On the perimeter length determination of the eight-centered oval

Jean-Claude Golvin; Jean-Marc Ginoux (LIS)

REVIEW 4 major objections 5 minor 20 references

On the perimeter length determination of the eight-centered oval

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The perimeter of the eight-centered oval, the classical architectural approximation of an ellipse, is exactly given by a closed-form arcsine formula in the two semi-axes.

desk verdict A sound, short derivation of the eight-centered oval perimeter with a genuine notational flaw in Eq. (17) that needs fixing, but the geometry and numerics hold up. read the letter →

arxiv 1908.00783 v1 pith:JIFCTUP3 submitted 2019-08-02 math.MG math.HO

Jean-Marc Ginoux (LIS) , Jean-Claude Golvin This is my paper · ORCID

classification math.MGmath.HO MSC 51M2551N20

keywords eight-centeredovalperimeterformulaellipseapproximationHoneyconstructionarclengthcentralanglesColosseumtrigonometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a closed-form formula for the perimeter of the eight-centered oval, a classical architectural curve that approximates an ellipse using eight tangent circular arcs. Working from Honey's 1908 construction, the authors derive the three central angles of one quarter of the oval using the laws of sines and cosines on the triangle formed by the three circle centers. The perimeter is then exactly four times the sum of each arc's radius times its angle, yielding $O(a,b) = 4[\arcsin(\gamma) a^{2}/b + \arcsin(\beta) (a+b)/2 + \arcsin(\delta) b^{2}/a]$ in terms of arcsines of algebraic expressions in the semi-axes $a$ and $b$. The formula matches the true ellipse perimeter to within 0.029% for axes from 1 to 10, and for the Colosseum ($a=94$, $b=78$) it gives 541.523 m versus 541.524 m from the elliptic integral. This gives architects and archaeologists a simple, direct way to compute perimeters of polycentric oval buildings without elliptic integrals.

What carries the argument

The load-bearing construction is Honey's system of three osculating circles per quadrant: the major circle with radius $R=a^{2}/b$ centered at $g(0,b-a^{2}/b)$, the minor circle with radius $r=b^{2}/a$ centered at $e(a-b^{2}/a,0)$, and the intermediate circle with radius $p=(a+b)/2$ whose center $k$ is the intersection of two auxiliary circles. The proof's engine is the triangle $(gek)$: applying the law of cosines twice yields the intermediate angle $\alpha'$, the law of sines yields the intermediate angle $\alpha$, and the right triangle $(geo)$ connects these to the two outer center angles $\gamma$ and $\delta$. The perimeter identity $O=4(R\gamma+p\beta+r\delta)$ then turns the sine expressions for the angles into a closed-form arcsine formula.

What would settle it

For an extreme ratio, say $a=10$ and $b=1$, evaluate the three arcsine arguments in Table I; if any lies outside $[0,1]$, or if the three resulting angles do not sum to $\pi/2$, the arcs cannot form a continuous tangent quarter-oval and the formula fails.

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Extended reading notes

Core claim

The central discovery is that the perimeter of the eight-centered oval, as constructed by Honey, is fully determined by the two ellipse semi-axes and is given by $O(a,b)=4[\arcsin(\gamma)a^{2}/b + \arcsin(\beta)(a+b)/2 + \arcsin(\delta)b^{2}/a]$, where the sine of each center angle is a rational-algebraic function of $a$ and $b$ (Table I). The three angles $\gamma$, $\beta$, $\delta$ are obtained geometrically: from the right triangle formed by the small and large osculating circles' centers and the ellipse center, and from the triangle formed by the three circle centers, using the law of cosines and the law of sines. Because the three arcs meet tangentially, the oval's perimeter is exactly the sum of the three arc lengths, giving a closed-form expression in $a$ and $b$. The paper verifies that when $a=b$ the formula collapses to the circle circumference $2\pi\rho$, and that for the Colosseum it reproduces the elliptic-integral perimeter to $1.85\times 10^{-4}\%$ relative error.

Load-bearing premise

The perimeter formula assumes the three arcs of each quadrant meet tangentially, so their lengths add without correction; this tangency is taken from Honey's construction and is not proved in this paper.

