= Arc Length =

For a given [[Calculus/ParametricEquation|parametric equation]] as a position function, '''arc length''' is speed. It can be evaluated for a given time as an arc length.

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== Description ==

For a position function expressed as a [[Calculus/ParametricEquation|parametric equation]] like ''r(t) = f(t)i + g(t)j + h(t)k'', it is well understood that the first derivative is velocity and the second derivative is acceleration.

The speed of movement along the function is given by:

{{attachment:speed.svg}}

For a given time, the speed function evaluates to the distance traveled by the original function in that time: arc length.

Note that notation is not standardized. Sometimes it uses a curly ''v'', to indicate its relation to the velocity vector (i.e., ''v(t) = r'(t)''). ''a'' is already reserved for the acceleration vector (i.e., ''a(t) = v'(t) = r'``'(t)'').

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== Arc Length Parameterization ==

In some cases, it is useful to re-parameterize a [[Calculus/ParametricEquation|parametric equation]] in terms of the arc length. For example, it can be useful to calculate the tangent vector for a given distance along an arc rather than for a given time.

This is accomplished by rewriting the above function of ''t''. Substitute ''t'' with ''du'', and [[Calculus/Integral|integrate]] arc length from time 0 to time ''t'' with respect to ''u''.

{{attachment:par.svg}}

This can be simplified down to a statement like ''s(t) = α du'', which can of course be rewritten as ''s = α t'', but more importantly can be rewritten as ''t = s/α''. This last formulation can be substituted back into ''r(t)'' to give ''r(s)'', the '''arc length parameterization'''.



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