# When we use arc length as a parameter

If we have a fly in a room, its position respect to some frame of reference will change with time, so if we want to describe the fly's movement with a parametrized curve, it is easy to see the convenience of taking time as the parameter.

I read that we can also take the length of the curve as a parameter and it is not dificult to follow formally the equations by which we put the original parameter -time- in terms of this new parameter. My question is: when or why is better to work with arc length as parameter, instead of time?

arildno
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From a geometrical point of view, it is simpler to use the arc length parameter in order to understand the concepts of curvature and torsion, for example.

Curvature of a curve is simply the rate of (locally planar) turning of the unit tangent when working with the arc length parameter.

HallsofIvy
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If we use arclength, s, as parameter, then the derivative, dr(s)/dx is the unit tangent vector. From that it follows that the second derivative, d2r(s)ds2, is the normal vector. With any other parameter, the derivative is tangent but not of unit length and the second derivative is not normal to the curve. Also, with arclength as parameter the length of the second derivative, |d2r(s)ds2| is the curvature of the graph.