  The analysis of curves has been routinely dealt with using tools from
functional data analysis. However its extension to multi-dimensional curves
poses a new challenge due to its inherent geometric features that are difficult
to capture with the classical approaches that rely on linear approximations. We
propose a new framework for functional data as multidimensional curves that
allows us to extract geometrical features from noisy data. We define a mean
through measuring shape variation of the curves. The notion of shape has been
used in functional data analysis somewhat intuitively to find a common pattern
in one dimensional curves. As a generalization, we directly utilize a geometric
representation of the curves through the Frenet-Serret ordinary differential
equations and introduce a new definition of mean curvature and mean shape
through the mean ordinary differential equation. We formulate the estimation
problem in a penalized regression and develop an efficient algorithm. We
demonstrate our approach with both simulated data and a real data example.
