---
license: mit
task_categories:
- feature-extraction
language:
- en
tags:
- mathematics
- topology
- genus
- knots
- Euler_characteristics
- genus_labeling
authors:
  - Eduardo Ivan Velazquez Richards
  - Rodrigo Fritz
  - Victor Mijangos
  - Anayanzi Martínez 
  - Pablo Suárez-Serrato
pretty_name: EuLearn
size_categories:
- 10B<n<100B
---
# 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, so each genus has its own subdirectory. Each surface comprises 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 surface reconstruction algorithm. An STL file of the same surface with smoothed faces.
    
    * Smoothed STL mesh: A geometrically smoothed version of the same 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.
    
    * Tubular radius 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 ("bup" stands for "blow-up").

## 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


**Example:**

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


Each component encodes specific geometric and topological information:

- **`g7`**: The topological genus of the surface. Here, genus 7 (i.e., the surface has 7 holes).
- **`lissajous` or `fibonacci`**: Parameterization type. Most examples are *Lissajous* (singular) knots; a few are *Fibonacci* knots.
- **`3,5,8`**: 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`**: Phase shifts (`φ_x`, `φ_y`, `φ_z`) in each coordinate. Notation like `pi2` represents multiples of π (e.g., `pi2` = π/2, `pi3` = π/3, etc.).
- **`03_1.1_0.2`**: Parameters of the cosine-based radius modulation for the tubular neighborhood around the curve:
  - `03`: frequency `f`
  - `1.1`: constant term `c`
  - `0.2`: amplitude `a`

  Thus, the radius function is:

  r(t) = 1.1 + 0.2cos(3t)

  
- **`r0.02_0.029`**: Minimum and maximum radius values of the tubular neighborhood.
- **`v90`**: Voxel resolution per axis. Here, the scalar field is discretized into a 90 × 90 × 90 grid (totaling 729,000 voxels).
- **`.stl`**: File extension. This denotes the *smoothed* version of the surface mesh.

Other file variants use suffixes to indicate their content:

- **`_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 (e.g., radius function samples or 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''