GRG Core (Turns)

GRG Core (Turns)

Aug 24, 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.

Core Turns

Angles are measured in turns (fractions of a full rotation). Primary 2D formulas use products only and avoid sexagesimal units and external constants. We write c(u), s(u) for cosine/sine with period 1 in turns (so $c(0)=1$, $s(0)=0$; $c(\tfrac{1}{4})=0$, $s(\tfrac{1}{4})=1$). Equivalently, $c(u)=\cos(2\pi u)$ and $s(u)=\sin(2\pi u)$.

Radial point

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

Perimeter (equal-angle chords)

$$ C_N \;=\; 2\cdot N \cdot r \cdot s\!\Bigl(\tfrac{1}{2N}\Bigr). $$

Area (equal-angle sectors)

$$ A_N \;=\; \tfrac{1}{2}\cdot N \cdot r^{2} \cdot s\!\Bigl(\tfrac{1}{N}\Bigr). $$

How to use

  • Pick a precision $N$ (e.g., $256$). If you store angles as $k/2^{B}$ turns, choose $N$ that divides $2^{B}$.
  • Dyadic refinement: with $b\in\mathbb{N}_{\ge 0}$, set $N=2^{b}$ so the turn step is $\Delta u=2^{-b}$. Quarter sampling uses $\Delta u=\tfrac{1}{4N}=2^{-b-2}$.
  • Optional check (step-doubling): compare $N$ vs. $2N$ to gauge the remaining gap.

Notes

  • Both $C_N$ and $A_N$ increase with $N$ and converge to the circle values $2\pi r$ and $\pi r^2$; the error typically scales like $O\!\bigl((\Delta u)^2\bigr)=O(N^{-2})$.
  • This block pairs with the Trapezoidal and Original blocks on the Applications pages.

Figure: turns, chord, and sector

Unit circle with radial point P(u), equal-angle step Δu=1/N, chord and sector Circle of radius r with angle u in turns; two rays separated by Δu illustrate the chord s(1/(2N)) and the associated sector s(1/N). u Δu = 1/N P(u) chord O
Radial point $P(u)$, equal-angle step $\Delta u=1/N$, chord $s(\tfrac{1}{2N})$, and sector $s(\tfrac{1}{N})$.

See also

Δx Trapezoidal · Original Projection · Applications / Circle · Applications / Sphere

Formalisation (2025 snapshot). GRG is fully angular: in 2D circumference via equal-angle Δu chords, and area via equal-angle Δu sectors (products only); in 3D, surface/volume are angular sums over meridian bands. No sexagesimal units, no π.

Generative Radial Geometry (GRG) — Formalisation (Turns)

GRG is a constructive, discrete framework based on full-turn angular partitions (turns). All angles are fractions of a full rotation $u\in[0,1)$. We use period-1 trigonometric functions $c(u), s(u)$ with $c(0)=1$, $s(0)=0$, $(c(\tfrac14), s(\tfrac14))=(0,1)$, and standard identities. Primary formulas contain no sexagesimal units and no external constants.

1) Setup and Notation (full turn)

  • Spaces. Plane $\mathbb{R}^2$ and space $\mathbb{R}^3$ with axes $x,y,z$.

  • Circle/Sphere. Center $O$, radius $r>0$. Circle: $x^2+y^2=r^2$. Sphere: $x^2+y^2+z^2=r^2$.

  • Angles. $u\in[0,1)$ turns. Turns-trigonometry: $c(u), s(u)$ with unit identities.

Fig. A — Radial construction over the full turn Axes, circle, and two radii separated by angle u in turns; point P(u) on the circle. u R
Fig. A — Angle $u$ (turns). Radial point $P(u)=(r\,c(u),\,r\,s(u))$.

2) Exact radial nodes (full turn)

For any $u\in[0,1)$:

Radial node

$$ P(u)\;=\;\bigl(r\,c(u),\; r\,s(u)\bigr). $$
Evenly-spaced angles: $u_k=k/N$ over the full turn.

3) 2D operators (products, no sums)

2D Circumference (inscribed regular N-gon)

$$ \boxed{\, C_N \;=\; 2\,N\,r\,s\!\Bigl(\tfrac{1}{2N}\Bigr) \,} $$
Sum of $N$ identical chords → product.

