Logic behind definition of Reparametrization

[Raman Choudhary]

What is the intuitive logic behind setting up the definition of reparametrization as being a bijective map and all that(the inverse map being smooth) and not alone that the reparametrisation must give us the same image curve.e.g if we see (t,t^2) as being describing the same curve as (t^3,t^6) but bijective map definition restricts(t^3,t^6) as being the reparametrisation even though physically it(t^3,t^6) describes that same curve??

 

[FactChecker]

Just to clarify your question, are you saying that (t³,t⁶) is not a reparametrization of (t,t²)? It seems like it is.

 

[Raman Choudhary]

Yeah that's the thing it is not a reparametrisation of (t,t^2)...this is why i am asking ??