The text we are using for Calculus III introduces the notion of unit tangent vector, principal unit normal vector, and curvature, for smooth curves , and it also mentions that circles and lines are planar curves of constant curvature. (A curve is planar if its image is contained in a plane, i.e., if it describes a two-dimensional trajectory.)
Surprisingly, though, the text does not explain (not even in the exercises) that circles and lines are the only smooth planar curves of constant curvature. The argument for this is simple enough, so I will show it here:
Recall that if is a smooth curve, we set
where denotes arc length. We can define the curvature of as
Assume that is constantly equal to zero. Then , so is constant, so for some constant vector . This is the parametric equation of a line, so we have shown that the only curves with zero curvature are straight lines. Notice that for this we did not need to assume that is a planar curve. From now on we may assume that this is the case and that is constant and different from zero.
In terms of , we can equivalently compute as
We then set to be the unit vector in the direction of , so
or, equivalently in terms of ,
so . Recall that , since is constant, so its derivative is zero.
Since is also constant, again we have that . Since is also perpendicular to and we are assuming that moves on a plane, this means that is a multiple of ,
where is some scalar function. To compute , notice that . But also , so , or . Since , we finally find that
Recall also that the osculating circle to the curve at the point is the circle of the same curvature as at and centered in the direction of the vector at , so its center is given by
since the curvature of a circle is the inverse of its radius. In particular, since is constant, all the osculating circles to have the same radius. Now we prove that is constant (from the assumptions that is planar and is constant). To see this, compute to find
But this completes the proof, because then , since , i.e., the curve lies on a circle of radius .
On the other hand, there are curves of constant curvature other than lines and circles, if the curve is not confined to a plane. The helix is an example.