SAME GEOMETRIC INTENT · DIFFERENT DISCRETE DATA

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Keep the transfer visible.

Begin with two tangent components at every vertex. Integrate them along edges, reconstruct one constant vector per face, then average those face vectors back to vertices. The pictures can resemble each other while their degrees of freedom and exact identities do not.

NATIVE VERTEX FIELDtwo tangent components at each mesh vertex

—

positive / source negative / curl reconstructedarrows live at vertices

displayed DOFs: —
triangle curl RMS: —
face fit residual: —
round-trip RMS: —

MANUFACTURED INPUT

Vertex fieldchoose an answer with known curl Grid resolutionvertices along one side9 × 9 Overlay the round tripgold = face averages transferred back to vertices

NATIVE UNKNOWN

Vertex tangent vectors

uᵢ ∈ TᵢM ≅ ℝ²

one value means
curl lives
do not assume

Read the transfer, not only the arrows

uᵢ → αᵢⱼ
Endpoint trapezoids integrate a vertex vector along an oriented edge.
α|∂f → w_f
A two-variable least-squares fit reconstructs a constant face vector. Nonzero circulation is exactly the incompatible third equation.
w_f → ũᵢ
Area averaging returns display arrows to vertices, but does not invert the earlier transfers.

WHY THIS IS A BREAKOUT

The course spine can remain vertex-first.

Students only need this comparison once: to understand why a reconstructed vertex picture is not a native vertex discretization. Then the live objective can keep one unknown association, one circulation operator, and one research question in view.

Continue with native vertex unknownsDrag handles and change the energy →Go down to forms and operatorsMove through d and the Hodge star →Read the representation source trailCourse notes, connections, and Directional →

Switch the address.Keep the transfer visible.

The course spine can remain vertex-first.

Switch the address.
Keep the transfer visible.