Description
Add a Lebedev Quadrature class that performs integration over the unit sphere for functions expressed in the three-dimensional Cartesian space.
The goal is to provide an accurate and efficient way to compute surface integrals of functions defined on the sphere: $\int_{S^2} f(x,y,z),d\Omega$
using Lebedev quadrature rules. The implementation should expose the quadrature points and weights, support several predefined Lebedev grids with different accuracy orders, and provide a simple interface for integrating scalar functions defined in Cartesian coordinates.
One of the targeted use cases is collision probability computation, where directional quantities must be integrated over all possible orientations or directions on the sphere. The high accuracy and efficiency of Lebedev quadrature make it particularly suitable for these calculations, especially when the evaluated function exhibits strong angular variations.
The implementation should be validated against analytical results such as:
$\int_{S^2} 1,d\Omega = 4\pi$
to ensure the expected accuracy of the selected quadrature rule.
Links and References
- Vyacheslav Lebedev, Dmitri Laikov, A Quadrature Formula for the Sphere of the 131st Algebraic Order of Accuracy, Russian Academy of Sciences Doklady Mathematics, Volume 59, Number 3, 1999, pp. 477-481.
Description
Add a Lebedev Quadrature class that performs integration over the unit sphere for functions expressed in the three-dimensional Cartesian space.
$\int_{S^2} f(x,y,z),d\Omega$
The goal is to provide an accurate and efficient way to compute surface integrals of functions defined on the sphere:
using Lebedev quadrature rules. The implementation should expose the quadrature points and weights, support several predefined Lebedev grids with different accuracy orders, and provide a simple interface for integrating scalar functions defined in Cartesian coordinates.
One of the targeted use cases is collision probability computation, where directional quantities must be integrated over all possible orientations or directions on the sphere. The high accuracy and efficiency of Lebedev quadrature make it particularly suitable for these calculations, especially when the evaluated function exhibits strong angular variations.
The implementation should be validated against analytical results such as:
to ensure the expected accuracy of the selected quadrature rule.
Links and References