Local response (concept only)
A single localized Gaussian patch: not necessarily C₂ symmetric, and not the R-ODD operator.
Geometry · perception · reproducible experiments
C₂ Visual Research Lab
Turn one drawing by 180°, separate what stays invariant from what changes, and inspect why geometric similarity does not guarantee two readable characters.
KanjiVG stroke paths · Japanese sourceExact C₂ algebra · finite-grid renderingNo human readability claim
01 / C₂
What survives a half-turn?
The same actual KanjiVG path drawing appears in three synchronized views. Move the rotation centre or interpolate between its original, C₂-even and 180°-rotated forms.
+1 = source; 0 = even component; −1 = 180° rotation.
Half the sum of the source and its half-turn.
Warm = positive; blue = negative differences.
PositiveNegative
—Binary rotational IoU (source vs turned)
—Normalized asymmetric ink
—Ink clipped by canvas
R²g = g | g₊ = (g + Rg)/2 | g₋ = (g − Rg)/2 | g = g₊ + g₋ | Rg = g₊ − g₋
The algebra is exact on the unbounded plane; the interactive image uses a padded 145×145 sampled raster. Rotation and pixel coverage are finite-grid approximations. Moving the centre may change overlap or clip edge ink. The even field is not a proof of a readable ambigram.
02 / PRC geometry
Why preserving the historical renderer matters
Move a *synthetic* measured median-to-contour radius. The historical v1.16 renderer divided that radius by two again; the corrected constructor uses the measured half-width. This schematic is not a licensed original outline.
0.4743Historical median source-outline IoU
0.9083Corrected median source-outline IoU
32/32Source-outline cases improved
Values are published geometric results from 32 pinned cases (PR #19), not values computed from the synthetic bands above and not measures of reading. The historical v1.16 code and reproduction remain unchanged.
03 / A/B fusion
Matching strokes is not the same as changing a drawing
In a frozen experiment of 32 new character pairs, relaxing the registration threshold found more candidate matches but no final accepted pixel changes. The bars show observed pairs with at least one stroke match, not numbers of readable ambigrams.
—New pairs with at least one match
0/32New pairs with changed raster pixels
This is a documented negative result, not a working A/B ambigram generator. At threshold 16, 31/32 new pairs had a stroke match but 0/32 produced changed output. The original acceptance rule returned the union baseline. Human readability was not tested.
Original two-pair forensic diagnosis · Independent 32-pair study (draft PR #21)
04 / research hypotheses
Local compensation and even harmonic fields
Two mathematical sketches of spatial thickness fields, not outputs of the R-ODD or harmonic construction algorithms. They illustrate the difference between a localized adjustment and a field deliberately invariant under 180° rotation.
C₂-even harmonic field (concept only)
An even cosine-product field, f(−x,−y)=f(x,y). Not an optimized stroke result.
PRs #14 (R-ODD) and #15 (harmonic) are separate, unmerged experiments. This visualization demonstrates an underlying distinction without claiming that either operator was experimentally superior.
05 / perception
A geometric match is a candidate, not a readable ambigram
There are three separate observations: source shapes can be close under half-turn, a constructor can change the drawing, and human readers may or may not identify both intended characters. The measurements on this page demonstrate geometry; they never substitute for a blinded reading study.