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
$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$
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
$$
\Delta x \;=\; \frac{r}{N}, \qquad
x_k \;=\; k\,\Delta x \ \ (k=0,\dots,N), \qquad
y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}.
$$
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.
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
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.
GRG — Looking from the Inside: a discrete, π-free language for shapes
Generative Radial Geometry (GRG) — Looking from the Inside
GRG conventions. Angles are measured in turns \(u\in[0,1)\).
Radial computations (perimeter and radial area) run on the full turn;
the Cartesian area uses the quarter (and then ×4).
Trigonometric forms are optional cross-checks, not inputs to the pipeline.
Work-in-progress. GRG is a discrete, π-free way to measure and compare shapes by radial generation. The turning point was this: I looked at the circle from its inside—a point that sends rays outward and measures what they meet. That inside→out switch is small to say and large to practice. The door is open; we will walk through it.
Why GRG matters
Discrete by nature: everything runs in finite steps (N)—heights, chords, bands.
π-free computation: the pipeline never needs π to run; π appears only when we compare to classical values.
Immediate comparability: two shapes at the same resolution are directly comparable with the same grammar.
Multi-scale: increase N; the very same procedure refines itself (coarse → fine).
GPU-friendly: short loops and sums, ideal for digital geometry, CAD/graphics, and simulation.
Looking from the inside
Not from a grid of squares, not from external coordinates—from a center that emits rays. A shape becomes a radial profile (a finite vector of values) rather than a formula to solve. From there, measurement and comparison happen discretely and immediately.
The joined method (circle)
Perimeter (radial, full turn): split the full turn \(u\in[0,1)\) into \(4N\) equal steps \(\Delta u=1/(4N)\).
Join successive GRG border nodes by chords; the quarter view in the figure is for clarity.
Area (Cartesian, quarter): split one radius into \(N\) parts; at each split, drop the perpendicular to the circle (the “height”).
Adjacent heights form a trapezoid; summing them tiles the quarter of a disk (the last one degenerates to a triangle). Multiply by 4 for the full disk.
Figure — joined method (circle)
Blue = angular sectors (perimeter, shown on the quarter). Red = single-radius trapezoids (area on the quarter). Dashed red rays start from the height intersections and generally do not coincide with the blue rays.
Figure 1. Quarter circle built from angular sectors (blue) and single-radius trapezoids (red); dashed rays start at height intersections.
Discrete comparability (the practical power)
At the same resolution N, any two shapes share the same grammar (heights, chords). This makes them comparable immediately—π-free inside the computation. We use simple, discrete indices relative to a baseline circle at the same N:
Area Index \(a_N\)
Perimeter Index \(c_N\)
Isoperimetric-like ratio \(q_N=a_N/c_N^2\) (the circle maps to 1 at the same N)
Minimal math — turns-first (π-free, node-based)
Quarter nodes (turns):
\[
\Delta u \;=\; \frac{1}{4N}
\]
\[
u_k \;=\; k\,\Delta u \quad (k=0,\dots,N)
\]
Cartesian area (quarter, Δx) — same as above, then ×4.
Figure — 3D comparator (hemisphere bands)
Clean comparator for surfaces: baseline hemisphere on the left, target surface on the right. Blue = GRG parallels on each latitude; red = bands (average of neighboring circumferences). Below, π-free indices (schematic bars).
Figure 2. Hemisphere bands comparator: left = baseline, right = target. Blue parallels indicate perimeter along latitude; red band shows area via average neighboring circumferences times slant. Bars below sketch π-free indices.
From the circle to the sphere (concept)
We stack latitude bands along the vertical axis and apply the same grammar on each parallel (GRG circumference on the band’s ends, averaged across the slant). Errors shrink fast with balanced vertical and angular splits. The comparator above is designed to show differences band-by-band without clutter.
Applications
Graphics/CAD: π-free arc lengths and areas that run fast and parallel.
Sampling/tiling on spheres: controllable bands, equal-area approximations.
Gear-like and elliptical profiles via directional radial generation.
Education: a concrete bridge from discrete geometry to classical values.
Status & invitation
GRG is young and grows with this site. The aim is simple: a practical, comparable, discrete language for shapes and surfaces. If the idea helps, let’s refine proofs, diagrams, and applications together.
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.
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 $A_{\text{circle}}(2N)$.
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.
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).
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 $s(u)=\sin(2\pi u)$.
$$ C_N \;=\; 2\cdot N \cdot r \cdot s\!\Bigl(\tfrac{1}{2N}\Bigr). $$
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
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.
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 — 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.
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.
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