SSplines

Simplex Splines on the Powell-Sabin 12-split

SSplines is a Python library for the evaluation of simplex splines over the Powell-Sabin 12-split of a triangle. The library provides efficient evaluation using the matrix recurrence relation for S-spline basis functions for constant, linear, and quadratic simplex splines as developed by Cohen, Lyche and Riesenfeld. The SSplines library was developed as part of the thesis: Simplex Splines on the Powell-Sabin 12-Split.

Features

• SplineFunction objects: Callable spline functions over a single triangle
• SplineSpace objects: Facilitate instantiation of multiple functions in the same spline space
• Evaluation and differentiation: Support for constant, linear, and quadratic simplex splines with convenient shortcuts for gradient, divergence, and Laplacian operators
• Hermite basis conversion: Conversion between quadratic S-spline basis and quadratic Hermite nodal basis for finite element methods
• Triangle sampling: Methods for sampling triangles for evaluation and visualization
• Numerical integration: Basic subdomain integration methods over the Powell-Sabin 12-split for finite element computations
• Polynomial pieces: Methods for returning polynomial restrictions of splines to each of the twelve sub-triangles
• Symbolic computation: Integration with SymPy for exact symbolic calculations

Installation

Using pip

pip install SSplines

Using uv (recommended for development)

# Install uv first (if not already installed)
curl -LsSf https://astral.sh/uv/install.sh | sh

# Install SSplines
uv pip install SSplines

# For development with testing and visualization
uv pip install "SSplines[dev]"

Local installation

git clone https://github.com/qTipTip/SSplines
cd SSplines
uv pip install -e ".[dev]"

Quick Start

Basic Usage

import numpy as np
from SSplines import SplineSpace, SplineFunction

# Define a triangle
triangle = np.array([
    [0, 0],
    [1, 0], 
    [0, 1]
])

# Create a quadratic spline space
space = SplineSpace(triangle, degree=2)

# Get the dimension of the space
print(f"Spline space dimension: {space.dimension}")  # 12 for quadratic

# Create a spline function with random coefficients
coefficients = np.random.rand(space.dimension)
spline_func = space.function(coefficients)

# Evaluate the spline at points
points = np.array([
    [0.3, 0.3],
    [0.1, 0.2],
    [0.5, 0.1]
])
values = spline_func(points)
print(f"Spline values: {values}")

Working with Basis Functions

# Get all basis functions
basis_functions = space.basis()

# Evaluate the first basis function
first_basis = basis_functions[0]
value_at_point = first_basis([0.3, 0.3])

# Compute derivatives
gradient = first_basis.grad([0.3, 0.3])
laplacian = first_basis.lapl([0.3, 0.3])

Hermite Basis

# For quadratic splines, you can use Hermite basis
hermite_basis = space.hermite_basis()

# Each Hermite basis function corresponds to nodal values or derivatives
h0 = hermite_basis[0]  # Value at first vertex
h1 = hermite_basis[1]  # x-derivative at first vertex
h2 = hermite_basis[2]  # y-derivative at first vertex

Symbolic Computation

from SSplines.symbolic import polynomial_basis_quadratic

# Get symbolic polynomial representations
symbolic_basis = polynomial_basis_quadratic(triangle)

# Each entry contains the polynomial pieces over the 12 sub-triangles
print(f"Number of polynomial pieces: {len(symbolic_basis[0])}")  # 12

Dependencies

• Core dependencies:

  • numpy ≥ 1.24.0: Numerical computations
  • sympy ≥ 1.10: Symbolic mathematics

• Development dependencies:

  • pytest ≥ 7.0.1: Testing framework
  • pytest-cov: Test coverage
  • matplotlib ≥ 3.5.1: Plotting and visualization

Development

Setting up a development environment

# Clone the repository
git clone https://github.com/qTipTip/SSplines
cd SSplines

# Install in development mode with all dependencies
uv pip install -e ".[dev]"

Running tests

# Run all tests
uv run pytest

# Run with coverage
uv run pytest --cov SSplines/

# Run specific test file
uv run pytest tests/test_spline_function.py -v

Code structure

SSplines/
├── __init__.py              # Main module exports
├── constants.py             # Mathematical constants and lookup tables
├── helper_functions.py      # Core computational functions
├── simplex_spline.py        # SimplexSpline class
├── spline_function.py       # SplineFunction class  
├── spline_space.py          # SplineSpace class
├── symbolic.py              # Symbolic computation utilities
└── dicts.py                 # Lookup dictionaries for sub-triangles

Mathematical Background

The library implements S-splines on the Powell-Sabin 12-split, which divides a triangle into 12 sub-triangles by:

1. Adding a point at the centroid
2. Adding midpoints on each edge
3. Connecting these points to create 12 sub-triangles

The S-splines are defined using a matrix recurrence relation that allows efficient evaluation without explicit polynomial representation. This approach is particularly useful for:

• Finite element methods
• Computer-aided geometric design
• Approximation theory
• Scientific computing applications requiring smooth basis functions

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

License

This project is licensed under the MIT License - see the LICENSE file for details.

Citation

If you use SSplines in your research, please cite the original theoretical work and the software implementation:

Original S-splines theory:

@article{cohen2013splines,
  title={S-splines},
  author={Cohen, Elaine and Lyche, Tom and Riesenfeld, Richard},
  journal={Mathematics of Computation},
  volume={82},
  number={283},
  pages={1577--1596},
  year={2013},
  doi={10.1090/S0025-5718-2013-02664-6}
}

SSplines software and implementation:

@mastersthesis{stangeby2018ssplines,
  title={Simplex Splines on the Powell-Sabin 12-Split},
  author={Stangeby, Ivar Haugal{\o}kken},
  school={University of Oslo},
  year={2018},
  url={http://hdl.handle.net/10852/64070},
  urn={URN:NBN:no-66606}
}

Software repository:

@software{stangeby2024ssplines,
  title={SSplines: Simplex Splines on the Powell-Sabin 12-split},
  author={Stangeby, Ivar},
  year={2024},
  doi={10.5281/zenodo.15742326},
  url={https://github.com/qTipTip/SSplines}
}

Author

Ivar Stangeby - istangeby@gmail.com

Links

• GitHub Repository
• Documentation
• PyPI Package
• Original Paper