GRG Trapezoidal

GRG Trapezoidal

Sep 6, 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.

Δx Trapezoidal

This GRG block computes the circle’s area from the inside using uniform $\Delta x=r/N$ on the quarter. On the graph $y(x)=\sqrt{r^{2}-x^{2}}$, adjacent samples form trapezoids strictly under the arc; multiply by $4$ for the full circle. No angles are required and no external constants appear in the primary formulas.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$
Quarter circle with Δx trapezoids (N=5) Axes, quarter arc of radius 100, and five Δx strips forming interior trapezoids strictly under the arc. x y
Fig. 1 — Δx trapezoids strictly inside the quarter arc (N=5).

Quarter-area sum (→ ×4 for the circle)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl( y_k + y_{k+1} \bigr), \qquad A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N). $$
Single Δx trapezoid between x_k and x_{k+1} One strip showing the trapezoid under the quarter arc; bracket indicates Δx. x_k x_{k+1} y_k y_{k+1} Δx
Fig. 2 — One Δx trapezoid; the dashed arc shows the exact curve above it.

Why use this block

  • Monotone under-approximation (trapezoids are strictly inside the quarter arc).
  • Smooth convergence $\sim O(N^{-2})$ and numerically stable (simple sums and square roots).
  • No angle handling; easy to implement with fixed-point or vectorized code.

How to use

  • Choose $N$ (e.g., $200$–$1000$) to balance speed and accuracy.
  • Step-doubling check: compare results at $N$ and $2N$ to gauge the remaining gap.
  • Optional diagnostic: plot the trapezoids on the quarter to see the under-tiling.

Notes

  • The last strip degenerates to a triangle since $y_N=0$ at $x_N=r$.
  • Perimeter pairs naturally with equal-angle chords; see the Core Turns block for the chord formula.

See also: Core Turns · Original Projection · Applications / Circle

Δx Trapezoidal

Area from the inside: sample the quarter with uniform $\Delta x = r/N$, use trapezoids under the arc on $y(x)=\sqrt{r^{2}-x^{2}}$, then multiply by $4$ for the full circle. No angles required.

Formula box — Quarter setup

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

Formula box — Quarter area (→ ×4)

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

Properties

  • Monotone under-approximation: each trapezoid lies strictly inside the quarter arc.
  • Convergence $\sim O(N^{-2})$. Numerically stable: sums and square roots only.
  • The last strip degenerates to a triangle because $y_N=0$ at $x_N=r$.

How to use

  • Choose $N$ (e.g., $200$–$1000$) based on speed/accuracy needs.
  • Step-doubling check: compare results at $N$ and $2N$ to estimate the remaining gap.
  • Optional: plot the trapezoids on the quarter to visualize the under-tiling.
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

ERG – Trapezoidal

ERG – Trapezoidal

Aug 24, 2025

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).
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.