  We provide statistical analysis methods for samples of curves when the image
but not the parametrisation of the curves is of interest. A parametrisation
invariant analysis can be based on the elastic distance of the curves modulo
warping, but existing methods have limitations in common realistic settings
where curves are irregularly and potentially sparsely observed. We provide
methods and algorithms to approximate the elastic distance for such curves via
interpreting them as polygons. Moreover, we propose to use spline curves for
modelling smooth or polygonal Fr\'echet means of open or closed curves with
respect to the elastic distance and show identifiability of the spline model
modulo warping. We illustrate the use of our methods for elastic mean and
distance computation by application to two datasets. The first application
clusters sparsely sampled GPS tracks based on the elastic distance and computes
smooth means for each cluster to find new paths on Tempelhof field in Berlin.
The second classifies irregularly sampled handwritten spirals of Parkinson's
patients and controls based on the elastic distance to a mean spiral curve
computed using our approach. All developed methods are implemented in the
\texttt{R}-package \texttt{elasdics} and evaluated in simulations.
