Is there any notion for a certain type of embedding of a smooth curve in a 2-d euclidean space? There is a smooth 1-manifold (a smooth curve of infinite arc length) embedded in a 2 dimensional euclidean space. This curve (of infinite arc length) is such that, there is one and only one point $P$ in the 2-d space such that any $\epsilon$ (arbitrarily small) neighbourhood of this point contains the entire curve except some parts of curve whose total arc length is finite. basically the arc length of the curve outside the neighbourhood is finite. I guess there certainly must be some notion in math that describes this type of geometry/topology or this type of embedding. I'd like to know where I can find such things. I am not interested in any other properties of the curve, except this situation. I'd like to know where I could find such a notion. Also if appropriate, I'd like to know similar things for higher dimensions such as a 'surface' embedded like this in a 4-d euclidean space.
I know almost nothing about topics such as topology, geometry or related topics, so please give some advice about where to look for such notions with terminology  that I  probably can understand in some intuitive sense. Also my tags could be inappropriate as I don't know much about it.
 A: "Locally rectifiable" is the nearest two-word term I know.
But more precisely, the situation can be described by saying that the pushforward of the measure $|F'|dx$ under the embedding $F$ is a locally finite measure on $\mathbb R^2\setminus \{P\}$. This pushforward, of course, is just the volume measure on the embedded manifold. 
Let's try to put this in a short sentence: "the volume of embedded manifold is locally finite outside of point P". 
A: Here's a simple example in polar coordinates:
$$[1,\infty)\ni t\mapsto \bigg(\frac{1}{t},t\bigg)\in\mathbb{R}^2.$$  It is the graph of the function $r(\theta) = \frac{1}{\theta}$.
The curve converges to $0$ as $t\to\infty$, so for all $\epsilon>0$ all but a compact interval $[1,T]$ maps into $B(0,\epsilon)$. 
The curve's arc length on $[0,T]$ is
$$\int_0^T\frac{\sqrt{t^2 + 1}}{t^2}dt.$$
Wolfram Alpha gives the indefinite integral as $\sinh^{-1}(t) - \frac{\sqrt{t^2 + 1}}{t} + C$, which diverges to $\infty$ as $t\to\infty$, so the curve has infinite length.
More sophisticatedly, and leaving $\mathbb{R}^2$'s Euclidean structure, take any curve converging to the origin in the punctured disk with its (unique) hyperbolic structure.  This is the prototype of a hyperbolic cusp: the (deleted) origin is infinitely far, and all but finitely much of the curve's length is .
Intuitively, it should not be difficult to cook up examples like this in the smooth category.  There is no notion of distance, so one can construct smooth objects as "stretched" or as "squeezed" as one likes, tailor-made to fit whatever notion of "distance" one has at hand.