2D Area (N equal-angle sectors)

$$ \boxed{\, A_N \;=\; \tfrac{N}{2}\,r^{2}\,s\!\Bigl(\tfrac{1}{N}\Bigr) \,} $$
Sum of $N$ identical sectors → product.

4) Sphere: meridian bands over a half turn (angular-only)

Meridian parameterization (angles only)

$$ \Delta v=\tfrac{1}{2M}, \qquad v_i=i\,\Delta v, \qquad \rho_i=r\,s(v_i). $$

4.1 Surface (angular sum over bands)

Meridian step (constant) & parallel circumference

$$ h=2\,r\,s\!\Bigl(\tfrac{\Delta v}{2}\Bigr)=2\,r\,s\!\Bigl(\tfrac{1}{4M}\Bigr), \qquad \mathrm{C}_{\mathrm{GRG}}(\rho;L)=2\,L\,\rho\,s\!\Bigl(\tfrac{1}{2L}\Bigr). $$

Surface via trapezoids (bands)

$$ S_{M,L}=h\left[ \tfrac12\,\mathrm{C}_{\mathrm{GRG}}(\rho_0;L) + \sum_{i=1}^{M-1}\mathrm{C}_{\mathrm{GRG}}(\rho_i;L) + \tfrac12\,\mathrm{C}_{\mathrm{GRG}}(\rho_M;L) \right]. $$

Surface — midpoints (often more accurate)

$$ S_{M,L}\;\approx\; h\sum_{i=0}^{M-1}\mathrm{C}_{\mathrm{GRG}}\!\bigl(\rho_{i+\tfrac12};L\bigr), \qquad \rho_{i+\tfrac12}=r\,s\!\bigl(v_i+\tfrac{\Delta v}{2}\bigr). $$
Fig. D — Meridian bands: parallels front/mid/back Three parallels drawn as ellipses to suggest a sphere; axes and reference labels included. v_i v_{i+1}
Fig. D — Meridian bands over a half turn; 2D angular operators on each parallel.

4.2 Volume (angular sum over sections)

Axial thickness & section area (angular)

$$ \Delta z_i = r\,\big|\,c(v_{i+1})-c(v_i)\,\big|, \qquad \mathrm{A}_{\mathrm{GRG}}(\rho;N)=\tfrac{N}{2}\,\rho^2\,s\!\Bigl(\tfrac{1}{N}\Bigr). $$

Volume via trapezoids (sections)

$$ V_{M,N}\;\approx\;\sum_{i=0}^{M-1} \frac{\Delta z_i}{2}\Bigl(\mathrm{A}_{\mathrm{GRG}}(\rho_i;N)+\mathrm{A}_{\mathrm{GRG}}(\rho_{i+1};N)\Bigr). $$

Volume — midpoints

$$ V_{M,N}\;\approx\;\sum_{i=0}^{M-1}\mathrm{A}_{\mathrm{GRG}}\!\bigl(\rho_{i+\tfrac12};N\bigr)\,\Delta z_i. $$

5) Compact “no-π” operators (summary)

  • 2D circumference: $\displaystyle \mathrm{C}_{\mathrm{GRG}}(r; N)=2\,N\,r\,s\!\bigl(\tfrac{1}{2N}\bigr)$.

  • 2D area: $\displaystyle \mathrm{A}_{\mathrm{GRG}}(r; N)=\tfrac{N}{2}\,r^2\,s\!\bigl(\tfrac{1}{N}\bigr)$.

Dyadic refinement (turns-first): $b\in\mathbb{N}_{\ge 0}$, $N=2^{b}$, and $\Delta u=2^{-b}$ (quarter sampling uses $\Delta u = \tfrac{1}{4N}=2^{-b-2}$).

Core Turns

Angles are measured in turns (fractions of a full rotation, u∈[0,1)). We write c(u) and s(u) for cosine/sine with period 1 in turns (so c(0)=1, s(0)=0; c(1/4)=0, s(1/4)=1). This block gives 2D, product-only formulas without sexagesimal units or external constants.

