Project: Shape Budget
Experiment: Control Knob
Status: Verified numerically (floating-point precision)
Let two sources be placed symmetrically on the real axis at positions (\pm c).
Define:
- (a) — semimajor axis (half of the total reach budget (S = 2a))
- (c) — focal half-separation (half of source separation (d = 2c))
- (b) — semiminor axis (transverse residue)
- (e = c / a) — eccentricity, the control knob
- (S = 2a) — total reach budget
- (d = 2c) — source separation
In the BGP reading of this base case, (e = c / a) is the normalized allocation readout: the fraction of total reach budget already committed to structural separation before transverse residue remains.
Assume (0 \le c \le a), equivalently (0 \le e \le 1). The case (0 \le e < 1) gives a proper ellipse; (e = 1) is the degenerate line-segment limit.
The constant-sum constraint requires that for any point (P = (x, y)) on the locus:
[ r_1 + r_2 = 2a ]
where (r_1 = |P - (-c, 0)|) and (r_2 = |P - (c, 0)|).
Starting from the constant-sum constraint (r_1 + r_2 = 2a), let (r_1 = \sqrt{(x+c)^2 + y^2}) and (r_2 = \sqrt{(x-c)^2 + y^2}).
Step 1. Set (r_1 = 2a - r_2) and square:
[ (x+c)^2 + y^2 = 4a^2 - 4a r_2 + (x-c)^2 + y^2 ]
Step 2. Expand and simplify:
[ 4cx = 4a^2 - 4a r_2 ]
[ r_2 = a - \frac{cx}{a} ]
Step 3. Square again and expand (r_2^2 = (x-c)^2 + y^2):
[ \left(a - \frac{cx}{a}\right)^2 = (x-c)^2 + y^2 ]
[ a^2 - 2cx + \frac{c^2 x^2}{a^2} = x^2 - 2cx + c^2 + y^2 ]
Step 4. Cancel (-2cx) from both sides and collect:
[ a^2 + \frac{c^2 x^2}{a^2} = x^2 + c^2 + y^2 ]
[ a^2 - c^2 = x^2\left(1 - \frac{c^2}{a^2}\right) + y^2 ]
Step 5. Substitute (b^2 = a^2 - c^2) and (e = c/a), so (1 - e^2 = b^2/a^2):
[ b^2 = x^2 \cdot \frac{b^2}{a^2} + y^2 ]
Dividing through by (b^2):
[ \boxed{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1} ]
This is the standard ellipse equation. The constant-sum construction is exactly equivalent to it. In the BGP reading, the equation describes the visible boundary implied by the budget split; it does not replace e = c/a as the governing control variable.
The transverse residue is defined as the normalized semiminor axis:
[ \frac{b}{a} = \sqrt{1 - e^2} ]
This follows directly from (b^2 = a^2 - c^2) and (e = c/a):
[ b = \sqrt{a^2 - c^2} = a\sqrt{1 - (c/a)^2} = a\sqrt{1 - e^2} ]
Dividing by (a):
[ \frac{b}{a} = \sqrt{1 - e^2} ]
Key property: this expression depends only on (e), not on the absolute scale (a). Any two ellipses sharing the same (e) will have the same normalized shape, regardless of how large or small (a) is.
Theorem (Scale Collapse). Let (\mathcal{L}(a, e)) denote the locus generated by the constant-sum two-source process with reach budget (2a) and eccentricity (e = c/a). Then the normalized locus
[ \hat{\mathcal{L}}(e) = \frac{1}{a} \mathcal{L}(a, e) = \left{ \left(\frac{x}{a},, \frac{y}{a}\right) : (x, y) \in \mathcal{L}(a, e) \right} ]
is independent of (a). That is, for any two semimajor values (a_1 \neq a_2):
[ \hat{\mathcal{L}}(e) \text{ is identical for both } a_1 \text{ and } a_2. ]
Proof. A point ((x, y) \in \mathcal{L}(a, e)) satisfies (x^2/a^2 + y^2/b^2 = 1) with (b = a\sqrt{1-e^2}). Substituting (\xi = x/a) and (\eta = y/a):
[ \xi^2 + \frac{\eta^2}{1-e^2} = 1 ]
This equation contains no (a). Therefore (\hat{\mathcal{L}}(e)) depends only on (e). (\square)
Experimental confirmation: Maximum pairwise scale-collapse error across all tested ((e, a_1, a_2)) pairs was (3.97 \times 10^{-8}) (dominated by floating-point roundoff in the circle-intersection sampling). Within fixed-(e) groups, the spread of normalized observables was:
| Observable | Max spread across scale |
|---|---|
normalized_width |
(2.22 \times 10^{-16}) |
normalized_perimeter |
(6.66 \times 10^{-16}) |
normalized_major_tip_curvature |
(1.60 \times 10^{-14}) |
normalized_minor_tip_curvature |
(2.22 \times 10^{-16}) |
All values are numerically negligible and consistent with floating-point roundoff in this computation. The largest spread, (1.60 \times 10^{-14}), is still tiny relative to the scale of the normalized observables themselves.
