appliedgeometry/EuLearn

Dataset Description

This dataset contains 3D objects representing a topologically diverse collection of surfaces, each generated from closed, parameterized curves with varying number of self-intersections (singular knots). The surfaces are organized by topological genus, ranging from 0 to 10. For each surface, we included the following four files:

• Non-Smoothed STL Mesh (*_ns.stl): A 3D mesh of the surface with sharp geometry and unmodified vertex positions. This version retains the raw output of the Marching Cubes algorithm.

• Smoothed STL Mesh (.stl): A geometrically smoothed version of the *_ns.stl surface, generated using Laplacian vertex smoothing in Blender. This process adjusts vertex positions to reduce sharp edges and improve regularity, modifying surface area and volume while preserving the mesh connectivity and topological features (e.g., genus and Euler characteristic).

• Scalar Field (*_sf.txt): A plain-text file encoding the scalar field used to generate the surface via the Marching Cubes algorithm. The isosurface at level 0 defines the mesh geometry in the *_ns.stl file.

• Blow-up Profile (*_bup.txt): A plain-text file containing the radius at each point along the discretized curve. These radii define the thickness of the tubular neighborhood used to generate the scalar field.

Filename Encoding Convention

Filenames follow the pattern:

g{genus}_{knot_type}_{n_x},{n_y},{n_z}_{φ_x},{φ_y},{φ_z}_{f}_{c}_{a}_r{r_min}_{r_max}_v{resolution}.stl

Each component encodes geometric or topological information.

For example:

g7_lissajous_3,5,8,0,pi2,0_03_1.1_0.2_r0.02_0.029_v90.stl

• g7: Topological genus of the surface. In this example, the genus is 7 (i.e., the surface has 7 holes).

• lissajous or fibonacci: Knot type. Most elements were generated from Lissajous (singular) knots, but just a couple from Fibonacci (singular) knots.

• 3,5,8: The frequencies n_x, n_y, n_z used in the parametric equations of the curve, corresponding to the x, y, and z coordinates.

• 0,pi2,0: The phase shifts φ_x, φ_y, φ_z in each coordinate. Notation like pi2 represents fractions of π (e.g., pi2 = π/2, pi3 = π/3, etc.).

• 03_1.1_0.2: Parameters of the (cosine-based) sinusoidal radius variation for the tubular neighborhood around the curve:

  • 03: frequency f,
  • 1.1: constant term c,
  • 0.2: amplitude a,

  from the radius variation function: r(t) = c ± a·cos(f·t), where t in [0,2π).

• r0.02_0.029: Minimum and maximum radius of the tubular neighborhood.

• v90: Voxel resolution per axis. While v90 nominally refers to a 90 × 90 × 90 voxel grid, an offset of 5 voxels per side was added to prevent surface clipping at the bounding box edges. As a result, the actual scalar field is discretized into 101 × 101 × 101 = 1,030,301 grid points, yielding 1,000,000 voxels in total.

• .stl: File extension. In this example, as there is no suffix, this filename ending corresponds to the smoothed version of the surface mesh.

  As mentioned earlier, the other file types use the following suffixes:

  • _ns.stl: The non-smoothed surface mesh, output directly from the Marching Cubes algorithm (raw geometry).
  • _sf.txt: The scalar field used to generate the surface.
  • _bup.txt: The blow-up profile, i.e., the radius function discretization for the analytical envelope.

Acknowledgements

• The authors thank DGTIC-UNAM for access to the Miztli HPC resources, grant LANCAD-UNAM-DGTIC-430.
• RF thanks CONAHCyT for a graduate fellowship.
• This work was supported by Universidad Nacional Autónoma de México Postdoctoral Program (POSDOC) for author EIVR, who also acknowledges the postdoctoral fellowship received during the production of this work.
• VM acknowledges the support from project PAPIIT TA100924 "Investigación de sesgos inductivos en aprendizaje profundo y sus aplicaciones."