Exploring Moduli of Tori: Isotopies and Area Distortion

Here we view different geometric structures on a torus. One aspect of this change is the metric distortion — how much the surface area of small neighborhoods stretches or shrinks during the transformation.

The color mapping uses coolwarm centered at 1.0. This means if a face has the same area as the baseline, it will appear grey/white. If it’s larger, it turns red; if smaller, it turns blue. The histogram on the right quantifies this distortion across all faces, showing the distribution of area change.

Note: The drift of the histogram away from 1.0 quantifies how much the geometry is stretching.

Min Ratio 1.000
Mean Ratio 1.000
Max Ratio 1.000

Geometric Distortion

Top: histogram representing the distribution of area change across all faces. An area-preserving map would result in a single spike at 1.0

Bottom: distribution of area change across the current surface

A tibble: 6 × 8

file r_idx r_idx_2 r_major r_minor mean max min 1 R0_r0.glb 0 0 5 1 0.169 0.190 0.154 2 R0_r1.glb 0 1 5 1.13 0.191 0.209 0.178 3 R0_r2.glb 0 2 5 1.27 0.213 0.225 0.204 4 R0_r3.glb 0 3 5 1.4 0.234 0.240 0.230 5 R0_r4.glb 0 4 5 1.53 0.255 0.257 0.253 6 R0_r5.glb 0 5 5 1.67 0.276 0.285 0.265