Abstract
<title>Abstract</title> <p>Online handwriting recognition systems typically represent pentrajectories through fixed-length Euclidean shape descriptors thatcapture the spatial outline of each stroke, but are insensitive to theorder in which that outline is produced.Two strokes that trace the same region of the plane in oppositedirections are indistinguishable to any such order-blind representation,yet their traversal directions may carry decisive class information incharacters where loop orientation and stroke sequencing matter.This paper introduces a Heisenberg-lift framework that addresses thisgap through a compact, interpretable, order-sensitive augmentation foronline pen-trajectory features.The simplest instance is the terminal signed area, a singleparameter-free scalar appended to an existing Euclidean descriptor atnegligible computational cost.Evaluated on two standard online handwriting benchmarks, this one-scalaraddition, consistently raises classifier accuracy over the Euclideanbaseline.On the hardest character pair in our study, the letters o and y, thesigned area alone achieves perfect separation while the Euclideanbaseline falls short.The advantage grows further under additive coordinate noise, apractically relevant degradation in pen-trajectory data.A richer fifteen-dimensional extension, derived from a noncommutativeHeisenberg-group subdivision scheme, provides additional gains in noisyand loop-structured conditions.Dimension-matched statistical controls confirm that all improvementsreflect geometric information rather than feature-count inflation.The resulting descriptor is lightweight, closed-form, and directlyinterpretable, making it a practical augmentation for online handwritingand related document-trajectory classification pipelines in which thedirection of stroke execution carries discriminative information.</p>