  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
develop an alternative characterization of a mean that reflects shape variation
of the curves. Based on a geometric representation of the curves through the
Frenet-Serret ordinary differential equations, we introduce a new definition of
mean curvature and mean torsion, as well as mean shape through the notion of
mean vector field. This new formulation of the mean for multi-dimensional
curves allows us to integrate the parameters for the shape features into the
unified functional data modelling framework. We formulate the estimation
problem of the functional parameters in a penalized regression and develop an
efficient algorithm. We demonstrate our approach with both simulated data and
real data examples.