Radial point

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

Perimeter (equal-angle chords)

$$ C_N \;=\; 2\,N\,r\, s\!\Bigl(\tfrac{1}{2N}\Bigr) $$

Area (equal-angle sectors)

$$ A_N \;=\; \tfrac{1}{2}\, N\, r^{2}\, s\!\Bigl(\tfrac{1}{N}\Bigr) $$

How to use

  • Choose a precision N (e.g., 256). If you store angles as k/2^B turns, pick N dividing 2^B.
  • For a quick accuracy check, compare results at N and 2N (step-doubling).

Notes

  • C_N and A_N increase with N and converge to the circle’s values.
  • This is one GRG method block. See also the Δx Trapezoidal and Original Projection blocks.

See also: Δx Trapezoidal · Original Projection · Applications / Circle · Applications / Sphere

Note: 3D (surface/volume) lives under Applications / Sphere. This page focuses on the 2D Core Turns block.

Circumference & Sphere

Circumference & Sphere

Aug 23, 2025

Why “Certainly on the Circumference” (and on the Sphere)

Category: Debate

Most ways to “draw a circle” are approximations: they split the arc into chords, use regular polygons, or compute points from arc lengths (where \(\pi\) eventually appears). In GRG we do the opposite: we do not measure the arc, we enforce the distance. We fix a radius \(R\) and choose a geometric direction; the generated point is the endpoint of a segment of length \(R\) from a center \(O\). By definition, that point is at distance \(R\) from \(O\): therefore it is certainly on the circumference. No approximation, no \(\pi\) needed.

What we prove, in plain words

In 2D, every point built as \((x,y)=(R\cos\varphi,\,R\sin\varphi)\) identically satisfies

$$x^{2}+y^{2}=R^{2}.$$

It does not “tend” to the circle: it is on the circle.

In 3D, take a unit direction \(\mathbf{u}\) and set \(\mathbf{P}=R\,\mathbf{u}\). By construction \(\|\mathbf{P}\|=R\): the point is certainly on the sphere

$$x^{2}+y^{2}+z^{2}=R^{2}.$$

The heart of it is: distance first. If the distance from the center is enforced to be \(R\), every generated point belongs to the locus of points at distance \(R\). That is the very definition of circle/sphere.

“Then what are sine and cosine for?”

Here they are not approximate numbers: they are ratios of sides in similar right triangles. They serve as a language to express the Cartesian projections of the chosen direction: \[ x=R\cos\varphi,\qquad y=R\sin\varphi. \] Even if, in a digital controller, you compute them with finite precision, the geometric law you implement remains exact: the commanded points are those of the ideal object defined by \(x^{2}+y^{2}=R^{2}\) (or \(x^{2}+y^{2}+z^{2}=R^{2}\)).

If you wish to avoid direct computation of \(\sin\) and \(\cos\) altogether, there is an equivalent incremental form: rotate the vector \((x,y)\) by multiplying it with an orthogonal matrix. Orthogonal matrices preserve the norm: every step remains on the circle by construction.

“What about error accumulating at each step?”

Distinguish carefully:

  • Geometric law (GRG): exact. The norm is preserved at each step (rotations), therefore there is no geometric drift.
  • Physical implementation: it can introduce errors (backlash, compliance, friction, encoder resolution, numerical rounding). These are real-world errors, not flaws of the idea.

In digital control, the small loss/gain of norm due to rounding is removed with three simple techniques:

\(\text{(i) adequate precision}\quad \text{(ii) occasional renormalization enforcing }x^{2}+y^{2}=R^{2}\quad \text{(iii) periodic recomputation from }\varphi.\)

In all cases, the reference object remains the perfect circle/sphere.

“But the machine moves in straight lines, not curves!”

What you often see is the visualization (a polyline between samples). It is not the definition of the path. The GRG law commands instantaneous poses that lie on the circle/sphere. The drive’s interpolation between samples may be linear or spline-based: that is a control choice. If you want a “fuller” trajectory, increase the sampling rate or use an arc interpolator; in any case, the nodes remain exactly on the geometric locus.

