Visualizing 1Forms without using Arrows

[lukem12345]

Currently, there are two strategies for visualizing 1Forms on a spherical mesh.

1. Render the divergence of the 1Form.
2. Render the 1Form as a collection of arrows

We would like a better method wherein we visualize 1Forms as colored triangles. The color along an edge will be the color of the corresponding value in the EForm. Interior colors will be linearly interpolated.

This is similar to Gouraud shading (except there, the values that inform interpolation are given at the vertices).

[lukem12345]

@jpfairbanks The new academic semester is about to start. This seems like a great opportunity to get a CS undergrad interested in computer graphics involved in research? I could imagine a slew of similar helpful visualization methods for 0-forms, 1-forms, and 2-forms on 2-manifolds. The new capability to do 3D sims presents new opportunities as well.

[jpfairbanks]

That is a great idea, can you reach out to Jorg and see if he has anyone in mind?

[lukem12345]

The concept that I was referring to with “bilinear Gouraud shading but defined on edges” can also be understood as Whitney maps. This feature is generally useful, not only for visualization purposes.

https://www.sciencedirect.com/science/article/pii/S0377042721001394

See also Chapter IV, Section 27 of Whitney’s GIT.