pith. sign in

arxiv: 2606.12802 · v1 · pith:7DYOF3FGnew · submitted 2026-06-11 · 🧮 math.NA · cs.NA

Local Consistency and Higher-Order Structure of Spherical Interpolation

Shingyu Leung This is my paper

Pith reviewed 2026-06-27 06:33 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords spherical interpolationSIDER-nlocal consistencySLERPhigh-order methodsgeodesic normal coordinatesNeville interpolation
0
0 comments X

The pith

SIDER-n recursive spherical interpolation achieves local consistency of order n+1 for smooth curves.

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

The paper analyzes SIDER-n, a recursive construction that builds high-order interpolants on the unit sphere from repeated spherical linear interpolation. Working in geodesic normal coordinates, it shows that the recursion preserves the polynomial degree structure required in the normalized stencil variable. Combined with the exact matches at the n+1 sample nodes, this preservation produces an error bound of order h to the power n+1. A reader would care because the result supplies a concrete way to obtain classical polynomial orders locally even though the underlying geometry is curved.

Core claim

SIDER-n is defined recursively from SLERP. Local Taylor expansions in geodesic normal coordinates first establish that SIDER2 attains third-order accuracy whose leading error shares the shifted nodal structure of Euclidean quadratic interpolation. The adjacent SIDER2 errors entering the SIDER3 step possess a common leading coefficient, so the cubic term cancels and fourth-order accuracy follows. Carrying the expansion one order higher yields the analogous cancellation for SIDER4 at fifth order. The degree-filtered formal expansion framework then proves that, for any fixed n, the recursion preserves the necessary polynomial degree structure in the normalized stencil variable; together with th

What carries the argument

The degree-filtered formal expansion framework that tracks polynomial degrees in the normalized stencil variable to establish preservation under the SIDER recursion.

If this is right

  • SIDER2 attains third-order accuracy whose leading error matches the shifted nodal structure of Euclidean quadratic interpolation.
  • Adjacent SIDER2 errors share a common leading coefficient, allowing the SIDER3 recurrence to cancel the cubic term and reach fourth-order accuracy.
  • Extending the expansion shows coefficient compatibility for SIDER3, proving fifth-order accuracy of SIDER4.
  • For each fixed n the method delivers the local consistency estimate O(h^{n+1}) under the stated smoothness and small-stencil assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The recursive structure could be inserted into numerical integrators for differential equations constrained to the sphere.
  • If analogous normal-coordinate expansions exist on other Riemannian manifolds, the same degree-preservation argument might apply there.
  • Local SIDER stencils could be assembled into global high-order approximations on spherical domains when partition-of-unity weights are introduced.

Load-bearing premise

The spherical curve is smooth enough and the stencil is small enough that geodesic normal coordinates permit valid local Taylor expansions of SLERP and the SIDER recursion.

What would settle it

For a concrete smooth spherical curve such as a small circle, compute the spherical distance error of the SIDER-n interpolant at a sequence of successively halved spacings h and check whether the observed convergence rate equals n+1.

read the original abstract

Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation construction for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper gives a local consistency analysis of SIDER for smooth spherical curves sampled at equally spaced parameter values. The analysis is carried out in geodesic normal coordinates, which allows the SIDER recursion to be compared with classical Neville interpolation while retaining the curvature-dependent corrections introduced by SLERP. We first derive local expansions of SLERP and show that SIDER2 has third-order accuracy; its leading error has the same shifted nodal structure as Euclidean quadratic interpolation. We then prove that the adjacent SIDER2 errors entering SIDER3 have a common leading coefficient, so that the SIDER3 recurrence cancels the cubic term and yields fourth-order accuracy. Carrying the expansion one order further gives the corresponding coefficient compatibility for SIDER3 and proves fifth-order accuracy of SIDER4. Finally, we introduce a degree-filtered formal expansion framework for the general SIDER recursion. This framework proves that, for each fixed $n$, SIDER-$n$ preserves the required polynomial degree structure in the normalized stencil variable. Together with the interpolation conditions at the $n+1$ nodes, this yields the local consistency estimate $d_{\mathbb{S}^2}\bigl(\gamma(\theta h),P_i^{[n]}(\theta;h)\bigr)=O(h^{n+1})$ under the stated smoothness and small-stencil assumptions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces SIDER-n, a recursive high-order interpolation scheme on the unit sphere S^2 constructed from repeated SLERP. It performs a local consistency analysis in geodesic normal coordinates, first deriving explicit Taylor expansions of SLERP to establish third-order accuracy for SIDER-2 (with leading error matching the shifted nodal structure of Euclidean quadratic interpolation), then showing that adjacent SIDER-2 errors share a common leading coefficient so that the SIDER-3 recurrence cancels the cubic term to yield fourth-order accuracy, and similarly verifying fifth-order accuracy for SIDER-4. A degree-filtered formal expansion framework is then introduced to prove that, for each fixed n, SIDER-n preserves the required polynomial degree structure in the normalized stencil variable; combined with the n+1 interpolation conditions, this yields the local consistency bound d_{S^2}(γ(θh), P_i^{[n]}(θ;h)) = O(h^{n+1}) under stated smoothness and small-stencil hypotheses.

