The Circle
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.
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.
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.