Lengths of Curves
If is a metric space with metric, then we can define the length of a curve by
where the sup is over all and all partitions of .
A rectifiable curve is a curve with finite length. A parametrization of is called natural (or unit speed or parametrised by arc length) if for any, in, we have
If is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of at as
and then
In particular, if is an Euclidean space and is differentiable then
Read more about this topic: Curve
Famous quotes containing the words lengths and/or curves:
“The minister’s wife looked out of the window at that moment, and seeing a man who was not sure that the Pope was Antichrist, emptied over his head a pot full of..., which shows to what lengths ladies are driven by religious zeal.”
—Voltaire [François Marie Arouet] (1694–1778)
“For a hundred and fifty years, in the pasture of dead horses,
roots of pine trees pushed through the pale curves of your ribs,
yellow blossoms flourished above you in autumn, and in winter
frost heaved your bones in the ground—old toilers, soil makers:
O Roger, Mackerel, Riley, Ned, Nellie, Chester, Lady Ghost.”
—Donald Hall (b. 1928)