REVIEW 1 major objections 3 minor 15 references
On cylindrical regression in three-dimensional Euclidean space
T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Fitting a cylinder to points reduces to solving a quartic system on a unit sphere, in coordinate-free form.
desk verdict The paper's main derivation ignores the constraint c⊥a, so the quartic system in Section 5 minimizes the wrong objective; the coordinate-free reproduction of Eberly's solution is flawed. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the replacement of point-to-cylinder distances by squared radial differences: after expanding $\rho_i^2 = |b|^2 - 2([r_i,a],b) + |[r_i,a]|^2$, the quartic and cubic terms in $b$ cancel, leaving an objective quadratic in the vector $c=[a,b]$. The quadratic part is the non-flatness form $Q(c,c) = (1/n)\sum_{i=1}^n (r_i - r_{\rm cm}, c)^2$, associated with a symmetric operator $Q$ on three-dimensional space. Inverting $Q$ in the non-degenerate case eliminates $c$ and leaves a quartic function of the unit axis direction $a$ on the unit sphere. The named objects that carry the argument are the vector $c$, the quadratic form $Q$ with its operator $Q$, and the symmetric quartic tensor $D_{ijkq}$ whose independent components appear in the final polynomial equations.
What would settle it
Take any non-coplanar point set, compute all real solutions of the quartic system (5.7)-(5.9) together with the unit condition (5.4), and compare their objective values with a brute-force numerical minimization of $\bar D^2$ over all unit vectors $a$. If the global minimum of the quartic objective occurs at a unit vector that does not satisfy the system, the claimed almost analytic solution is incomplete; if the system has solutions that are not stationary points of the constrained objective, it is over-inclusive.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that minimizing the biquadratically averaged deflection $\bar D^2 = (1/n)\sum_{i=1}^n (\rho_i^2 - \rho^2)^2$ can be reduced to an almost analytic procedure. With $a$ the unit direction of the cylinder axis and $c=[a,b]$ the cross product of $a$ with the axis-position vector $b$, the deflection takes the form $\bar D^2 = 4Q(c,c) - 4(L,c) + M$, where $Q$ is a quadratic form independent of $a$. In the non-degenerate case where $Q$ is positive definite, minimizing over $c$ yields $c = \frac{1}{2} Q^{-1} L$, so $c$ and then $b$ are explicit functions of $a$. The remaining task is to minimize a quartic polynomial in $a$ on the unit sphere, which is equivalent to solving the homogeneous quartic system (5.7)-(5.9) together with the unit-length condition (5.4). This reproduces the earlier almost analytic solution of the problem in a coordinate-free form; the solution is called almost analytic because the quartic system is not resolved by explicit formulas.
Load-bearing premise
The whole solution rests on the quadratic form $Q$ being positive definite, meaning the data points cannot all lie in a single plane; if they do, $Q$ cannot be inverted and the explicit formula for $c$ breaks down.
Editorial extensions
If this is right
- For any non-planar point cloud, the best-fitting cylinder under the biquadratic error can be obtained by solving a quartic system and then evaluating explicit formulas for the axis position and radius.
- The axis direction is determined before the radius: the unit vector $a$ solves the quartic system, and the squared radius $\rho^2$ is the average of the squared distances from the data points to that axis.
- The coordinate-free formulation removes the need to pick a center point on the axis, fixing an ambiguity that appears in earlier semi-coordinate solutions.
- Planar, collinear, or coincident data are excluded by the non-degeneracy assumption, and in those degenerate cases infinitely many cylinders tie for best fit.
Reading between the lines
- The same cancellation of high-order terms in the position vector may extend to other quadratic surfaces, so similar reductions might hold for sphere, ellipsoid, or cone fitting under biquadratic errors.
- Since $\bar D / \bar d \to 2\rho$ as the residuals tend to zero, the biquadratic optimum should approach the standard least-squares cylinder for tightly cylindrical data, making the two errors nearly interchangeable in the low-residual regime.
- The quartic system can have multiple real solutions; a resultant or homotopy solver could enumerate all stationary cylinders, converting the almost analytic solution into a complete decision procedure.
