please explain in detail the arc length reparametrization (Arc length reparametr
ID: 3374381 • Letter: P
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please explain in detail the arc length reparametrization
(Arc length reparametrization.) Let be a curve in space. Define the function of t Assume that the velocity of c does not vanish at any time. Show that .s is a strictly increasing function of t. If L is the total length of ft show that .s takes the interval [a, b] to the interval [0,L] in an one-to-one manner. If .s-1 : [0,L] rightarrow [a, b] is the inverse function of s, explain why is another way of describing the same curve (i.e. a re-parametrization of ). Show that the length of the curve from its beginning to the point is alpha.Explanation / Answer
(a) Since You are integrating the magnitude of c ' (t), and the magnitude is always positive , means that integrating it will always return a positive value. Now on increasing the time, the upper limit of the integration will always increase and hence s(t) will also increase
HEnce s(t) is increasing function of time
(b) L is the total length of c. which means s(b) = L
THis portion will be alittle to hard. s(t) is the length of c(t) between time values 0 and t. since t belongs to [a,b] and and L is the total length of c , hence it follows that s(b) = L
also s(a) = 0 as both the limits of integration become same
Hence s(t) transform (a,b) to (0,L). Now since s(t) is an increasing function of time.
So s takes the intervak [a,b] to [0,L] in a one to one manner.
(c) s inverse will give back the time for the length of the curve. that is s (inv) (a) = t means at time t. the arc length is a
so c (t) will again give the same curve. Note that had s not been a one to one function, this would not have been valid
d) Total length of curve ,
the final value of alpha is L, initial value is 0
total length is L which is again the total length of the curve (hence proved)
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