The Circle

Aug 18, 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.

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: $C_N = 2N r\, s(1/(2N))$ (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-approximation on a concave arc; area error of order N^(-3/2) for this complete profile.

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 area error is of order N^(-3/2); the midpoint/right corridor width is of order N^(-1) and 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; multiply by 4 for the full circle.

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.