Evaluate vtu data at arbitrary points with Python scripting (especially for higher order Lagrange elements)

I have a .vtu file for which finite element data is written on an unstructured grid in vtk’s xml format. Using the python scripting capability, I’d like to read data at a set of arbitrary points, being interpolated by the Lagrange elements which are included in the data (as I understand it, one can write arbitrary lagrange elements to .vtu files).

I suspect that this is actually easy, but I have no idea where to look for it in the documentation.