Editorial extensions

If this is right

The perimeter of any structure built on Honey's eight-centered oval can now be computed directly from its two measured axes, with no elliptic integrals.
For the Colosseum's semi-axes $a=94$, $b=78$, the formula gives 541.523 m, within one millimeter of the elliptic-integral value 541.524 m.
The numerical tests show the relative error between the oval and ellipse perimeters stays below 0.029% for semi-axes between 1 and 10.
Setting $a=b$ recovers the circle's circumference $2\pi a$, confirming the formula's behavior at the circular limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the arcsine expressions are algebraic, the formula could serve as a fast, closed-form approximation of the complete elliptic integral of the second kind, with the 0.029% bound providing a known worst-case error.
Honey's construction could be iterated by inserting additional intermediate circles between adjacent osculating circles, yielding N-centered ovals whose perimeters follow from the same trigonometric method.
The symmetry noted in the sine expressions (swapping $a$ and $b$ exchanges $\gamma$ and $\delta$) suggests the approximation error depends mainly on the axis ratio, so a one-variable error bound might be derivable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick read: this note does what it says. It derives the three center angles for Honey's eight-centered oval and gives a perimeter formula. The trigonometric derivation checks out, and the numerical agreement with the ellipse perimeter is real — under 0.03% relative error for a,b in [1,10] and 1.85e-4% for the Colosseum. That's the useful part for archaeologists, architects, and CAD users who want a closed-form perimeter rather than an elliptic integral.

The paper's main defect is in Eq. (17). As printed, it applies arcsin to the angles γ, β, δ themselves, but those are angles in radians, not sines. For a=10, b=1, δ≈1.09, so arcsin(δ) is undefined. The intended meaning is clear from Table I and the circle-limit check: they mean arcsin(sinγ), etc. All three angles are in fact acute, so that recovers the angle, but the branch condition is never stated or proved, and the formula as written is not well-defined over the claimed domain. This is a notation/typo problem, not a geometric one, but it is load-bearing in the sense that Eq. (17) is the central claim. A referee should insist on restating it as O = 4[γ R + β p + δ r] or equivalently 4[arcsin(sinγ)R + ...] with an explicit acuteness proof.

There is also a minor arithmetic typo in the circle-limit remark: the identity should be 2 arcsin((4−√2)/6) = arcsin(7/9), not 2 arcsin(7/9). The final 2πρ is correct, so this is cosmetic.

Novelty is modest. The calculation is a routine application of the law of cosines and sines once Honey's construction is given. The authors do not verify whether Herrera-Samper (2015) or Mazzotti (2017) already contains a perimeter formula; if either does, the novelty claim collapses. They also withhold the algebraic derivation, which is a small annoyance but not a flaw.

Who is this for? People working with polycentric ovals in historical architecture or technical drawing. It is a short note, not a major contribution. But it is correct in substance, and the presentation issues are fixable in one round. I would send it to a specialized venue and let a referee require the Eq. (17) restatement and a citation check. A desk rejection would be too harsh.

Referee Report

4 major / 5 minor

Summary. The manuscript treats Honey's 1908 eight-centered oval as a piecewise-circular approximation to the ellipse with semi-axes a and b. For one quadrant it uses three circular arcs with radii R = a^2/b, p = (a+b)/2, r = b^2/a and center angles gamma, beta, delta. The paper derives trigonometric expressions for the sines of these angles (Eqs. (11), (14), (16); Table I) using the triangles (gek) and (geo), and then writes the perimeter of the oval as O = 4[arcsin(gamma) a^2/b + arcsin(beta)(a+b)/2 + arcsin(delta) b^2/a] in Eq. (17). It reports that for the Colosseum dimensions a = 94, b = 78 the oval perimeter is 541.523 m versus 541.524 m from the elliptic integral, a relative error of 1.85 x 10^-4%, and that for 1 <= a,b <= 10 the relative error stays below 0.029%. It also claims that the formula reduces to 2*pi*rho when a = b = rho.

