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.