How can arc length start from this point?

[cosmic badger]

I understand the entire process of arc length reparametrization, but how can the point start from (0,8,0)? What t value would that even correspond to? I can't figure out the lower bound of the integration without knowing what t value that point corresponds to.

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[vale vapor]

where is the curve defined, for starters

you need that to calculate arc length

[cosmic badger]

isn't it defined for all t?

[vale vapor]

when you do arclength parametrization you'll have r(s) = (x(s), y(s), z(s)) and you will pick the parameter domain such that the "starting point" is (0,8,0)

[vale vapor]

so it's a closed curve?

[cosmic badger]

im not sure, i can solve all these types of problems when it starts at t = 0, its just the initial point thats confusing me

[vale vapor]

parametrizations are always given such that r(t) = ... , where t in...

[cosmic badger]

yes

but can't i write the arc length as a function of t

[vale vapor]

mhm, as usual integral 0^t r'(s)ds

[cosmic badger]

yeah, but would it start at t = 0?

[vale vapor]

yes

[cosmic badger]

thats the thing t = 0 doesn't pass through the point (0, 8, 0)

oh i see

[vale vapor]

what do you think, is (0,8,0) on the curve in the first place?

[cosmic badger]

hmm, i would say no

[vale vapor]

t=0 seems forced and we have (0,-3,0) instead

[cosmic badger]

yeah exactly

[vale vapor]

so shift the curve

but don't change the curve

what do you do

[cosmic badger]

ok i was thinking something like that

shift 11 down the y axis?

wouldnt z be up

[vale vapor]

just shift 😄

[cosmic badger]

ok so

would i find the reparametrization

if its something like

this for example

just add 11 to the second term would that work?

[vale vapor]

that seems kind of arbitrary

[cosmic badger]

ah sorry then where do i incorporate the shift

how can you do arc length measured from a point thats not on the arc

as you move along the arc do you just apply the distance formula again and again?

[vale vapor]

in current form, the problem is illposed

when you reparametrize, you do it starting from a fixed point t = a (implying, said point is on the curve)

the point (0,8,0) is not on the curve, but the initial curve can be shifted such that this would be true

[cosmic badger]

i see

so can i change the function in the middle to 8t + 8?

in that case the arc length formula would still be the same right

[vale vapor]

yes, because when you differentiate, the constants disappear anyway

[cosmic badger]

that honestly makes a lot of sense

[cosmic badger]

and at the end when i reparametrize t with s, i would have added the 11 back anyways

[vale vapor]

the curve is still the same shape, it's just shifted

[cosmic badger]

so this would work right?

[cosmic badger]

got it! thank you so much

[pastel prism]

@cosmic badger has given 1 rep to @vale vapor

[vale vapor]

lost me there, just calculate the reparametrization from 0 to t as you normally would

[cosmic badger]

thank you man, I think I can piece it together now