Why \(\pi\) is not needed

\(\pi\) is needed when you measure the arc (lengths, areas, times along an arc). In GRG we do not measure the arc: we enforce the distance from the center and the direction. Therefore \(\pi\) is irrelevant to guarantee that points lie on a circle/sphere. You may use \(\pi\) later if you care about arc length, but not to construct the point.

Natural extension to the sphere (including the \(Z\) axis)

What holds in \(X\!-\!Y\) holds in \(X\!-\!Y\!-\!Z\). A unit direction in 3D (obtained via Cartesian projections, without “walking on the surface”) is scaled by \(R\). The result has norm \(R\). That is all. Azimuth–elevation formulas are just a convenient way to obtain \((x,y,z)\). The key property remains the preservation of distance.

Practical consequences for robotics, CNC, and 3D printers

If your machine moves on Cartesian axes and you command \((x,y)\) (or \((x,y,z)\)) according to the GRG law, the ideal path is exactly the circumference (or the sphere).

Any deviations you might measure on a real part come from mechanics and numerics: backlash, compliance, calibration, friction, quantization. Improve the machine, the controller, and the metrology, and you approach the ideal without changing the GRG law.

Typical objections, short answers

  • “Without \(\pi\) I cannot make a circle.” False: a circle is the locus of points at distance \(R\) from a center. No \(\pi\) is needed to guarantee that distance.
  • “Sine and cosine are approximate numeric functions.” In geometry they are exact ratios; in control you can use incremental orthogonal rotations that preserve the norm by construction.
  • “Steps accumulate error.” Rotations do not change the norm. In digital control, rounding errors are contained with renormalization or periodic recomputation.
  • “Points on the sphere are not uniformly distributed.” Different topic: sampling uniformity does not affect the fact that each point lies on the sphere.

The key takeaway

GRG does not approximate the circle or the sphere: it imposes them. With a radius \(R\) fixed and a direction chosen, the distance from the center is \(R\) by construction: therefore the point is certainly on the circumference (or on the sphere). In a Cartesian system, the geometric precision of the trajectory does not depend on \(\pi\) nor on arc approximations: it depends only on your machine and numerics.

Certainly on the Circumference — and the Sphere

Category: Rigor of GRG

GRG conventions. Angles in turns \(u\in[0,1)\) with step \(\Delta u=1/N\). Radial computations (perimeter & radial area) on the full turn; Cartesian area on the quarter (×4). Trig appears only as classical cross-check.

Certainly on the Circumference

  • Circle \(R\): \(x^2+y^2=R^2\).

  • Direction (turns): \(u\in[0,1)\) → radial pose (no arc lengths).

  • GRG border: certificate \(x(u)^2+y(u)^2=R^2\) (Pythagoras on quarter or tangent intersections on turn).

Classical (optional): with \(\varphi=2\pi u\): \(x=R\cos\varphi,\ y=R\sin\varphi\).

Theorem

Every GRG point \(P(u)\) satisfies \(x(u)^2+y(u)^2=R^2\) ⇒ it lies on the circle.

Fig. — Radial construction / orthogonal update preserves norm.

Certainly on the Sphere

  • Sphere \(R\): \(x^2+y^2+z^2=R^2\).

  • Unit direction: build \(\mathbf{u}\) constructively; \(\mathbf{P}=R\,\mathbf{u}\) certifies \(\|\mathbf{P}\|=R\).

Classical (optional): \((\varphi,\theta)\) → \(x=R\cos\varphi\,\sin\theta,\ y=R\sin\varphi\,\sin\theta,\ z=R\cos\theta\).

Theorem

\(\|R\,\mathbf{u}\|^2=R^2\|\mathbf{u}\|^2=R^2\) ⇒ point on the sphere.

Fig. — 3D orthogonal rotations preserve the norm.

GRG · Operative summary (turns & quarter)

Params. Pick \(N\). Full turn: \(u_j=j/N\) with \(\Delta u=1/N\). Generate \(P_j=(x_j,y_j)\) with \(x_j^2+y_j^2=R^2\). Quarter: \(\Delta x=R/N\).