Significance. If the central claims hold, the work supplies a rigorous, curvature-aware error analysis for manifold interpolation that extends classical Neville schemes while retaining SLERP corrections. The explicit low-order coefficient comparisons (n=2,3,4) and the general degree-filtered framework constitute reproducible, parameter-free derivations of the consistency orders; these are genuine strengths that could support subsequent development of high-order methods on spheres and other manifolds in numerical analysis and applications.

minor comments (2)
  1. [Introduction] §1 (Introduction) and the abstract: the notation P_i^{[n]}(θ;h) is introduced without an explicit recursive definition or reference to the precise stencil indexing; a short displayed equation or pointer to the definition in §2 would improve readability.
  2. [Abstract] The final paragraph of the abstract states the smoothness and small-stencil assumptions under which the geodesic-normal Taylor expansions are valid; repeating these hypotheses verbatim at the start of the general-framework section would make the scope of the O(h^{n+1}) claim immediately clear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The manuscript's local consistency analysis and degree-filtered framework are correctly characterized. With no specific major comments provided, we note that any minor editorial adjustments will be incorporated in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds via explicit local Taylor expansions of SLERP in geodesic normal coordinates, direct coefficient matching to verify error cancellation for SIDER-2/3/4, and a degree-filtered formal expansion framework that preserves polynomial degree structure for general n. These steps rest only on the paper's stated smoothness and small-stencil hypotheses; no parameters are fitted, no result is renamed as a prediction, and no load-bearing premise reduces to a self-citation. The O(h^{n+1}) consistency bound follows from the n+1 interpolation conditions plus the preserved degree structure, rendering the argument self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard local analysis tools for manifolds; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Geodesic normal coordinates allow the SIDER recursion to be compared with classical Neville interpolation while retaining curvature-dependent corrections from SLERP
    Invoked to derive the local expansions of SLERP and the subsequent error cancellations (abstract, second paragraph).
  • standard math Taylor expansions of the curve and of SLERP are valid to the required orders under the smoothness and small-stencil assumptions
    Used throughout the low-order cases and the general framework (abstract, final paragraph).

pith-pipeline@v0.9.1-grok · 5798 in / 1429 out tokens · 29599 ms · 2026-06-27T06:33:57.601454+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

300 extracted references · 12 canonical work pages · 2 internal anchors

  1. [1]

    and Leung, S

    Lee, Y.K. and Leung, S. , date-added =. J. Sci. Comput. , number =

  2. [2]

    and Qian, J

    Wei, Y. and Qian, J. and Leung, S. , date-added =. SIAM J. Sci. Comput. , number =

  3. [3]

    Benamou, J. D. and Luo, S. and Zhao, H. , date-added =. J. Computational Mathematics , number =

  4. [4]

    , date-added =

    Cybenko, G. , date-added =. Approximation by superposition of a sigmoidal function , volume =

  5. [5]

    Near optimal signal recovery from random projections: universal encoding strategies? , volume =

    Cand. Near optimal signal recovery from random projections: universal encoding strategies? , volume =. IEEE Transactions on Information Theory , number =

  6. [6]

    and Zuo, W

    Zhang, K. and Zuo, W. and Chen, Y. and Meng, D. and Zhang, L. , date-added =. IEEE Trans. Image Process. , number =

  7. [7]

    A Fast Marching Level Set Method for Monotonically Advancing Fronts

    Sethian, J A , date-added =. A Fast Marching Level Set Method for Monotonically Advancing Fronts. , urldate =. 1996 , bdsk-url-1 =. doi:10.1073/pnas.93.4.1591 , journal =

    work page doi:10.1073/pnas.93.4.1591 1996

  8. [8]

