Sum of small displacement vectors.

[STAii]

Suppose we have an object.
The object is moving in a 3D world.
Now, let S_{t1 - t2} denote the displacement vector from moment t1 to moment t2.
Now, let's say that t0 is the moment of the beginning of the motion, and (tf) is the last moment of movement.
We can split the time from t0 to tf into small bands, each x seconds long.
Now, let's add up the length of the vectors |S_{t0 - t0+x}| + |S_{t0+x - t0+2x}| + |S_{t0+2x - t0+3x}| + ... + |S_{(something) - tf}| = Y
Now, it is obivous (at least for me) that if you make x smaller and smaller (x->0) then the value of Y will get nearer and nearer to the Distance passed by the object.
First of all, am i right ? Secondly (if so), how can it be prooved ?
Thanks !

 

[Hurkyl]

That is (typically) how the arclength is defiend! What you described is just the riemann sum for the line integral

∫|ds|

The only other way of which I know that one could define the arclength of a path is if you can find a differential mapping from [0, t] to your curve such that the derivative is always a unit vector. (intuitively this map would thus preserve length), then the arclength of your curve would be t.

 

[plus]

Of course this approach would not work in certain cases. e.g. tracing the mandlebrot curve, or a similar fractal.