Perimeter — full turn (radial)

Inscribed (lower)

\[ L_{\mathrm{in}}(N)=\sum_{j=0}^{N-1}\sqrt{(x_{j+1}-x_j)^2+(y_{j+1}-y_j)^2}. \]

Circumscribed (upper) — tangents at \(P_j\): \(x\,x_j+y\,y_j=R^2\); intersections \(T_j\) with \(\det_j=x_jy_{j+1}-y_jx_{j+1}\), \(T_j^x=\frac{R^2(y_{j+1}-y_j)}{\det_j}\), \(T_j^y=\frac{R^2(x_j-x_{j+1})}{\det_j}\).

\[ L_{\mathrm{out}}(N)=\sum_{j=0}^{N-1}\lVert T_{j+1}-T_j\rVert. \]

Radial area — full turn

\[ A_{\mathrm{in}}(N)=\tfrac12\sum_{j=0}^{N-1}(x_jy_{j+1}-x_{j+1}y_j),\qquad A_{\mathrm{out}}(N)=\tfrac12\sum_{j=0}^{N-1}(T_j^xT_{j+1}^y-T_{j+1}^xT_j^y). \]

Cartesian area — quarter (Δx) → ×4

Abscissas \(x_i=i\Delta x\), \(\Delta x=R/N\); heights \(y_i=\sqrt{R^2-(i\Delta x)^2}\).

\[ A_{\mathrm{right,q}}=\sum_{i=1}^{N}y_i\Delta x,\quad A_{\mathrm{mid,q}}=\sum_{i=1}^{N}y\!\left(x_{i-\tfrac12}\right)\Delta x,\quad A_{\mathrm{trap,q}}=\sum_{i=1}^{N}\tfrac{y_{i-1}+y_i}{2}\Delta x. \]

Bounds: \(A_{\mathrm{right,q}}\le A_{\mathrm{quarter}}\le A_{\mathrm{mid,q}}\); disk via ×4. π remains outside the computation; trig only as cross-check.

GRG for ellipses

GRG for ellipses

Aug 22, 2025

GRG for Ellipses — Constructive Area, Perimeter, and Equal-Area Sectors

The “Variable Radius” in GRG

The effective radius depends on the chosen direction.

  • From the center (angle \(\phi\) from the major axis):

    $$ r_c(\phi)=\frac{ab}{\sqrt{(b\cos\phi)^2+(a\sin\phi)^2}}. $$

  • From the focus (true anomaly \(\theta\)):

    $$ r_f(\theta)=\frac{p}{1+e\cos\theta},\qquad e=\sqrt{1-\frac{b^2}{a^2}},\qquad p=\frac{b^2}{a}. $$

  • Without trigonometry (slope \(m=\tan\phi\)):

    $$ r_c(m)=\frac{ab\,\sqrt{1+m^2}}{\sqrt{\,b^2+a^2m^2\,}}. $$

Ellipse Area in GRG

A1) Cartesian strips in the first quadrant

Split the semi-axis \(a\) into \(N\) parts: \(x_i=i\,a/N\) with \(\Delta x=a/N\). Heights \(y_i=b\sqrt{1-x_i^2/a^2}\).

$$ A_{1/4}^{(N)}=\sum_{i=1}^{N} y_i\,\Delta x,\qquad A^{(N)}=4\,A_{1/4}^{(N)}. $$
A1 — Cartesian strips in the first quadrant x–y axes, ellipse centered at (400,250) with a=240, b=160; shaded vertical strip inside Q1. +y +x Cartesian strips in Q1; then ×4
Quarter-by-strips: sum \(y_i\,\Delta x\) in Q1 and multiply by 4 for the ellipse area.

A2) Radial wedges from the center (by quarters)

With \(\Delta\phi=90^\circ/N\) and \(\phi_i=i\,\Delta\phi\):

$$ A_{1/4}^{(N)}=\sum_{i=0}^{N-1}\tfrac12\,r_c(\phi_i)\,r_c(\phi_{i+1})\,\sin\Delta\phi, \qquad A^{(N)}=4\,A_{1/4}^{(N)}. $$
A2 — Center-based wedges (quarter symmetry) Rays from center to ellipse boundary; the wedge between two rays approximates an area piece. C +x +y Center-based wedges (quarter symmetry)
Sum center-based wedges over a quarter (then ×4). Summing the chords between successive boundary points gives the corresponding perimeter.