    Sussman, Mark and Smereka, Peter and Osher, Stanley , date-added =. A. Journal of Computational Physics , month = sep, number =. 1994 , bdsk-url-1 =. doi:10.1006/jcph.1994.1155 , issn =

    work page doi:10.1006/jcph.1994.1155 1994

  9. [9]

    Computing Minimal Surfaces via Level Set Curvature Flow , year =

    Chopp, David Layne and Sethian, James , date-added =. Computing Minimal Surfaces via Level Set Curvature Flow , year =

  10. [10]

    and Caflisch, R

    Merriman, B. and Caflisch, R. and Osher, S. , booktitle =. Level Set Methods, with an Application to Modeling the Growth of Thin Films , volume =

  11. [11]

    , date-added =

    Litman, A. , date-added =. Reconstruction by Level Sets of N-Ary Scattering Obstacles , volume =. doi:10.1088/0266-5611/21/6/S10 , journal =

    work page doi:10.1088/0266-5611/21/6/s10

  12. [12]

    A multilayer level-set method for eikonal-based traveltime tomography

    Li, W.B. and Hung, K.K.T. and Leung, S. , date-added =. arXiv:2510.16413 , title =

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    and Cheng, L.-T

    Osher, S. and Cheng, L.-T. and Kang, M. and Shim, H. and Tsai, Y-H , booktitle =. J. Comput. Phys. , number =

  14. [14]

    , date-added =

    Larios, A. , date-added =. arXiv:2510.02761 , title =

    work page arXiv

  15. [15]

    , date-added =

    Boritchev, A. , date-added =. Commun. Math. Phys. , pages =

  16. [16]

    , date-added =

    Bulinskii, A.V. , date-added =. Annales Academie Scientiarum Fennice , pages =

  17. [17]

    and Aslam, T.D

    Lozano, E. and Aslam, T.D. , date-added =. Implicit fast sweeping method for hyperbolic systems of conservation laws , volume =. J. Comp. Phys. , number =

  18. [18]

    and Zhang, Y

    Hu, R. and Zhang, Y. T. , date-added =. J. Sci. Comput. , number =

  19. [19]

    and Chen, S

    Zhang, Y.T. and Chen, S. and Li, F. and Zhao, H. K. and Shu, C. W. , date-added =

  20. [20]

    , date-added =

    Sederberg, T.W. , date-added =. Computer aided geometric design , year =

  21. [21]

    Equivalent extensions of

    Martin, Lindsay and Tsai, Yen-Hsi Richard , date-added =. Equivalent extensions of. Journal of Scientific Computing , number =

  22. [22]

    and Qian, J

    Luo, S. and Qian, J. and Burridge, R. , date-added =. High-order factorization based high-order fast sweeping methods for point-source eikonal equations , volume =

  23. [23]

    and Qian, J

    Luo, S. and Qian, J. , date-added =. Fast sweeping methods for factored anisotropic eikonal equations: multiplicative and additive factors , volume =

  24. [24]

    and Rector, J

    Zhang, L. and Rector, J. W. and Hoversten, G. M. , date-added =. Eikonal solver in the celerity domain , volume =

  25. [25]

    , booktitle =

    Pica, A. , booktitle =. Fast and accurate finite-difference solutions of the

  26. [26]

    Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms , volume =

    Wei, Yuxiao and Cheng, Jin and Leung, Shingyu and Burridge, Robert and Qian, Jianliang , date-added =. Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms , volume =. J. Comput. Phys. , number =

  27. [27]

    Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing , volume =

    Wei, Yuxiao and Cheng, Jin and Burridge, Robert and Qian, Jianliang , date-added =. Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing , volume =. J. Comput. Phys. , pages =

  28. [28]

    and Chau, W.M

    Leung, S. and Chau, W.M. and Lee, Y.K. , date-added =. J. Sci. Comput. (arXiv:2410.10420) , number =

    work page arXiv

  29. [29]

    , date-added =

    Leung, S. , date-added =. arXiv preprint arXiv:2503.17618 , title =

    work page arXiv

  30. [30]

    and Niesen, J

    Moan, P.C. and Niesen, J. , date-added =. On an asymtotic method for computing the modified energy for sympletic methods , volume =. Discrete and Continuous Dynamical Systems , number =

