Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2511.01621

Jacobi's solution for geodesics on a triaxial ellipsoid

Charles F. F. Karney
View original source ↗Read source document ↗Export metadata ↓

Abstract

On Boxing Day, 1838, Jacobi found a solution to the problem of geodesics on a triaxial ellipsoid, with the course of the geodesic and the distance along it given in terms of one-dimensional integrals. Here, a numerical implementation of this solution is described. This entails accurately evaluating the integrals and solving the resulting coupled system of equations. The inverse problem, finding the shortest path between two points on the ellipsoid, can then be solved using a similar method as for biaxial ellipsoids.

Explore related subjects

physics.geo-phmath.DGphysics.comp-ph

Keep this discovery

BibTeXRIS

Charles F. F. Karney. 2025-11-03. Jacobi's solution for geodesics on a triaxial ellipsoid. https://doi.org/10.1007/s00190-026-02030-3

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Holistic law of aftershocks

The paper is devoted to the phenomenological theory of aftershocks occurring in the source of a tectonic earthquake following the main shock. The theory was developed by the author jointly with A.D. Zavyalov and O.D. Zotov during the course of a long-term study of aftershocks. The theory is based on the concepts of source deactivation and the source's proper time. The holistic law governing the decay of aftershock activity over proper time follows from the theory. The damping decrement is equal to the source deactivation coefficient. The main focus of this paper is the analysis of the logical structure of the theory. The paper also contains a brief description of the experimental results obtained using the theory. Keywords: earthquake source, aftershocks, Omori's law, Utsu's law, deactivation coefficient, proper time, underground clock, foreshock convergence, aftershock divergence.

physics.geo-ph↗
arXiv2026

Bayesian deep learning integration of geophysical and drilling data for 3D prediction of copper mineralization and drill targeting: a case study from the Kogodai prospect, Rudny Altai

Exploration drill targeting in structurally complex terranes is hindered by sparse sampling, heterogeneous datasets, and the ambiguity of geophysical inversions. Here, we present an uncertainty-aware 3D workflow for the acceleration of time-to-discovery in brownfield explorations and apply it to the Kogodai prospect in the Rudny Altai metallogenic province. We jointly analyse existing drilling and geophysical data in a comprehensive approach, revealing hidden patterns in already available data. Drillholes and trenches were desurveyed to a common 3D reference frame, and assays were composited to a consistent spatial support to facilitate joint modelling with geophysical inputs. We develop Bayesian deep-learning models to predict 3D fields of Cu grade together with chargeability and apparent resistivity while quantifying epistemic uncertainty via Monte Carlo sampling. The original contribution of this work is to treat the problem not as pointwise regression between co-located observations, but as joint learning of spatially continuous 3D fields from sparse, heterogeneous exploration evidence. The resulting 3D predictions delineate a principal mineralized trend and several localized candidate zones that coincide with elevated induced polarization (IP) responses, while uncertainty mapping highlights where predictions are robust versus where additional drilling would be most informative. The continuous Cu-grade field can also be thresholded to produce binary prospectivity maps, allowing the sensitivity of target delineation to the chosen cutoff to be evaluated. The outputs are intended for qualitative interpretation and risk-aware drill targeting rather than resource estimation, and we discuss key limitations arising from incomplete provenance metadata for geophysical products and heterogeneity of historical sampling.

physics.geo-ph↗
arXiv2026

PyelogP: Automated Energy-Based Determination of Preconsolidation Pressure in Clay Deposits

Estimating the preconsolidation pressure ($\sigma'_p$) from one-dimensional consolidation (oedometer) tests is critical in geotechnical engineering for settlement analysis. Traditional graphical methods, such as the Casagrande procedure, may introduce uncertainties, particularly when interpreting rounded $e$-log($P$) curves typical of disturbed specimens of soft clays and silt deposits. This paper introduces PyelogP, an open-source Python library designed to calculate $\sigma'_p$ using the strain-energy method proposed by Becker et al. (1987) as an automated and reproducible alternative. The algorithm combines natural cubic spline interpolation, knee-point detection via the Kneedle algorithm, and split-point linear regression within the work-pressure space. Physically informed thresholds, including overconsolidation ratio limits and second-derivative maxima (${d^2 e}/{d(\log \sigma')^2}$), are incorporated to establish pre-yield and post-yield fitting boundaries. The performance of PyelogP is evaluated against a suite of 22 experimental consolidation datasets covering various clay deposits, including Saint-Alban clay and San Francisco Old Bay Clay. The results demonstrate strong agreement with the published values ($R^2$ = 0.912, RMSE = 0.374, MBE = -0.080), while the $O(N^2)$ algorithm requires only a few milliseconds per curve for typical oedometer datasets and less than 150 milliseconds for the largest datasets.

physics.geo-ph↗