- Near-planar data will make $Q$ ill-conditioned, so a perturbation analysis of the recovered axis and radius would indicate how close to planar a point set may be before the non-degenerate recipe becomes numerically unreliable.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper treats the cylindrical regression problem in three-dimensional Euclidean space: given points r_1,...,r_n, find a cylinder (axis direction a, axis offset b, radius ρ) minimizing the biquadratic averaged deflection (3.1). After reducing the objective to (4.21), the authors minimize with respect to c=[a,b] for fixed a, obtaining the unconstrained solution c=1/2 Q^{-1}L in (4.28), substitute it into (4.29), and then reduce the remaining minimization over the unit vector a to the polynomial system (5.7)-(5.9) together with (5.4). The paper claims this reproduces Eberly's solution in coordinate-free, “almost analytic” form, with only the non-degenerate case treated.
Significance. If the derivation were correct, the paper would give a useful coordinate-free companion to Eberly's semi-coordinate solution, with a transparent step-by-step reduction from the biquadratic error to a quartic system and a machine-readable polynomial file. The derivation is self-contained and does not rely on circular reasoning. However, the central minimization step is incorrect: the paper minimizes (4.21) over c without imposing the constraint c⊥a that is part of the definition c=[a,b]. The resulting equations do not solve the cylindrical regression problem, so the main claimed result is not established.
major comments (1)
- [Section 4, Eqs. (4.15)-(4.16), (4.27)-(4.29)] The minimization over c is performed without the constraint c⊥a. Since c is defined by c=[a,b] and b⊥a, every admissible c is orthogonal to a; conversely, only vectors c⊥a correspond to an axis. Equation (4.27) is the unconstrained stationarity condition, so the solution (4.28) need not lie in the admissible plane. For the four-point example (1,0,0), (−1,0,0), (0,2,0), (0,−1,0.1) with a=(0,0,1), a direct calculation gives Q^{-1}L=(0,1.5,15), and hence (Q^{-1}L,a)=15≠0. Thus (4.28) is inadmissible, (4.29) is not the constrained minimum, and the full system (5.7)-(5.9)+(5.4), which is derived from (4.29), solves a different optimization problem. The constrained minimization yields the additional term (a^TQ^{-1}L)^2/(a^TQ^{-1}a) in the reduced objective, so the quartic system in Section 5 would need to be re-derived.
minor comments (3)
- [Eq. (4.29)] The last term is written as (e3,L)^3; the preceding terms and dimensional consistency require (e3,L)^2. This is a typo, but it appears in the central formula and should be corrected in any revision.
- [Section 5, after Eqs. (5.7)-(5.9)] The paper should state explicitly that the quartic system yields candidate stationary points and that the global minimum must be identified by comparing the objective values at all real solutions, including cases where one component of a vanishes and the parameterization leading to (5.5) is not valid.
- [Throughout] There are several typographical and presentation issues: the title contains 'SP ACE', the phrase 'greaterorequalslant' appears in place of a symbol, and the proof of Lemma 4.2 contains the duplicated word 'the the'.
Circularity Check
No circularity: the biquadratic cylinder-fit derivation is self-contained; prior-work citations are comparisons, not load-bearing imports.
full rationale
The paper's claim is a coordinate-free rederivation of the biquadratic cylindrical regression solution. The derivation chain is self-contained: (4.5) defines the objective, (4.15)-(4.21) rewrite it algebraically in terms of c=[a,b], (4.27)-(4.29) minimize the quadratic in c by solving the gradient equation, and Section 5 substitutes this into the remaining quartic in the unit vector a to obtain the polynomial system (5.7)-(5.9) with (5.4). No fitted parameter is renamed as a prediction, and no target result is assumed as an input. Citations to the authors' earlier work (e.g., [3], [6], [13]) are used for terminology or for comparison of the quadratic form Q, while the needed formulas for Q, L, and M are re-derived from the data. The self-citations are therefore not load-bearing. The skeptical observation that the minimization over c omits the constraint c orthogonal to a is a correctness concern about whether (4.28) satisfies the geometric constraint, not a circularity: the derivation does not hide its input as its output. The paper also explicitly restricts to the non-degenerate case where Q is invertible; this is a stated limitation, not a circular step.
Assumptions & free parameters
assumptions (4)
- standard math Spectral theorem for symmetric positive semidefinite operators on a 3D Euclidean space.
- standard math Weierstrass extreme value theorem ensures existence of a minimum for the continuous quartic on the compact unit sphere.
- domain assumption The biquadratic error (3.1), equivalently (4.5), is the objective to optimize.