Significance. If Eq. (17) is interpreted as intended, namely O = 4(R*gamma + p*beta + r*delta), this is a useful closed-form perimeter for a classical polycentric oval. The derivation is elementary, uses no fitted parameters, and is checked against an independent elliptic-integral benchmark for the Colosseum example. The paper also identifies a plausible historical gap, since Honey supplied radii but not center angles. However, the central formula is miswritten in a way that makes it ill-posed over part of the claimed domain, and the branch and tangency conditions needed for rigor are not explicitly established. These issues are local and fixable, but they affect the main result.

major comments (4)

[III, Eq. (17)] The central formula (17) is not well-defined as printed. The quantities gamma, beta, delta are center angles, but only their sines are derived (Eqs. (11), (14), (16); Table I), and the notation arcsin(gamma) in Eq. (17) applies arcsin to an angle rather than to its sine. For a = 10, b = 1, which lies inside the stated test range, delta is about 1.09 rad > 1, so arcsin(delta) is not real; more generally arcsin(angle) is not equal to the angle. The intended perimeter is O = 4(R*gamma + p*beta + r*delta), or equivalently O = 4[R arcsin(sin gamma) + p arcsin(sin beta) + r arcsin(sin delta)] once the acute-angle branch is proved. This restatement is required before Eq. (17) can be evaluated.
[II, Eqs. (11)-(16)] The identities arcsin(sin gamma) = gamma, arcsin(sin beta) = beta, and arcsin(sin delta) = delta used implicitly in Eq. (17) are valid only if gamma, beta, delta lie in (0, pi/2). The manuscript neither states nor proves this branch condition. It follows from the geometry of Fig. 2, for example from theta > alpha' and theta' > alpha for a,b > 0, but that verification is absent, so the domain of validity of Eq. (17) is not established.
[II, Eqs. (3)-(4)] The perimeter calculation as the sum of three arc lengths assumes the three arcs meet tangentially at their junction points. This is not proved in the paper. It is an immediate consequence of the quoted Honey relations |ek| = p - r and |gk| = R - p, since internal tangency is equivalent to the distance between centers being the difference of the radii; the manuscript should state this explicitly, because it is essential to the exactness of O.
[III, Remark] The circle-limit check contains a false displayed identity: '2 arcsin((4 - sqrt(2))/6) = 2 arcsin(7/9)' is not true. The correct identity is 2 arcsin((4 - sqrt(2))/6) = arcsin(7/9), obtained from sin(2 arcsin x) = 2x sqrt(1 - x^2). As printed, the subsequent cancellation that yields O = 2*pi*rho does not follow; this is probably a typographical slip but should be corrected.

minor comments (5)

[III] The numerical error statements ('below 0.029%' for 1 <= a,b <= 10 and '1.85 x 10^-4%' for the Colosseum) are reported without a table or sample of computed values; a small table would make the claim verifiable.
[II] In the sentence describing the large circle, 'Pour le cercle mineur (en rouge sur la Fig. 1), il emploie le centre g...' should presumably read 'cercle majeur'; as written it contradicts the preceding assignment of the blue circle to e.
[I] The sentence 'il ne nous semble pas que ce calcul ... n'ait jamais encore été effectué ni publié' contains a double negative that makes the intended meaning ambiguous.
[II] The derivation of sin(alpha') in Eq. (8) and sin(alpha) in Eq. (13) silently selects positive square roots; since the angles are meant to be acute, a one-line justification of the sign choice would improve clarity.
[III] The paper does not specify the numerical method or software used to evaluate the elliptic integral (2); adding this information would make the reported comparisons reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the perimeter formula is a direct trigonometric consequence of Honey's 1908 construction and is checked against an independent elliptic-integral benchmark.

full rationale

The derivation chain is self-contained once Honey's construction is accepted. Sections II and III compute the three center angles from the fixed triangle (geo) and (gek) via the law of cosines and the law of sines (Eqs. 7-16), then form the perimeter as four times the sum of the three arc lengths (Eq. 17). Nothing is fitted to the elliptic-integral value: the comparison L(94,78)=541.524 versus O(94,78)=541.523 is an external numerical check, not an input to the formula. The only external ingredient is Honey's 1908 construction of the radii and centers (Eqs. 3-4), which is prior published work and not a self-citation. The citation to Golvin [5] supplies archaeological dimensions and historical context, but it is not load-bearing for the trigonometric derivation. The apparent arcsin-argument issue in Eq. (17) — where arcsin should be applied to the sines of the angles rather than to the angle variables themselves, with an unstated acute-angle branch assumption — is a correctness or typographical concern, not evidence that the result reduces to its own inputs. Accordingly, no genuine circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data: a and b are inputs describing the ellipse, and Honey's radii are taken from a cited construction. The main assumptions are standard Euclidean geometry, the curvature radii of the ellipse, and the unstated requirement that the three center angles lie in the principal branch of arcsin.

