by Luis Eduardo Muñoz Guerrero
ABSTRACT
The distortion of Ptolemy’s coordinates has been explained by transformation units, groups of places sharing
a scale and a shift, and by gradual distortion across a map. This study tests whether blocks with their own
similarity transformation predict held-out places better than a smooth field. The catalogue was linked to the
Pleiades gazetteer without Ptolemy’s coordinates, and half the entries were sealed behind a hash commitment
and a frozen plan. Segmenting a map into similarity units, a strongly NP-hard minimum-cost cover with
outliers, was solved by column generation and integer programming with a certified interval for the optimum.
Units, a moving-least-squares field and one frame per map were fitted on the exploration half of 16 maps and
compared on sealed places by an exact randomization test. On 897 sealed places, the smooth field predicted
better (pooled median difference −6.1 km; 95% interval −10.3 to −3.3; p = 0.006), and units beat one frame per
map by 16.9 km. Optimality was certified in 8 maps. For identifiable places, a smooth field describes the
distortion better than similarity units, by a margin close to the rounding of the coordinates; the published
units, with shared scales, were not tested.