- domain assumption Non-degenerate data: the quadratic form Q is positive definite.
Cite this review
Pith. "Pith review of On cylindrical regression in three-dimensional Euclidean space." pith.science (2026). https://pith.science/paper/I3ZPC7ZL
@misc{pith2026190802215,
author = {Pith},
title = {Pith review of: On cylindrical regression in three-dimensional Euclidean space},
year = {2026},
howpublished = {\url{https://pith.science/paper/I3ZPC7ZL}},
note = {Machine review of arXiv:1908.02215}
}
read the original abstract
The three-dimensional cylindrical regression problem is a problem of finding a cylinder best fitting a group of points in three-dimensional Euclidean space. The words best fitting are usually understood in the sense of the minimum root mean square deflection of the given points from a cylinder to be found. In this form the problem has no analytic solution. If one replaces the root mean square averaging by a certain biquadratic averaging, the resulting problem has an almost analytic solution. This solution is reproduced in the present paper in a coordinate-free form.
Reference graph
Works this paper leans on
-
[1]
Gander W., Golub G. H., Strebel R., Least-squares fitting of circles and ellipses , BIT Numer- ical Mathematics 34 (1994), no. 4, 558–578
work page 1994
- [2]
-
[3]
Algorithms for laying points optimally on a plane and a circle
Sharipov R. A., Algorithms for laying points optimally on a plane and a circl e, e-print arXiv:0705.0350
-
[4]
Least square ellipsoid fitting using iterative orthogonal transformations
Reza A., Sengupta A. S., Least square ellipsoid fitting using iterative orthogonal t ransforma- tions, e-print arXiv:1704.04877
-
[5]
Direct ellipsoidal fitting of discrete multi-dimensional data
Anwar R., Hamilton M., Nadolsky P. M., Fast ellipsoidal fitting of discrete multidimensional data, e-print arXiv:1901.05511
work page Pith review arXiv 1901
-
[6]
On linear regression in three-dimensional Euclidean space
Ageev O. V., Sharipov R. A., On linear regression in three-dimensional Euclidean space , e-print arXiv:1907.06009
work page Pith review arXiv 1907
-
[7]
Sherrod Ph., Cylindrical regression – fit a cylinder to data points , online resource cylinder.htm at nlreg.com
-
[8]
Abeles P., Fit cylinder to points , java class FitCylinderToPoints F64 at georegression.org, 2011, 2019
work page 2011
Show all 15 references
-
[9]
Nievergelt Y., Fitting cylinders to data , Journal of Computational and Applied Mathematics 239 (2013), 250–269
2013
-
[10]
XLII-1/W1, 2017, pp
Nurunnabi A., Sadahiro Yu., Lindenbergh R., Robust cylinder fitting in three-dimensional point cloud data , proceedings of ISPRS Hannover W orkshop: HRIGI, CMRT, ISA, EuroCOW, 6–9 June 2017, The International Archives of the Photogramm etry, Remote Sensing and Spatial Informati...
2017
-
[11]
Nurunnabi A., Sadahiro Yu., Lindenbergh R., Robust cylinder fitting in laser scanning point cloud data , Measurement 138 (2019), 632–651
2019
-
[12]
Eberly D., Least squares fitting of data by linear or quadratic structur es, online resource LeastSquaresFitting.pdf at geometrictools.com, July 1999, February 2019
1999
-
[13]
A., Course of analytical geometry , Bashkir State University, Ufa, 2010; see also arXiv:1111.6521
Sharipov R. A., Course of analytical geometry , Bashkir State University, Ufa, 2010; see also arXiv:1111.6521
2010 arXiv
-
[14]
A., Course of linear algebra and multidimensional geometry , Bashkir State Uni- versity, Ufa, 1996; see also arXiv:math.HO/0405323
Sharipov R. A., Course of linear algebra and multidimensional geometry , Bashkir State Uni- versity, Ufa, 1996; see also arXiv:math.HO/0405323
1996
-
[15]
Extreme value theorem , Wikipedia, Wikimedia Foundation Inc., San Francisco, USA . Self-employed individual, Ufa, Russia E-mail address : ageev-ufa@yandex.ru Bashkir State University, 32 Zaki V alidi street, 450074 Ufa , Russia E-mail address : r-sharipov@mail.ru URL: http:/ /...
Reviewed August 14, 2026 · model on record in the stance chip above.
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