https://github.com/mhdeeb/spherical-quadrature
This project attempts to compare multiple numerical quadrature methods in an accessible way
https://github.com/mhdeeb/spherical-quadrature
quadrature-integration website
Last synced: 11 months ago
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This project attempts to compare multiple numerical quadrature methods in an accessible way
- Host: GitHub
- URL: https://github.com/mhdeeb/spherical-quadrature
- Owner: mhdeeb
- License: mit
- Created: 2025-08-05T23:13:13.000Z (11 months ago)
- Default Branch: main
- Last Pushed: 2025-08-18T18:50:10.000Z (11 months ago)
- Last Synced: 2025-08-18T20:39:02.980Z (11 months ago)
- Topics: quadrature-integration, website
- Language: TypeScript
- Homepage: https://www.mhdeeb.com/grad
- Size: 11.7 MB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
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Metadata Files:
- Readme: README.md
- License: LICENSE.txt
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README
# 3D Spherical Quadrature Visualization
A high-performance [Three.js](https://threejs.org/) implementation for interactive visualization of spherical quadrature methods and numerical integration on the unit sphere. Features dynamic 3D rendering with real-time computation and error analysis.
[Demo](https://www.mhdeeb.com/grad)
## Features
### Quadrature Methods
- **Monte Carlo**: Random points on the sphere
- **Lebedev**: Spherical quadrature with octahedral symmetry
- **Product Quadrature**: Cartesian product of 1D quadratures (Gauss-Legendre × Trapezoidal)
- **Spherical Design**: Points with optimal spherical distribution properties
### Test Functions
- **Polynomial**: f = 1 + x + y² + x²y + x⁴ + y⁵ + x²y²z²
- **Gaussian Peaks**: Sum of Gaussian functions with different centers
- **Hyperbolic Tangent**: Smooth step function
- **Sign Functions**: Step function
## Usage
### Controls
- **Mouse**: Drag to rotate the sphere manually.
### Quadrature Methods
1. **Monte Carlo**: Statistical sampling approach
- Uniform: Correctly distributed points
- Clustered: Demonstrates improper sampling
2. **Lebedev**: Exact integration of spherical harmonics up to degree 2n-1
3. **Spherical Designs**: t-designs integrate exactly all spherical harmonics up to degree t
4. **Product Quadrature**: Combines 1D methods for φ and θ directions
## Technical Implementation
### Coordinate Systems
- Spherical coordinates: (r, θ, φ) where θ is azimuthal, φ is polar
- Cartesian coordinates: (x, y, z) = (sin φ cos θ, sin φ sin θ, cos φ)
- Three.js world coordinates with proper depth sorting and camera controls
### Numerical Methods
- Associated Legendre polynomials computed via recurrence relations
- Gauss-Legendre quadrature points and weights
- Random number generation for Monte Carlo methods
- Golden spiral for spherical design approximation
- High-performance GPU-accelerated rendering with Three.js
## Educational Value
This visualization helps understand:
- How different quadrature methods distribute points on the sphere
- The relationship between point distribution and integration accuracy
- Convergence behavior of numerical integration methods
## Future Plans
- **Schema Editor**: Allow users to edit grid schema manually or generate them with functions and export those schemas.
- **Function Input**: Allow users to input functions have them be integrated on and tested against analytical solution if provided.