At present, mobile computation is developing rapidly for location-based services. For the online GIS or cartography of
moving objects, the techniques of progressively refined details greatly reduce the overhead of storage, computation, display,
and communication resources. Again, human cognition about the reality is made at multi-scales of abstraction. Multi-scale
representations of the moving object trajectory are required upon human hierarchical cognition and progressively refined
details modeling. To this end, wavelet-based multi-scale representations of the motion trajectory are presented in this paper.
We consider parametric motion and a sequence of spatio-temporal coordinates as database representations of the motion
trajectory. Correspondingly, multi-scale wavelets representations are considered as computing representations of the motion
trajectory. The wavelet transform is applied to time-series of spatial coordinates for finding dramatic or gradual changes of
motion speed at each scale. Multi-scale wavelets representations of the motion trajectory have revealed global motion trends
at the large scale and local motion details at the small scale. Inherently, spatio-temporal coordinates, motion, and dynamics
are three-scale representations of spatio-temporal semantics. Multi-scale representations of motion trajectory are
experimentally illustrated. Our work is devoted not only to mobile computation but also to extension of wavelet analysis into
geometry data processing.