Question

Can anyone explain me what different between "reparametrized a curve r(t) (2t,1-3t,5+4t) with respect to arclength measured from the point (1,0,0) in the direction of increasing t " and "reparametrized a curve r(t) (2t,1-3t,5+4t) with respect to arclength measured from the point where t=0 in the direction of increasing t"? and how to get forward and backward between them. I mean if I get point, how to convert to t, and if i have t, how to figure out the point

Answer 1

In the first case, we now r(t) satisfies the point (1,0,0) thus we need to find a time to such that r(t) = (1,0,0). On the second case, we know that we start with t=0.

What I can't figure out though is there is no such t that satisfies r(t) = (1,0,0). The fact that it satisfies must mean 2(t) = 1 and 1-3(t) = 0. Solving the first gives t = 1/2 which does not satisfy the second equation.