrm{right}} \;\le\; A_{\mathrm{true}} \;\le\; A_{\mathrm{mid}} $$
$$ \textbf{Estimate (optional):}\quad A_{\mathrm{avg}} \;=\; \frac{A_{\mathrm{mid}}+A_{\mathrm{right}}}{2} \quad\text{(use half-gap as an error bar).} $$

Notes

  • $y(x)=\sqrt{r^{2}-x^{2}}$ is concave on $[0,r]$; hence midpoint gives an upper estimate and right-endpoint a lower one.
  • Increase $N$ to tighten the corridor; publish $A_{\mathrm{avg}}$ with its half-gap if you need an error bar.

See also: Δx Trapezoidal · Core Turns · Applications / Circle

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.