Logic behind definition of Reparametrization

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SUMMARY

The discussion centers on the definition of reparametrization in the context of curves, specifically addressing why a bijective map is required for a valid reparametrization. The participants clarify that while the curves represented by (t, t²) and (t³, t⁶) may visually appear the same, the latter does not qualify as a reparametrization due to the lack of a bijective mapping. This distinction is crucial for maintaining the integrity of the mathematical definition of reparametrization, ensuring that the inverse map remains smooth.

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Raman Choudhary
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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??
 
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Just to clarify your question, are you saying that (t3,t6) is not a reparametrization of (t,t2)? It seems like it is.
 
Yeah that's the thing it is not a reparametrisation of (t,t^2)...this is why i am asking ??
 

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