assumptions (5)

standard math Euclidean law of cosines and law of sines apply to triangle (gek).
Used in Section II-A through II-D to derive angles α', α, and β.
domain assumption The osculating circle at (a,0) has center e=(a-b^2/a,0) and radius b^2/a; at (0,b), center g=(0,b-a^2/b) and radius a^2/b.
Standard curvature formulas for the ellipse, used to define |ek| and |gk| in Eqs. (3)-(4).
domain assumption Honey's construction defines k as an intersection of the circles centered at e and g with radii |ek| and |gk|, and the intermediate arc has radius p=(a+b)/2 with tangency at the junctions.
Taken from Honey (1908); the perimeter formula depends on the three arcs being tangent and covering one quadrant.
domain assumption The angles γ, β, δ lie in (0, π/2), so that arcsin(sinγ)=γ and similarly for β and δ.
Required to turn the derived sine values into angles in Eq. (17); not stated or proved, and checked numerically only for 1 ≤ a,b ≤ 10.
standard math The perimeter of the reference ellipse is given by the elliptic integral in Eq. (2).
Used as the benchmark for the relative error calculations in Section III.

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Cite this review

Pith. "Pith review of On the perimeter length determination of the eight-centered oval." pith.science (2026). https://pith.science/paper/JIFCTUP3

@misc{pith2026190800783,
  author       = {Pith},
  title        = {Pith review of: On the perimeter length determination of the eight-centered oval},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JIFCTUP3}},
  note         = {Machine review of arXiv:1908.00783}
}

read the original abstract

On the perimeter length determination of the eight-centered oval. Several studies have shown that an eight-centered oval coincides almost perfectly with the ellipse constructed on the same axes and can be considered as a representation of the latter provided that the radii of the arcs of circles that compose it had been suitably chosen. Its perimeter's computation is then reduced to the simple sum of arc lengths of circles. However, it doesnot seem to us that this calculation, which could prove to be useful, has never been performed nor published. This note aims thus to present a geometric demonstration of the perimeter length determination of the eight-centered oval.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

loi des cosinus

Calcul de l’angle α ′ dans le triangle ( gek) avec la “loi des cosinus”

work page

[2] [2]

Calcul de l’angle γ = θ − α ′ avec θ = ˆg dans le triangle rectangle ( geo)

work page

[3] [3]

loi des sinus

Calcul de l’angle α dans le triangle ( gek) avec la “loi des sinus”

work page

[4] [4]

loi des sinus

Calcul de l’angle β dans le triangle ( gek) avec la “loi des sinus”

work page

[5] [5]

loi des cosinus

Calcul de l’angle δ = θ′ − α avec θ′ = ˆe dans le triangle rectangle ( geo). o k e g i Γ Α ' Β Α ∆ /Minus2 2 4 6 /Minus6 /Minus4 /Minus2 2 Figure 2: Triangles ( geo) et ( gek) pour la détermination des trois angles aux centres. Au préalable, en appliquant le théorème de Pythagore au tria ngle rectangle ( geo), il est facile de déduire l’expression de la l...

work page

[6] [6]

Remarque

029%. Remarque. En posant a = b = ρ, l’expression (17) permet de retrouver le périmètre du cerc le de rayon ρ. L’expression (17) s’écrit alors : O (a, b ) = 4 ρ [arcsin (γ) + arcsin (β ) + arcsin (δ)] (18) Dans ce cas, les trois sinus (11, 14 and 16) deviennent égaux à sin (γ) = 4 − √ 2 6 , sin (β ) = 4 √ 2 9 et sin (δ) = 4 − √ 2 6 . En remplaçant ces sin...

work page

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Berndt, Ramanujan’s Notebooks, vol

B. Berndt, Ramanujan’s Notebooks, vol. 3, New York, Spri nger-Verlag, 1991

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Bronstein & K.A

I.N. Bronstein & K.A. Semendiaev, Aide-mémoire de Mathé matiques, Paris, Eyrolles, 1990

work page 1990