  A time series is a sequence of data items; typical examples are streams of
temperature measurements, stock ticker data, or gestures recorded with modern
virtual reality motion controllers. Quite some research has been devoted to
comparing and indexing time series. Especially, when the comparison should not
be affected by time warping, the ubiquitous Dynamic Time Warping distance
function ($\texttt{DTW}$) is one of the most analyzed time series distance
functions. The Dog-Keeper distance ($\texttt{DK}$) is another example for a
distance function on time series which is truely invariant under time warping.
  For many application scenarios (e.$\,$g. motion gesture recognition in
virtual reality), the invariance under isometric spatial transformations
(i.$\,$e. rotation, translation, and mirroring) is as important as the
invariance under time warping. Distance functions on time series which are
invariant under isometric transformations can be seen as measurements for the
congruency of two time series. The congruence distance ($\texttt{CD}$) is an
example for such a distance function. However, it is very hard to compute and
it is not invariant under time warpings.
  In this work, we are taking one step towards developing a feasable distance
function which is invariant under isometric spatial transformations and time
warping: We develop four approximations for $\texttt{CD}$. Two of these even
satisfy the triangle inequality and can thus be used with metric indexing
structures. We show that all approximations serve as a lower bound to
$\texttt{CD}$. Our evaluation shows that they achieve remarkable tightness
while providing a speedup of more than two orders of magnitude to the
congruence distance.
