How to cut a part out of a path?

Let us assume I have a path that is mathematically parametrized by a parameter t in [0,T]. 

How do I take only a sub path of this bath, parametrized by t in [a,b] where a => 0 and b <= T ?

Does do I get the reduced object when I have swept an object along the path t in [0,T]?

In both situations, one can either take a sub path or delete the other parts of the path.