Let there be a *displacement* vector **r** that is a parametric function of arc time *t* so that **r** = **r**(*t*). Then define *s* as the arc length of **r** so that

*s* = *s*(*t*) = ∫ || **r**′(*τ*) || *dτ*,

where the integral is from 0 to *t*. Let us further assume that *s* is bijective so that the inverse function is *t*(*s*).

Let there also be a *distimement* vector **w** that is a parametric function of arc length *s* so that **w** = **w**(*s*). Then *t* is the arc time of **w** if

*t* = *t*(*s*) = ∫ || **w**′(*σ*) || *dσ*,

where the integral is from 0 to *s*.

Now the arc time derivative of **r**, that is, the derivative of *s* with respect to *t* is

*s†*(*t*) = *ds*/*dt* = || **r***†*(*t*) || = || **v**(*t*) ||,

where **v**(*t*) is the curve’s speed function. And the arc length derivative of **w**, that is, the derivative of *t* with respect to *s* is

*t′*(*s*) = *dt*/*ds* = || **w**′(*s*) || = || **u**(*s*) ||,

where **u**(*s*) is the curve’s pace function.

Next put these equations in terms of a generic variable, *x*, to get

*s′*(*x*) = *ds*/*dx* = || **r**′(*x*) || = || **v**(*x*) || and

*t′*(*x*) = *dt*/*dx* = || **w**′(*x*) || = || **u**(*x*) ||.

From the inverse function theorem we have that

*s′*(*x*) = 1/|| **w**′(*x*) || = 1/|| **u**(*x*) || = 1/*t′*(*x*) and

*t′*(*x*) = 1/|| **r**′(*x*) || = 1/|| **v**(*x*) || = 1/*s†*(*x*).

Putting these together we find

*s′*(*x*) = || **r**′(*x*) || = || **v**(*x*) || = 1/|| **u**(*x*) || = 1/|| **w**′(*x*) || = 1/*t′*(*x*) and

*t′*(*x*) = || **w**′(*x*) || = || **u**(*x*) || = 1/|| **v**(*x*) || = 1/|| **r**′(*x*) || = 1/*s′*(x).

We then have

*s* = *s*(*x*) = ∫ || **r**′(*τ*) || *dτ* = ∫ || **v**(*τ*) || *dτ* = ∫ 1/|| **u**(*σ*) || *dσ* = ∫ 1/|| **w**′(*σ*) || d*σ*,

where the integrals are from 0 to *x*. And also that

*t* = *t*(*s*) = ∫ || **w**′(*σ*) || *dσ* = ∫ || **u**(*σ*) || *dσ* = ∫ 1/|| **v**(*τ*) || *dτ* = ∫ 1/|| **r**′(*τ*) || *dτ*,

where the integrals are from 0 to *x*.

Because of the difficulty of inverting *s*(*x*), this shows a bypass is available. That is, from **r**(*x*) we find

*t*(*x*) = ∫ 1/|| **r**′(*τ*) || *dτ* = ∫ 1/|| **v**(*τ*) || *dτ*.

and from **w**(*x*) we find

*s*(*x*) = ∫ 1/|| **w**′(*σ*) || *dσ* = ∫ 1/|| **u**(*σ*) || *dσ*.

The interpretation is that the space vector, **r**, is a position vector function of the arc time, *t*, and the time vector, **w**, is a time vector function of the arc length, *s*.