Task: I have the two given curve, in different intervals for t:

$x_1(t)=t$ , $y_1(t)=t$

$x_2(t)=t^5$ , $y_2(t)=t^5$

for $t ∈[-1,0]$


$x_1(t)=2t$ , $y_1(t)=2t$

$x_2(t)=16t^4$ , $y_2(t)=16t^4$

for $t ∈[0,1/2]$

I can think of many different ways to approach this task. Maybe the easiest is to substitute with $u=t^5$ in the first and $p=8t^4$ in the second. I don't know if that is enough. Is it?

But let us say that the task was not this easy to subsitute, for instance if I have two parametrizations for a circle, one with cos and sin and the other one with square roots or something like that.

I can look at the derivative (in my case: monotonically increasing), but it is not easy to determine if the curve follows the same path. I can also look at the start and end point, and the length of a vector with same vector function. Yet, I do not see how: (1) start and end point (2) derivative (3) length of the vector if the parametrization is a vector function, can give an exact answer. It is very difficult to see these kinds of things.

Summary of my quiestions: (1) Is substitution enough to answer the task. (2) How can I use (1) start and end point (2) derivative (3) length of the vector (if vector function) to give a complete answer?



Some helps, (1) the sustitution (exp. $u=t^5$) must be a diffeomorphism, (2) start point and end point (if I understand your question) depends at least on the orientation of the two curves.


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