A3) Radial wedges from the focus (natural for orbits)

With \(\Delta\theta=360^\circ/M\) and \(\theta_i=i\,\Delta\theta\):

$$ A^{(M)}=\sum_{i=0}^{M-1}\tfrac12\,r_f(\theta_i)\,r_f(\theta_{i+1})\,\sin\Delta\theta. $$
A3 — Focus-based sectors (equal areas → equal times) Sectors with apex at a focus; each filled sector has the same area increment ΔA (Kepler-2). F F′ periapsis → Equal-area sectors from the focus (constant ΔA; angle widths vary)
Equal-area stepping from the focus: constant \(\Delta A\) gives uniform time steps (Kepler’s 2nd law). Near periapsis the angular gaps shrink; away they widen.

Perimeter (sum of chords)

P1) Chords from the center (by quarters)

With \(\Delta\phi=90^\circ/N\):

$$ P_{1/4}^{(N)}=\sum_{i=0}^{N-1}\sqrt{\,r_c(\phi_i)^2+r_c(\phi_{i+1})^2-2\,r_c(\phi_i)\,r_c(\phi_{i+1})\cos\Delta\phi\,}, \qquad P^{(N)}=4\,P_{1/4}^{(N)}. $$

P2) Chords from the focus (full ellipse)

$$ P^{(M)}=\sum_{i=0}^{M-1}\sqrt{\,r_f(\theta_i)^2+r_f(\theta_{i+1})^2-2\,r_f(\theta_i)\,r_f(\theta_{i+1})\cos\Delta\theta\,}. $$

GRG Error Estimate (after-the-fact)

With step doubling the error typically decays ~quadratically. Method-agnostic (π-free):

  • Compute \(S_N\) and \(S_{2N}\) with the same operator (A1–A3, P1–P2).

  • Accelerated approximation:

    $$ S_\star \approx S_{2N}+\frac{S_{2N}-S_N}{3}. $$

  • Error estimate:

    $$ \varepsilon \approx \frac{|S_{2N}-S_N|}{3}. $$

Practical Pipeline (geometry → orbits)

  • Geometry: use A1 or A2 for area and P1 for perimeter with quarter symmetry.

  • Orbits: use A3. For uniform time steps, enforce constant-area sectors:

    $$ \tfrac12\,r_f(\theta_k)\,r_f(\theta_{k+1})\,\sin(\theta_{k+1}-\theta_k)=\Delta A. $$

  • The arc length per step is the P2 chord between successive points.

Rigor of GRG

Rigor of GRG

Aug 22, 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.

Generative Radial Geometry (GRG)

A Constructive Alternative to Classical Circle Measurement

Introduction

Generative Radial Geometry (GRG) is a digital, constructive approach: it does not assume π in the build step. Circles are generated from the radius with finite operators (Pythagoras, chords, tangents); classical values (e.g. π) then emerge as limits or comparisons.

Two complementary builds

  • Quarter / Cartesian (Δx): work on one quarter of the disk, sum rectangles/trapezoids with \(\Delta x=R/N\), then ×4. Certified right/midpoint corridor.
  • Full turn / Radial (Δu): work on the full loop in turns \(u\in[0,1)\), step \(Δu=1/U\); equal-angle chords/tangents give in/out polygons (inscribed/circumscribed).

Notation in turns: \(c(u)=\cos(2\pi u),\ s(u)=\sin(2\pi u)\). π appears only when comparing to classical formulas.

