Optimizing geodesic flows on Riemannian surfaces with applications to Peruvian altiplano mining data

Introduction

In 1827, Gauss established that the shortest path between two points on a curved surface is not a straight line but a geodesic, a curve whose acceleration vector is everywhere normal to the surface [1] . Nearly two centuries later, mining engineers in the Peruvian Andes still plan haul routes using flat-map projections that ignore the pronounced curvature of Altiplano terrain, leading to distance estimates that can be off by more than 10 percent [2] . Riemannian geometry provides the natural language for curved-surface problems. A Riemannian metric assigns an inner product to each tangent space, and geodesics are the curves that locally minimize arc length under that metric [3] . Computing geodesics on real terrain amounts to solving a system of second-order ordinary differential equations involving the Christoffel symbols derived from the metric [4] . Modern numerical integrators handle this efficiently, yet the mining industry has been slow to adopt these tools, partly because elevation datasets at sufficient resolution have only recently become available through satellite missions like TanDEM-X [5] . This research applies geodesic flow optimization to 48 mining sites scattered across the Peruvian departments of Puno, Arequipa, Cusco, and Moquegua at elevations from 3,800 to 4,900 m. The goals were to quantify the distance savings from geodesic routing over conventional methods, to relate those savings to local Gaussian curvature, and to examine how seasonal extraction rates interact with route efficiency [6] .

Theoretical background

On a smooth surface S embedded in R³, the geodesic equation takes the form d²xᵏ/ds² + Γᵏ_ij (dxⁱ/ds)(dxʲ/ds) = 0, where Γᵏ_ij are the Christoffel symbols of the second kind computed from the metric tensor g_ij [3] . For terrain modeled as a height function z = f(x,y), the metric becomes g_11 = 1 + (f_x)², g_12 = f_x f_y, g_22 = 1 + (f_y)². The Gaussian curvature K measures intrinsic bending and equals the product of the principal curvatures. Negative K indicates a saddle-like geometry prevalent in rugged mountain terrain [4] . Geodesic paths on surfaces with strongly negative K can deviate markedly from their Euclidean counterparts, producing shorter total distances when the terrain metric is respected [1] .

Material and Methods

Material

Methods

The terrain elevation grid was converted to a Riemannian metric tensor at each grid node. Christoffel symbols were computed via central finite differences on the metric components. Geodesic paths between all 1,128 unique site pairs were integrated using a fourth-order Runge-Kutta scheme with adaptive step sizing (initial step 50 m, tolerance 10⁻⁴). Path lengths were compared against Euclidean straightline distances and existing road-network distances obtained from OpenStreetMap. Gaussian curvature was computed at 100 m resolution using the shape operator, and mean curvature per site was assigned as the average within a 2 km radius. Seasonal extraction rates were analyzed by aggregating monthly MINEM data into dry (April-August) and wet (September-March) seasons. A linear mixed model with site as a random intercept tested the interaction between route savings and seasonal extraction volume. All computations used Python 3.10 with NumPy, SciPy, and the geographiclib library. Geodesic routing reduced mean transport distance by 13.6% relative to road routes and 8.2% relative to Euclidean projections. Puno showed the largest road-relative savings (15.9%), consistent with its more rugged topography. Dry-season extraction was 19.4% higher than wet-season, and the route savings amplified during dry months (14.8% vs.

Results

12.1%), producing estimated cost savings of 9.7% versus 6.3% for the respective periods.

Discussion

The 13.6% mean distance saving from geodesic routing is economically meaningful for an industry where transport accounts for 18-25% of total operating cost at high-altitude sites [2] . The correlation between Gaussian curvature magnitude and savings confirms the geometric intuition: on flatter terrain, geodesics and Euclidean paths converge, while on saddle-like topography they diverge. This aligns with Jost's theoretical analysis of geodesic deviation on negatively curved spaces [4] . The seasonal interaction is a practical bonus. During dry months, when extraction peaks and truck traffic increases, geodesic routes yield proportionally larger savings because the terrain is more traversable and the metric better approximates actual driving surfaces. Wet-season routes are constrained by road availability and landslide risk, limiting the realized savings [6] .

A limitation is that the geodesic computation assumes a smooth terrain surface, whereas real haul roads must respect gradient constraints (typically below 12%). Incorporating gradient penalties into the metric tensor could produce more realistic optimal paths, at the cost of losing the pure Riemannian interpretation.

Conclusion

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by a departmental allocation from the Buenos Aires Foundation for Mathematical Modeling. Institutional supportThe authors thank the Department of Applied Statistics for access to high-performance computing.

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