What is meant by "a curve $r(t)$ of constant length $|r(t)|$"? I am working on an assignment for Calculus III. This particular one has to do with some vector calculus, but I am caught on some wording by my professor and he is not holding office hours at the moment.
In a few questions he asks us to show certain properties of "a curve $r(t)$ with constant length $|r(t)|$." I have been operating under the assumption that this means any curve with a scalar as its length, such as $\langle 0,0,t\rangle$ over $0\le t\le 1$, but am now questioning that assumption, as he asks us to show a property is true "for all of $t$."
Does anyone have any ideas what he means here? I have been working these by creating examples with small bounds such as the one above for simple integration for the lengths.
 A: First off, a couple of nit-picks:

*

*The function $r$ is a curve in $\mathbb{R}^3$.  That means that it has the form $$ r : \mathbb{R} \to \mathbb{R}^3 : t \mapsto (x(t), y(t), z(t)), $$
where $x(t)$, $y(t)$, and $z(t)$ denote the $x$-, $y$-, and $z$-coordinates of some object at some time $t$.  That is, $r(t)$ is a point in space.  This may be a subtle point, but I think that it is wise to regard the the value of $r$ at a time $t$ to be a point in three-dimensional space, rather than a three-dimensional vector (some books call this a "position vector", but I'm not a huge fan of that terminology, either).  The codomain of $r$ is (presumably) endowed with a smooth structure, but the additional vector space structure which $\mathbb{R}^3$ possesses is largely irrelevant to the material in an introductory calculus class, and is nonexistent in a more general setting (i.e. the study of smooth manifolds).


*I am not a fan of the description of $r$, which is "a curve with constant length".  The length of a curve typically refers to the distance that a particle traverses while traveling along that curve.  Instead, what is being described here is something else about the geometry of the curve.  Specifically, there is some fixed number $\varrho>0$ such that for all $t$,
$$ |r(t)| = d(0,r(t)) = \rho. \tag{$\ast$}\label{dist}$$
This is not a curve of constant length, but rather a curve where each point on the curve is some fixed distance from the origin.  I think that the notation in (\ref{dist}) is quite clear, but if one wants a description of this identity in English, I might say "$r$ is a curve on a sphere" or "$r$ is a curve such that each point on $r$ is a fixed distance $\varrho$ from the origin."


*The term "curve" is a little overloaded.  We actually typically mean two different things by the word:  first, a curve is the image of an interval with respect to a continuous function, i.e. the "curve" $r$ is, as a set,
$$ \{ r(t) : t \in \mathbb{R} \}. $$
On the other hand, we also thing of the function itself as a curve, i.e. a "parameterized curve"—the curve consists not only of the set described above, but also the parameterization which tells you which point in the set corresponds to which moment in time.  That is, a "curve" can be seen both as a static object and as a dynamic object.  It might be worth noting that there may be many functions which correspond to the same static curve, but each corresponds to a different dynamic.  For example
$$ r_t(t) = \left( \cos(t), \sin(t), 0 \right) \ (t\in [0,2\pi])
\quad\text{and}\quad
r_2(t) = \left( \cos(-t), \sin(-t), 0 \right) \ (t\in [0,2\pi]) $$
are identical as sets, but have different dynamics.  These are distinct parameterizations of the same set.
This is, perhaps, a rather minor point, but I think that it is worth noting when one word is used to simultaneously represent two distinct, though related, ideas.
As I said, I think that the identity in (\ref{dist}) explains what is going on very clearly.  However, if more intuition is required, imagine the following scenario:
Grab a sphere off of your shelf (you have several, right?).  Declare that the center of this sphere is the origin and pick out some preferred coordinates (that is, mark out three coordinate axes).  Next, find an ant or other small insect, then dip this ant in paint (you monster), and drop it on the sphere.
From the moment you place the ant on the surface of the sphere, keep track of the ant's position.  After $t$ seconds have elapsed, the ant's position is denoted by $r(t)$—that is, $r$ is a function which takes a real number as input (a time) and spits out a position in three-dimensional space as an output (the ant's position in space).
Because the ant is on the surface of the sphere, its distance from the center of the sphere (the origin) is constant.  Therefore
$$ |r(t)| = \text{ the distance from center of the sphere to position of the ant at time $t$} = \text{constant}. $$
Now, remember that you dipped the ant in paint (you sadist) before you put it on the sphere, so it has been painting a trail of paint on the sphere which corresponds to the path that it has taken.  This path, which we may regard as the static curve described by the ant's position, is a curve with the property that each point is a fixed, constant distance from the origin.
