Patch area and uniform sampling on the surface of any ellipsoid
Callum Robert Marples1, Philip Michael Williams1
1Molecular Therapeutics and Formulation, School of Pharmacy, University of Nottingham, Nottingham, NG7 2RD UK.
Numerical Algorithms
|June 30, 2025
Summary
Generating uniform random points on triaxial ellipsoids is challenging. This study derives a surface area formula to evaluate sampling algorithms, finding the best method depends on whether Cartesian or polar coordinates are needed.
Area of Science:
- Geometry
- Computational Mathematics
- Numerical Analysis
Background:
- Generating uniform random points on complex surfaces like triaxial ellipsoids is difficult due to the lack of analytical surface area formulas.
- Existing algorithms for sampling points on ellipsoids require robust verification methods.
Purpose of the Study:
- To derive a formula for the surface area of an ellipsoidal patch using numerical integration and elliptic integrals.
- To evaluate and compare the efficiency of different algorithms for generating uniform random points on a triaxial ellipsoid.
Main Methods:
- Derivation of a surface area formula for an ellipsoidal patch via one-dimensional numerical integration.
- Analytical formula derivation for the special case of a spheroid.
- Investigation of surface sampling algorithms using the derived triaxial ellipsoid surface area formula.
Main Results:
- The derived formula enables patch area calculations for algorithm verification.
- The efficiency of sampling algorithms is dependent on the desired output coordinate system.
- For Cartesian coordinates, Chen and Glotzer's gradient rejection sampling with Marsaglia's method is most efficient.
- For polar coordinates, a surface area rejection sampler is preferable.
Conclusions:
- The study provides a validated method for assessing random point generation algorithms on triaxial ellipsoids.
- Algorithm selection for uniform point distribution on ellipsoids should consider the target coordinate system for optimal efficiency.
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