Quarter coordinates (Δx, no trig in the step)

On the quarter, sample \(x_k=k\,\Delta x\) with \(\Delta x=R/N\) and heights by Pythagoras:

$$ y_k=\sqrt{R^2-x_k^2},\qquad k=0,\dots,N. $$

Area — Cartesian quarter (GRG)

Δx Trapezoidal (lower)

$$ A_{\text{quarter}}^{\text{trap}}(R,N)=\sum_{k=1}^{N}\frac{y_{k-1}+y_k}{2}\,\Delta x,\quad A_{\text{GRG}}^{\text{trap}}=4\,A_{\text{quarter}}^{\text{trap}}. $$

Projection corridor (certified bounds)

$$ A_{\text{quarter}}^{\mathrm{right}}=\!\sum_{k=1}^{N} y_k\,\Delta x,\qquad A_{\text{quarter}}^{\mathrm{mid}}=\!\sum_{k=1}^{N} y\!\left(x_{k-\tfrac12}\right)\Delta x, $$
$$ A_{\text{quarter}}^{\mathrm{right}}\le A_{\text{quarter}}\le A_{\text{quarter}}^{\mathrm{mid}},\qquad 4A_{\text{quarter}}^{\mathrm{right}}\le A_{\text{GRG}}\le 4A_{\text{quarter}}^{\mathrm{mid}}. $$

Circumference — Radial full turn (GRG)

With \(U\) equal sectors per full turn (step \(Δu=1/U\)), all chords are equal. The inscribed polygon length is:

$$ C_U \;=\; 2\,U\,R\,s\!\Big(\tfrac{1}{2U}\Big) \;=\; 2\,U\,R\,\sin\!\Big(\tfrac{\pi}{U}\Big)\quad\text{(classical cross-check)}. $$

On a quarter with \(N\) steps set \(U=4N\) and get \(C=8NR\,s\!\big(\tfrac{1}{8N}\big)\).

Radial area — in/out corridor on the full turn

Inscribed polygon area (lower, shoelace)

Let \(P_j=(x_j,y_j)\) be border points in CCW order, \(j=0,\dots,U-1\) and \(P_U=P_0\) with \(x_j^2+y_j^2=R^2\).

$$ A_{\text{in}}(R,U)=\tfrac12\sum_{j=0}^{U-1}\bigl(x_jy_{j+1}-x_{j+1}y_j\bigr)\ \le\ A. $$

Circumscribed polygon area (upper, tangents)

The tangent at \(P_j\) is \(x\,x_j+y\,y_j=R^2\). Intersecting tangents at \(P_j,P_{j+1}\) yields \(T_j\):

$$ \det_j=x_jy_{j+1}-y_jx_{j+1},\qquad T_j=\Bigl(\frac{R^2\,(y_{j+1}-y_j)}{\det_j},\ \frac{R^2\,(x_j-x_{j+1})}{\det_j}\Bigr), $$
$$ A_{\text{out}}(R,U)=\tfrac12\sum_{j=0}^{U-1}\bigl(T_j^xT_{j+1}^y-T_{j+1}^xT_j^y\bigr)\ \ge\ A. $$

Hence \(A_{\text{in}}\le A \le A_{\text{out}}\) and the corridor collapses as \(U\) increases.

Answering common objections

“This is nothing new”

GRG constructs perimeter and area from the radius alone in two certified ways: quarter/Δx (right–mid corridor) and full-turn/Δu (inscribed–circumscribed corridor).

“It is only an approximation”

Accuracy is controlled by \(N\) or \(U\); both builds provide bounding pairs that converge.

“It lacks rigor”

Only algebraic steps (Pythagoras, chords, tangents, polygon sums). π is used solely for external comparison, not as an input.

“It is less practical”

Finite sums with elementary operations are ideal for digital computation and come with a transparent error model.

Conclusion

GRG complements classical geometry with a generative language. On the quarter, Δx gives fast areas with bounds; on the full turn, equal-angle chords/tangents give circumference and radial area—letting π emerge as a limit.

Circle with inscribed (lower) and circumscribed (upper) polygons; arrows suggest convergence. A_in A_out U ↑ ⇒ A_in ↑, A_out ↓ inscribed (lower) circumscribed (upper)
ERG Cartesian

ERG Cartesian

Aug 20, 2025

ERG Cartesian — Quarter Chords to Sphere (GRG)

From a quarter construction we derive circumference, sphere surface and volume using discrete, angle-free operators. All figures below are drawn in a strict Cartesian frame (linear x–z axes, equal scale).

