Alpine Ski Track Design Combining Conjugated Harmonic Functions and Terrain Surfaces

by Jesús Mateo-Lázaro^1,2,4*, Salvador. M. Galve-Martín³, Enrique García-Vicente⁴, Jorge Castillo-Mateo^6,7, Andrea Marín-Ostáriz⁴, Rafael de Guadalfajara-Senra⁵

¹ Department of Earth Sciences, University of Zaragoza. Spain

² Ilustre Colegio Oficial de Geólogos de España

³ Colegio Oficial de Ingenieros Industriales de Aragón y La Rioja

⁴ Estrategias de Ingeniería y Desarrollo (EID Consultores)

⁵ Garona Estudios Territoriales

⁶ Department of Statistical Methods, University of Zaragoza, Spain

⁷ Department of Statistical Science, Duke University, Durham, USA

*  Corresponding author: jesmateo@unizar.es

Abstract

This article proposes a methodology that allows for optimized adaptation of the terrain surface on steep mountain slopes for the implementation of alpine ski tracks. It is a mathematically based stepwise model that generates bands of harmonic surfaces by studying the best adaptation to the natural terrain, considering and assessing earthwork balances, longitudinal and transverse drainage (stream crossings), allowable slope ranges, and comfort criteria for alpine skiing. Conjugated harmonic surfaces have important characteristics that make them attractive for multiple technical applications. This is the first time that these surfaces have been applied within a methodological procedure to optimize the diversity of characteristics described in the design of alpine ski tracks. A case example at a ski resort in the Spanish Pyrenees will be illustrated in the article.

Keywords

Alpine Ski Tracks, Conjugated Harmonic Functions, Slope of the ground surface, Earthworks Balances, SHEE software

Note:

Papers published in this special issue of the European Geologist journal have undergone a thorough peer-review process but have not been copy-edited. Authors bear full responsibility for the linguistic accuracy of their contributions.

This work is licensed under a Creative Commons Attribution 4.0 International License.

1. Introduction

For several years now, physics-based methodologies have been published to investigate the inertial forces to which skiers are subjected along trajectories on ski tracks surfaces. The trajectories measured with a GPS antenna and applying Newton’s second law, we can estimate the trajectory of the skier’s center of mass between two points [1-2]. Other authors propose alpine skiing tracking methods based on deep learning and correlation filter [3].

To carry out this procedure using a computer application, a module has been created within the SHEE program of the University of Zaragoza that has multiple applications, initially of a hydrological type [4-6] and extended to the design of dams, optimization of large earthmoving areas, hydrodynamic and groundwater models and other applications.

The methodology presented here would link with other harmonic-based techniques for earthwork surfaces that involve the application of Fourier analysis principles, spherical harmonics, or harmonic functions to model, analyze, and optimize ground surfaces, slope stability, and geophysical data [9-12].

2. Background

2.1. Multiapplication and Multiplatform SHEE Interface

By combining the functionalities of the SHEE package, it is possible to construct, adjust, analyze, and compare the hydrological processes occurring in a watershed. The program, developed at the Department of Earth Sciences at the University of Zaragoza, provides fast and high-quality OPENGL graphics in both RASTER and VECTOR formats. The software offers numerous applications for managing DEMs and simulating hydrological processes. New coverages can also be obtained by combining DEMs and simulated processes. Figure 1 shows a screenshot of the SHEE program interface.

Figure 1: Multiapplication and Multiplatform Interface of the SHEE Software of the University of Zaragoza.

2.2. DEM Management

DEM management is performed using GDAL (Geospatial Data Abstraction Library), which allows importing and exporting different file formats and creating new coverages from multiple files as global DEMs, e.g., SRTM30 represented in the Figure 2.

The program can combine coverages with different coordinate systems thanks to the USGS PROJ4 library. Thousands of astronomical and terrestrial geodetic systems can be represented, transformed, and converted between them. To do this, the program can obtain the necessary parameters from the Spatial Reference Organization (SRO) through internet server transfer. It is also possible to download information from a remote WMS server.

Figure 2: Global DEM displayed in three geodetic projections.

3. Application

Harmonic equations are real-valued functions that satisfy Laplace’s equation ∇²u = 0. For every harmonic equation, there exists a conjugate harmonic function. Both functions can form a holomorphic (analytical) function of a complex variable. Harmonic functions and their conjugates are fundamental in complex analysis and physics, specifically in the study of potential fields. They are also essential in groundwater models for defining the potential field and streamlines, respectively. In the case at hand, a terrain surface representing the altimetry can be analytically defined, along with its conjugate, which represents the paths of steepest descent of the first function.

The method we propose has three main stages. The first involves an analysis of terrain slopes and trajectories along which a main axis and a maximum amplitude are drawn.

In the second stage, the digital terrain model of the strip defined in the previous stage is subjected to an iterative smoothing filter with a four-neighbor kernel.

The target solution can optimize several characteristics such as maximum and minimum slopes, minimum earthwork volume, and the orientation vector function. Figure 3 shows the layout scheme with a main reference and control axis, the trajectory function and its conjugate, the elevation function.

Figure 3: Layout scheme with a main axis for reference and control, the trajectory function and the its conjugate, the elevation function. Note that at all points in the domain, the path lines are perpendicular to the elevation lines (contour lines).