Theorem (Process Equivalence). Let (P = (x, y)) be a point generated by the circle-combination process: choose radius (r \in [a-c,, a+c]), draw a circle of radius (r) centered at ((-c, 0)) and a circle of radius (2a - r) centered at ((c, 0)), and record an intersection. Then (P) lies exactly on the analytic ellipse (x^2/a^2 + y^2/b^2 = 1).
Proof. By construction, (r_1 + r_2 = r + (2a - r) = 2a). Since (P) is an intersection of the two circles, (r_1 = |P - (-c,0)|) and (r_2 = |P - (c,0)|). The condition (r_1 + r_2 = 2a) is exactly the definition of the ellipse with semimajor axis (a) and foci at (\pm c). From Section 2, this is equivalent to (x^2/a^2 + y^2/b^2 = 1). (\square)
Experimental confirmation:
- Maximum absolute ellipse-equation residual across all 95 reconstructed loci: (1.59 \times 10^{-14})
- Maximum RMS ellipse-equation residual: (3.14 \times 10^{-15})
Both are consistent with finite-precision floating-point evaluation at double precision.
Each geometric observable, once normalized by the appropriate scale factor built from (a), is a function of (e) alone. The following closed forms hold:
[ W(e) = \frac{b}{a} = \sqrt{1 - e^2} ]
Monotonically decreasing from (W(0) = 1) (circle) to (W(1) = 0) (degenerate segment).
[ A(e) = \frac{\pi a b}{a^2 \pi} = \frac{b}{a} = \sqrt{1 - e^2} ]
Identical to (W(e)); area and width residue track together under the normalization convention used here.
The ellipse perimeter has no elementary closed form. In exact form:
[ \widetilde{P}(e) = \frac{P}{2\pi a} = \frac{2}{\pi}E(e) ]
where (E(e)) is the complete elliptic integral of the second kind with modulus (e). Using Ramanujan's approximation, the same normalized quantity is
[ \widetilde{P}(e) \approx \frac{1}{2}\left(3(1 + \sqrt{1-e^2}) - \sqrt{(3 + \sqrt{1-e^2})(1 + 3\sqrt{1-e^2})}\right) ]
The normalized perimeter decays more slowly than (b/a) and tends to (2/\pi \approx 0.6366) as (e \to 1), corresponding to the degenerate limit (P \to 4a).
At the major-axis tip ((a, 0)), curvature of the ellipse is:
[ \kappa_{\text{major}} = \frac{a}{b^2} ]
The scale-normalized form used in the experiment is:
[ a \cdot \kappa_{\text{major}} = \frac{a^2}{b^2} = \frac{1}{1 - e^2} ]
This diverges as (e \to 1), confirming the observed sharp curvature amplification in Experiment 3.
At the minor-axis tip ((0, b)):
[ \kappa_{\text{minor}} = \frac{b}{a^2} ]
The scale-normalized form:
[ a \cdot \kappa_{\text{minor}} = \frac{b}{a} = \sqrt{1 - e^2} ]
This decays monotonically to zero as (e \to 1), consistent with the purple curve in Experiment 3.
In the separation-budget plane ((d, S) = (2c, 2a)), the control knob is the separation-to-budget ratio that labels rays from the origin:
[ e = \frac{c}{a} = \frac{d/2}{S/2} = \frac{d}{S} ]
For (0 < e < 1), constant-(e) loci are straight rays through the origin with slope:
[ S = \frac{1}{e} \cdot d ]
The special case (e = 0) is the vertical axis (d = 0). The physically realizable region in the first quadrant is (0 \le d \le S), with the diagonal (d = S) marking the degenerate boundary where (b = 0). The width residue field (b/a) in this plane is:
[ \frac{b}{a} = \sqrt{1 - \left(\frac{d}{S}\right)^2} ]
This is constant along each ray, confirming that the phase map's iso-color bands align exactly with constant-(e) rays, as observed in Experiment 4.
Corollary (Control Knob Sufficiency). Under the constant-sum two-source process, the normalized geometry is a function of (e = c/a) alone. Specifically:
- The normalized locus is (\hat{\mathcal{L}}(e)), determined solely by (e).
- All scale-normalized geometric observables (width, area, perimeter, tip curvatures) are one-dimensional functions of (e).
- The process-level construction recovers the analytic ellipse up to floating-point precision.
Therefore, (e = c/a) is a sufficient control variable for normalized geometry under this process model.
| (e) | Geometry |
|---|---|
| (0) | Circle: (b = a), (\kappa_{\text{major}} = \kappa_{\text{minor}} = 1/a) |
| ((0, 1)) | Proper ellipse with all observables smooth in (e) |
| (1^-) | Degenerate: (b \to 0), (\kappa_{\text{major}} \to \infty), (\kappa_{\text{minor}} \to 0) |
The sufficiency result holds for the symmetric, isotropic, equal-weight case. Natural generalizations that break these symmetries and require a higher-dimensional control space include:
- Weighted asymmetry — unequal budget splits or growth rates; the single ratio (e) is no longer sufficient and an additional weight parameter is required.
- Anisotropic media — replacing Euclidean distance with a directional cost function; the locus is generally no longer an ellipse and (e) is replaced by a richer directional descriptor.
- Multi-source generalization — for (n > 2) sources with a shared budget constraint, the control space requires multiple normalized allocation variables rather than a single ratio. One natural candidate is an ((n-1))-simplex, with (e) as the special two-source case.