Quarter circumference (equal-angle chords)

For a radius $\rho$, divide the $90^\circ$ arc into $N$ equal angles. Replace each arc step with a chord. The discrete GRG quarter circumference is:

$$ C_{1/4}^{\mathrm{GRG}}(\rho;N)=2\,N\,\rho\,\sin\!\Bigl(\tfrac{45^\circ}{N}\Bigr) \;=\;2\,N\,\rho\,s\!\Bigl(\tfrac{1}{8N}\Bigr). $$
Quarter arc with equal-angle chords — strict Cartesian Axes and a true quarter circle of radius 100; chords join consecutive points at 15° steps (N=6). Coordinates are (x=ρ cosθ, z=ρ sinθ). xz
Equal-angle chords on a true quarter circle (example $N=6$). Axes and lengths are Cartesian.

Quarter section → sphere surface (horizontal slices + quarter chords)

Slice the sphere horizontally into $N$ levels of thickness $\Delta z=\tfrac{r}{N}$. At level $k$:

$$ z_k=\frac{k}{N}\,r, \qquad \rho_k=\sqrt{\,r^2-z_k^2\,}, \qquad S_{1/8}(N)=\sum_{k=1}^{N} C_{1/4}^{\mathrm{GRG}}(\rho_k;N)\,\frac{r}{N},\quad S(N)=8\,S_{1/8}(N). $$
Unwrapped surface S1/8(N) — strict Cartesian Bars of constant height Δz with width proportional to ρ_k (and hence to C_{1/4}^{GRG}(ρ_k;N) for fixed N). Axes are linear. local width ∝ ρk z
Area under the bars equals $S_{1/8}(N)$ up to the constant factor $2N\,s(\tfrac{1}{8N})$ common to all strips for fixed $N$.

Quarter area and sphere volume (stack quarter areas)

Quarter area for radius $\rho$ (base split into $N$ parts; rectangles on circle heights), and octant volume via the same $\Delta z$:

$$ A_{1/4}^{\mathrm{GRG}}(\rho;N)=\sum_{i=1}^{N}\Bigl(\tfrac{\rho}{N}\Bigr)\sqrt{\,\rho^{2}-\Bigl(\tfrac{i\,\rho}{N}\Bigr)^{2}\,}, \qquad V_{1/8}(N)=\sum_{k=1}^{N} A_{1/4}^{\mathrm{GRG}}(\rho_k;N)\,\tfrac{r}{N},\quad V(N)=8\,V_{1/8}(N). $$
Octant volume from stacked quarter-disk slabs — strict Cartesian Quarter disks of radii ρ_k = sqrt(r^2 - z_k^2) stacked along z; each slab has thickness Δz. x z
Stacking quarter-disk slabs of radius $\rho_k$ (Cartesian x–z plane). The sum of their quarter areas times $\Delta z$ gives $V_{1/8}(N)$.

ERG — Original Projection (Cartesian)

$$ \Delta x=\frac{r}{N},\quad x_k=k\,\Delta x,\quad x_{k+\tfrac12}=\Bigl(k+\tfrac12\Bigr)\Delta x,\quad y(x)=\sqrt{\,r^{2}-x^{2}\,}. $$
$$ A_{\text{quarter, mid}}(N)=\sum_{k=0}^{N-1} y\!\bigl(x_{k+\tfrac12}\bigr)\,\Delta x,\qquad A_{\text{quarter, right}}(N)=\sum_{k=0}^{N-1} y\!\bigl(x_{k+1}\bigr)\,\Delta x. $$

Below: true Cartesian rectangles for $r=100$, $N=5$ (so $\Delta x=20$). Heights are exactly $y=\sqrt{r^2-x^2}$ at the mid/right samples.

Quarter — midpoint (upper) vs right-endpoint (lower), strict Cartesian Quarter circle r=100. Δx=20. Green bars at x_mid = (k+1/2)Δx with exact heights sqrt(r^2 - x_mid^2). Orange outlines at right endpoints x=(k+1)Δx. x y
Midpoint (filled, upper) vs right-endpoint (outlined, lower) — heights computed from $y=\sqrt{r^2-x^2}$ with $r=100$, $\Delta x=20$.
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.