ERG – Trapezoidal

Aug 24, 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.

Document Status and Current Reference

This page preserves an earlier formulation or research method. Its notation, formulas, and embedded demonstrations belong to that version and should not be treated as the specification of the current binary radial kernels.

Read the current formulation and its analytical verification. Historical performance statements and implementation behaviour require separate validation. Earlier references to circular constants remain in this historical document; they are not inputs to the current geometric kernel definitions.

Δx Trapezoidal (Cartesian)

Area from the inside on the quarter of a circle. Sample uniformly on the x-axis with $\Delta x=r/N$, build trapezoids strictly under the arc $y(x)=\sqrt{r^2-x^2}$, then multiply by 4 for the full circle. No angles and no external constants appear in the primary operators.

Setup (quarter)

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

Quarter area (→ ×4 for the circle)

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

Because $y(x)$ is concave on $[0,r]$, these trapezoids give a monotone under-approximation. The error typically decays like $O(N^{-2})$. A quick check is step-doubling: compare $A_{\text{circle}}(N)$ vs $A_{\text{circle}}(2N)$.

Quarter with Δx trapezoids (Cartesian x–y) Strict Cartesian axes (equal units). r=100, N=5 ⇒ Δx=20. Trapezoids are inside the quarter arc. x y
Fig. 1 — Δx trapezoids strictly inside the quarter arc (example: $r=100$, $N=5$).

Single strip geometry

Between $x_k$ and $x_{k+1}$ the trapezoid has bases $y_k$ and $y_{k+1}$ and width $\Delta x$; the arc segment lies strictly above it.

One Δx trapezoid under the arc Strip between x=60 and x=80 for r=100. Heights are y_k=80 and y_{k+1}=60. Dashed arc segment shown above the trapezoid. x y
Fig. 2 — One Δx strip (here $k=3$): the arc is strictly above the trapezoid.

Pseudocode

Input: r, N
Δx = r / N
Aq = 0
for k = 0..N-1:
    yk   = sqrt(r*r - (k*Δx)*(k*Δx))
    yk1  = sqrt(r*r - ((k+1)*Δx)*((k+1)*Δx))
    Aq  += 0.5 * Δx * (yk + yk1)
A_circle = 4 * Aq

Notes

  • Monotone: $A_{\text{circle}}(N)$ increases with $N$ and approaches the true area from below.
  • Accuracy: smooth $O(N^{-2})$ decay; step-doubling is a simple diagnostic.
  • Implementation: works cleanly with fixed-point or vectorized code (just sums and square roots).