## Reparamaterization, Regular Curves, and Unit Speed Curves (Pt. II)

**Point of Post: **This is a continuation of this post.

As stated in the motivation there is another naturally desirable type of curve, namely we call a curve *unit speed *if for every . Or, said differently, if . These curves can be thought of as carving it’s image into pieces and move along them at a given speed, moving so long each ‘step’. In other words, they are the continuous analogue of the following situation: measure the length of a hallway, divide the total length by and each step move units.

So, what exactly do regular curves have to do with unit speed curves? Well, it should be clear that every unit speed curve is regular, but in a sense the converse is true. Namely, every regular curve is, up to reparamaterization, unit speed. So, every regular curve is equivalent (in the sense defined above) to a unit speed curve. Why this should intuitively make sense is that if then we should be able to ‘normalize’ it to get a curve who is unit speed. The real proof is:

**Theorem: ***Let be a smooth curve. Then, is regular if and only if it posses a unit speed reparamaterization.*

**Proof: **Assume that is a unit speed curve such that there exists a diffeomorphism such that . We see then that and so, in particular, if then which is impossible since .

So, assume now that is regular. Fix some and let be with respect to . Since for all we know that is a smooth function with the property that for all . We know then is injective and a local diffeomorphism, and so from analysis we know that is an open map and so for some open interval . So, consider the reparamaterization given by . We claim that is unit speed. Indeed:

from where the conclusion follows.

Note that we used the arc length to paramaterize the curve. Thus, often times people will say that, given a curve in (in the sense of a subset of ) that it is ‘paramaterized by arc length’.

We end this post by showing that choosing arc length as our reparamaterization map was necessary, in the sense that every map which reparamaterizes a given regular curve must be very close to the arc length curve. In particular:

**Theorem: ***Let be a smooth regular curve. Then, if is an interval and a diffeomorphism such that is unit speed then for any one has that for some constant and is with respect to . Moreover, .*

**Proof: **We note that

and so which obviously implies that .

*Remark: *Note that this ‘makes ok’ a brushed over fact. Namely, when proving that every regular curve has a unit speed reparamaterization we fixed some point in the interval and measured arc length from that point. In particular, it didn’t matter what point we picked. Thus, the arc length function started from any point will reparamaterize a regular curve to unit speed. What makes this ‘ok’, in the sense that it’s consistent with the above is that fooling around with the starting point of the integral one can get that any two arc length functions (measuring from different points) differ at most by a sign and a constant. In other words, not trying to insult the readers intelligence,

Now, arc length is an intuitively geometric concept. By this, I mean that we are intuitively measuring the arc length of the image of the curves we are dealing with. In particular, if then and so we would expect that arc length from point to (on the curve, think geometrically!) should be the same as the length from to . And, in deed, this is the case:

**Theorem: ***Let a smooth curve and a reparamaterization of with reparamaterization map . Then, picking some one has that the arc length along from to is equal, up to a sign, to the arc length of along to .*

**Proof: **We merely note that

The sign discrepancy can be thought of as saying that the ‘absolute length’ is invariant under reparamaterization, but recall that our definition of arc length ‘signs’ the ‘absolute length’ between two points according to whether you are measuring ‘backwards’ or forwards (i.e. if you are measuring from a point on a curve to a point on the curve with so you are ‘going backwards’) and this relation of backwards to forwards is not preserved under diffeomorphism (i.e. think of curves (e.g. and decreasing diffeomorphisms (e.g. )).

**References:**

1. Carmo, Manfredo Perdigão Do. *Differential Geometry of Curves and Surfaces*. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.

2. Montiel, Sebastián, A. Ros, and Donald G. Babbitt. *Curves and Surfaces*. Providence, RI: American Mathematical Society, 2009. Print.

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