GRG Trapezoidal

Sep 6, 2025

Scientific correction — 13 October 2026

For the complete quarter-circle profile sampled uniformly along x, midpoint and trapezoidal area errors have order N^(-3/2), not N^(-2), because the endpoint derivative is singular. The right-endpoint error and midpoint/right corridor width have order N^(-1). The geometric upper/lower corridor remains valid in exact arithmetic. These rates concern the historical Cartesian rules, not the current weighted binary radial kernels.

Current reference: Circle and Area from Binary Radial Modules.

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).
  • Area error of order N^(-3/2) for this endpoint-singular profile; numerical rounding requires separate analysis.
  • 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.