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 $r=100$, $\Delta x=20$.