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Implement an accurate and efficient spherical triangle area calculation using Eriksson’s formula #1585
Description
Proposed new feature or change:
UXarray’s current spherical triangle area calculation uses a Quadrature-based implementation adapted from the existing TempestExtreme software.
Based on our benchmarks, the current approach has two main limitations:
- Its computational performance is not ideal.
- It remains vulnerable to numerical errors in difficult geometric configurations, despite using quadruple-precision arithmetic.
We recommend replacing the current implementation with Eriksson’s formula for calculating the area of a spherical triangle.
With an appropriate implementation, our benchmarks show that Eriksson’s formula provides:
- better computational performance;
- better numerical accuracy; and
- more reliable behavior for challenging spherical triangles.
This formula has also already been adopted by YAC for its spherical triangle area calculations.
Please refer to the following section of the paper for the algorithm, numerical analysis, and benchmark results:
Page 30, Eqn (23), Spherical Excess
https://egusphere.copernicus.org/preprints/2026/egusphere-2026-636/
Activity
Sevans711 commentedon Jul 20, 2026
Hi @hongyuchen1030, thank you for posting this and pointing me to it to take a look! Some initial thoughts:
Would
face_node_anglesfrom #1566 possibly enable more accurateface_areascomputations for all faces, not just triangles, as per @rajeeja's suggestion there to compare spherical excess with the computed areas? The process could shift from "use quadrature" to "sum angles in face, apply formula to compute face area from spherical excess."Alternatively, I wonder, is there any nice way to generalize Eriksson's formula to apply to non-triangles as well?
Either way, for triangles Eriksson's formula looks more efficient than computing all the angles individually, because there is only one trig function called here.
hongyuchen1030 commentedon Jul 20, 2026
@Sevans711,
These are all great questions. I provide a more detailed discussion in Section 5.10, “Face-level: Spherical face triangulation and face area,” of my paper:
https://egusphere.copernicus.org/preprints/2026/egusphere-2026-636/egusphere-2026-636.pdf
To summarize the main points mentioned in the issue description above:
That said, it is possible to calculate the area of an arbitrarily shaped spherical face, provided that all boundary edges are great-circle arcs, using the pre-quantization method described in Theorems 3.4 and 3.5 of the paper shown below. I am including it in case you are interested. As noted in my paper, these formulas are relatively recent, and their performance and numerical accuracy have not yet been thoroughly benchmarked.
https://epubs.siam.org/doi/full/10.1137/23M1565255
Sevans711 commentedon Jul 20, 2026
Thank you for these extra details!
Ah, I didn't realize it had poor accuracy for these purposes, that is helpful to know.
The paper's notation is a bit difficult for me to parse at a glance, but it is exciting that such formulas exist! If ends up making sense to try implementing them, instead of just using triangulation plus Eriksson's formula, I would be interested to take a closer look. I will keep you posted if that ends up happening, especially if I get stuck on any parts of the paper at that point.
hongyuchen1030 commentedon Jul 20, 2026
It would probably help streamline many things if you could take a look at the paper: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-636/egusphere-2026-636.pdf
This paper provides the most up-to-date overview of spherical geometry algorithms for geoscience regridding. It explains the definition of each geometric object and the recommended algorithms for each step. The goal is for UXarray to eventually implement all of the recommended algorithms from this paper. In other words, this paper serves as UXarray’s algorithmic documentation. And feel free to let me know if you have any problems with the paper.
For the area calculation, I would recommend proceeding with the triangulation + Eriksson formula implementation first. Once we have more time, we can look into the more advanced pre-quantization method.