From the generated surface, the smoothed edges of the ski track are analytically defined with their three spatial coordinates.

In the third stage, a numerical fitting model is applied using the harmonic Laplace equation to the slope boundary conditions defined in the previous surface.

This harmonic function represents the geometry of the skiable surface, while a conjugated harmonic function analytically linked to the previous one describes the trajectories of maximum slope.

The harmonic function Elevation (H) and its harmonic conjugate, the Trajectory function (T) are defined in Equations 2 and 3.

In an optional fourth stage for the final layout, the mass balance can be optimized according to environmental requirements, material quality, and other constraints external to the model.

Figure 4a shows the coverage of excavation depths (orange) and backfill depths (blue) across the entire earthworks area. Figure 4b shows the slope coverage of the ski track surface, which is very useful for identifying the difficulty level of each stretch.

Figure 4: Basibe Ski Track showing (a) the earthmoving area and (b) the slope distribution in the ski domain. Note that on turns, the slope on the inside is always steeper than on the outside due to the difference in distance between the two sections.

4. Conclusions and Outlook

The SHEE package, initially conceived to study hydrological processes using digital terrain models, allows database management from multiple perspectives, but always based on digital terrain models (DTM o DEM). Thus, the application of this package allows the Surface Design to which specific conditions can be applied. In the case presented of ski tracks, it has allowed the generation of a smoothed mesh of the existing terrain model, and on it, the design of surfaces fitted to harmonic functions and their conjugate harmonic functions that represent trajectories with infinitesimal parallelism that follow the maximum slope and are perpendicular to the elevation gradient.

Among the method’s limitations is the resolution of the digital terrain model, although this limitation is inherent to all methodologies. Terrain complexity is another limitation that may sometimes require the introduction of specific definitions, which can be implemented as boundary conditions. An example of this would be the passage of a ski slope under a chairlift.

As a future perspective, radial basis functions (RBFs) can be applied with the general equation for grid nodes j and n i weights and neurons:

These functions allow to apply specific conditions at strategic nodes of the grid and of varied typologies (e.g., excavation/backfill depth, maximum or minimum slope, concavity and convexity, the latter being very interesting for surface and platform drainage, as well as combinations of inertial, gravitational, and skier acceleration forces).

These functions are precursors, or a specific class, of artificial intelligence (AI) models that are currently very promising, especially because they allow for the interaction of very different types of data.

References

1. Wu, Y.; Wu, X. Optimal Design of Ski Tracks in Construction Projects: Taking the Warm-Up and Training Ski Track of the South Area in the Yanqing Competition Zone of the Beijing 2022 Winter Olympic Games as an Example. Buildings 2023, 13, 659. https://doi.org/10.3390/buildings13030659
2. Žvan, M.; Lešnik, B. Correlation between the length of the ski track and the velocity of top slalom skiers. Acta Gymnica 2007, 37, 37–44. [Google Scholar]
3. Qi, J.; Li, D.; Zhang, C.; Wang, Y. Alpine Skiing Tracking Method Based on Deep Learning and Correlation Filter. IEEE Access 2022, 10, 39248–39260. [Google Scholar] [CrossRef]
4. Mateo-Lázaro, J., Sánchez-Navarro, J.A., García-Gil, A., Edo-Romero, V. (2014). 3D-geological structures with digital elevation models using GPU programming. Computers & Geosciences, 70, 147-153. DOI: http://10.1016/j.cageo.2014.05.014
5. Mateo-Lázaro, J., Sánchez-Navarro, J.A., García-Gil, A., Edo-Romero, V., 2015. A new adaptation of linear reservoir models in parallel sets to assess actual hydrological events. J. Hydrol. 524, 507–521. https://doi.org/10.1016/j.jhydrol.2015.03.009.
6. Mateo-Lázaro, J, Sánchez-Navarro, JA, García-Gil, A, Edo, V, Castillo-Mateo, J, 2016. Modelling and layout of drainage-levee devices in river sections. EngGeol. 214, 11–19. https://doi.org/10.1016/j.enggeo.2016.09.011.
7. Kharchenko, S.V. (2017). Application of harmonic analysis for the quantitative description of Earth surface topography. Geomorfologiya, 2, 14-24. Doi:10.15356/0435-4281-2017-2-14-24.
8. Haghshenas, S. S., Haghshenas, S. S., Geem, Z. W., Kim, T.-H., Mikaeil, R., Pugliese, L., & Troncone, A. (2021). Application of Harmony Search Algorithm to Slope Stability Analysis. Land, 10(11), 1250. https://doi.org/10.3390/land10111250.
9. Popadyev, V. (2024).Harmonic analysis of the earth`s surface points` horizontal movements in the ITRF. Geodesy and Cartography. 1002, 10-16. Doi:10.22389/0016-7126-2023-1002-12-10-16.
10. Shen, Z., Shen, W., Xu, X., Zhang, S., Zhang, T., He, L., Cai, Z., Xiong, S., & Wang, L. (2023). A Method for Measuring Gravitational Potential of Satellite’s Orbit Using Frequency Signal Transfer Technique between Satellites. Remote Sensing, 15(14), 3514. https://doi.org/10.3390/rs15143514.

This article has been published in European Geologist journal 60 – 5th IPGC Special Edition 1