  31. [31]

    , date-added =

    Hernandez, D.M. , date-added =. Monthly Notices of the Royal Astronomical Society , number =

  32. [32]

    and Murua, A

    Chan, R.P.K. and Murua, A. , date-added =. Appl. Num. Math. , pages =

  33. [33]

    and Farago, I

    Bayleyegn, T. and Farago, I. and Havasi, A. , date-added =. Periodica Mathematica Hungarica , number =

  34. [34]

    , date-added =

    Richardson, L.F. , date-added =. The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam , volume =. Phil. Trans. Roy. Soc. A , number =

  35. [35]

    , date-added =

    Wennsch, J. , date-added =. Numerische Mathematik , pages =

  36. [36]

    , date-added =

    Orel, B. , date-added =. BIT , number =

  37. [37]

    and He, W

    Guo, H. and He, W. and Sea, S. and Shen, H.-W. and Constantinescu, E.M. and Liu C. and Peterka, T. , date-added =. Extreme-scale stochastic particle tracing for uncertain unsteady flow visualization and analysis , volume =. IEEE Transactions on Visualization and Computer Graphics , number =

  38. [38]

    , date-added =

    Rockafellar, R.T. , date-added =. Mathematics of Operations Research , number =

  39. [39]

    and Le Tallec, P

    Glowinski, R. and Le Tallec, P. , date-added =

  40. [40]

    and Mercier, B

    Lions, P.L. and Mercier, B. , date-added =. Splitting algorithms for the sum of two nonlinear operators , volume =. SIAM J. Numer. Analy. , number =

  41. [41]

    and Wajs, V.R

    Cambettes, P.L. and Wajs, V.R. , date-added =. Multiscale Model. Simul. , number =

  42. [42]

    , date-added =

    Nesterov, Y. , date-added =. Dual extrapolation and its applications to solving variational inequalities and related problems , volume =. Mathematical Programming , pages =

  43. [43]

    and Bobin, J

    Becker, S. and Bobin, J. and Candes, E.J. , date-added =. SIAM J. Imaging Sciences , number =

  44. [44]

    , date-added =

    Nesterov, Y. , date-added =. Mathematical Programming , pages =

  45. [45]

    and Blanc-Feraud, L

    Weiss, P. and Blanc-Feraud, L. and Aubert, G. , date-added =. SIAM J. Sci. Comput. , number =

  46. [46]

    and Teboulle, M

    Beck, A. and Teboulle, M. , date-added =. SIAM J. Imaging Sciences , number =

  47. [47]

    and Bertsekas, D.P

    Eckstein, J. and Bertsekas, D.P. , date-added =. Mathematical Programming , pages =

  48. [48]

    and Mercier, B

    Gabay, D. and Mercier, B. , date-added =. A dual algorithm for the solution of nonlinear variational problems via finite element approximation , volume =. Computers & Mathematics with Applications , number =

  49. [49]

    and Bresson, X

    Goldstein, T. and Bresson, X. and Osher, S. , date-added =. J. Sci. Comput. , pages =

  50. [50]

    and Aujol, J.F

    Duval, V. and Aujol, J.F. and Gousseau, Y. , date-added =. SIAM Multiscale Model. Simul. , number =

  51. [51]

    and Esedoglu, S

    Chan, T. and Esedoglu, S. , date-added =. UCLA CAM 03-77 , title =

  52. [52]

    and Qian, J

    Lu, W. and Qian, J. , date-added =. A local level-set method for 3D inversion of gravity-gradient data , volume =

  53. [53]

    and Qian, J

    Li, W. and Qian, J. and Li, Y. , date-added =. Joint inversion of surface and borehole magnetic data: A level-set approach , volume =

  54. [54]

    and Lu, W

    Li, W. and Lu, W. and Qian, J. , date-added =. A level-set method for imaging salt structures using gravity data , volume =

  55. [55]

    and Leung, S

    Kwan, W. and Leung, S. and Wang, X.P. and Qian, J. , date-added =. J. Comput. Phys. , pages =

  56. [56]

    and Faucheux, L.O

    Braun, E. and Faucheux, L.O. and Libchaber, A. , date-added =. Strong self-focusing in nematic liquid crystals , volume =. Phys. Rev. A , number =

  57. [57]

