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

The Circle

The Circle

Aug 18, 2025

The Circle — Radius-First (Three Methods)

Principle

The circle is generated from the center. The radius is the only primitive; the perimeter is the path traced by the radius, and the area is the surface swept by its rotation. We work in turns (fractions of a full rotation), not in sexagesimal degrees.

Angles in turns & radial node

Let \(u\in[0,1)\) be an angle in turns (period \(1\)). We use period-1 trigonometric functions \(c(u), s(u)\) with \(c(0)=1, s(0)=0\) and \((c(\tfrac14), s(\tfrac14))=(0,1)\). The exact radial node is:

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

Bit-friendly turns

Use fixed-point angles \(u = k/2^B\). Picking \(N\) that divides \(2^B\) makes equal-angle steps exact (\(\Delta u=1/N\)).
Optional: LUT + interpolation for \(c(u), s(u)\).

Method scope & scaling (quick key)

  • Radial Sectors (equal-angle, full turn) → no ×4.
  • Δx Trapezoidal (quarter tiling) → ×4 to get the full circle.
  • Original Projection (midpoint/right on a quadrant) → ×4.
Perimeter (equal-angle chords) is naturally full-turn (no ×4) and shared across methods.

Perimeter (shared across methods)

Equal-angle chords (full turn)

$$ \boxed{\, C_N \;=\; 2\,N\,r\,s\!\Bigl(\tfrac{1}{2N}\Bigr) \,} $$
Inscribed regular \(N\)-gon in turns. Monotone ↑ in \(N\). No quadrant scaling.
Perimeter by equal-angle chords (Δu = 1/N turns) Cartesian axes; circle r=100; sample radii and dashed chords (example N=12).
Perimeter by equal-angle chords: \(C_N = 2N r\, s(1/(2N))\) (full turn).

Area — three interchangeable builds (pick one)

Δx Trapezoidal (quarter → ×4)

$$ \Delta x=\frac{r}{N},\quad x_k=k\,\Delta x,\quad y_k=\sqrt{r^2-x_k^2} $$
$$ A_{\text{quarter}}(N)=\frac{\Delta x}{2}\!\left(y_0+2\sum_{k=1}^{N-1}y_k+y_N\right),\qquad A(N)=4\,A_{\text{quarter}}(N). $$
Under-approx on a concave arc; smooth convergence \(O(N^{-2})\).

Original Projection (midpoint/right corridor, quarter → ×4)

$$ \Delta x=\frac{r}{N},\ \ x_{k+\tfrac12}=(k+\tfrac12)\Delta x,\ \ y(x)=\sqrt{r^2-x^2} $$
$$ A^{\text{quad}}_{\text{mid}}=\sum_{k=0}^{N-1} y\!\bigl(x_{k+\tfrac12}\bigr)\,\Delta x,\qquad A^{\text{quad}}_{\text{right}}=\sum_{k=0}^{N-1} y\!\bigl(x_{k+1}\bigr)\,\Delta x, $$
$$ A_{\text{mid/right}}=4\,A^{\text{quad}}_{\text{mid/right}},\qquad A_{\text{right}}\le A_{\text{true}}\le A_{\text{mid}}. $$
Midpoint is second-order; corridor shrinks with \(N\).

Radial Sectors (equal-angle sectors, full turn)

$$ \boxed{\, A_N = \tfrac{N}{2}\,r^2\,s\!\Bigl(\tfrac{1}{N}\Bigr) \,} $$
Sum of \(N\) congruent isosceles triangles with included angle \(1/N\) turns (inscribed lower; monotone ↑ in \(N\)). No ×4.
Quarter area by Δx trapezoids (under the arc) Cartesian x–y axes; quarter arc with Δx verticals and interior trapezoids. Δx trapezoids
Quarter area by Δx trapezoids; multiply by 4 for the full circle.
Radial sector tiling (equal-angle sectors, full turn) Two example sectors on a unit circle (illustrative N=8).
Radial sectors: \(A_N=\tfrac{N}{2}r^2\,s(1/N)\) (full turn, no ×4).

Practical notes

  • Pick \(N\). If angles are \(k/2^B\) turns, choose \(N\mid 2^B\). Use step-doubling (\(N\!\to\!2N\)) to gauge the gap.
  • Monotonicity. \(C_N\) ↑ with \(N\). Δx trapezoids ↑ toward area (under-tiling). Original corridor shrinks. Radial sectors ↑ (inscribed).
  • All in turns. No degrees, no π in the constructive steps.