    , date-added =

    Brown, C.V. , date-added =. Physical Properties of Namatic Liquid Crystals , year =

  58. [58]

    , date-added =

    Leung, S. , date-added =. Proceeding of International Conference on Spectral and High Order Methods , title =

  59. [59]

    and Nelson, D.R

    Le Doussal, P. and Nelson, D.R. , date-added =. Statistical Mechanics of Directed Polymer Melts , volume =. Europhys. Lett. , number =

  60. [60]

    and Bruinsma, R.F

    Selinger, J.V. and Bruinsma, R.F. , date-added =. Hexagonal and namatic phases of chains. I. Correlation functions , volume =. Phys. Rev. A , number =

  61. [61]

    , date-added =

    De Gennes, P.G. , date-added =. Polymeric liquid crystals: Frank elasticity and light scattering, , volume =. Molecular Crystals and Liquid Crystals , number =

  62. [62]

    and Leok, M

    Gawlik, E.S. and Leok, M. , date-added =. Embedding-based Interpolation on the Special Orthogonal Group , volume =. SIAM J. Sci. Comput. , number =

  63. [63]

    and Podgonik, R

    Svensek, D. and Podgonik, R. , date-added =. Correlation functions of main-chain polymer nematics constrained by tensorial and vectorial conservation laws , volume =. J. Chem. Phys. , number =

  64. [64]

    and Podgonik, R

    Svensek, D. and Podgonik, R. , date-added =. Generalized conservation law for main-chain polymer nematics , volume =. Phys. Rev. E , number =

  65. [65]

    and Grason, G.M

    Svensek, D. and Grason, G.M. and Podgonik, R. , date-added =. Tensorial conservation law for nematic polymers , volume =. Phys. Rev. E , number =

  66. [66]

    and Campbell, S.L

    Brenan, K.E. and Campbell, S.L. and Petzold, L.R. , date-added =

  67. [67]

    , date-added =

    Magnus, W. , date-added =. On the exponential solution of differential equations for a linear operator , volume =. Comm. Pure Appl. Math. , pages =

  68. [68]

    and Grossman, R.G

    Crouch, P.E. and Grossman, R.G. , date-added =. Numerical integration of ordinary differential equations on manifolds , volume =. J. Nonlinear Sci. , pages =

  69. [69]

    and Prince, P.J

    Dormand, J.R. and Prince, P.J. , date-added =. J. Comput. Appl. Math. , pages =

  70. [70]

    and Wanna, G

    Hairer, E. and Wanna, G. and Norsett, S.P. , date-added =. Solving Ordinary Differential Equations I: Nonstiff Problems , year =

  71. [71]

    and Chu, K.L

    Ying, N. and Chu, K.L. and Leung, S. , date-added =. arXiv:2407.06467 , title =

    work page arXiv

  72. [72]

    , date-added =

    Ying, N. , date-added =. HKUST PhD Thesis , title =

  73. [73]

    and Wanner, G

    Hairer, E. and Wanner, G. , date-added =. J. Comput. and Appl. Math. , number =

  74. [74]

    and Lubich, C

    Hairer, E. and Lubich, C. and Wanner, G. , date-added =

  75. [75]

    Splitting Methods in Communication, Imaging, Science, and Engineering , year =

    Glowinski, Roland and Osher, Stanley J and Yin, Wotao , date-added =. Splitting Methods in Communication, Imaging, Science, and Engineering , year =

  76. [76]

    and Ross, S.D

    Lekien, F. and Ross, S.D. , date-added =. The computation of finite-time. Chaos , pages =

  77. [77]

    and Vrolijk, B

    Post, F.H. and Vrolijk, B. and Hauser, H. and Laramee, R.S. and Doleisch, H. , date-added =. Comput. Graph. Forum , pages =

  78. [78]

    and Hauser, H

    Laramee, R.S. and Hauser, H. and Doleisch, H. and Vrolijk, B. and Post, F.H. and Weiskopf, D. , date-added =. Comput. Graph. Forum , pages =

  79. [79]

    and Peikert, R

    Pobitzer, A. and Peikert, R. and Fuchs, R. and Schindler, B. and Kuhn, A. and Theisel, H. and Matkovic, K. and Hauser, H. , date-added =. Comput. Graph. Forum , pages =

  80. [80]

    , date-added =

    Higham, D.J. , date-added =. SIAM J. Numer. Anal. , pages =

Showing first 80 references.