Curves With Infinite Length
As mentioned above, some curves are non-rectifiable, that is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to "measure" the size of such curves.
Read more about this topic: Arc Length
Famous quotes containing the words curves, infinite and/or length:
“At the end of every diet, the path curves back toward the trough.”
—Mason Cooley (b. 1927)
“What is man, when you come to think upon him, but a minutely set, ingenious machine for turning, with infinite artfulness, the red wine of Shiraz into urine?”
—Isak Dinesen [Karen Blixen] (1885–1962)
“It is the vice of our public speaking that it has not abandonment. Somewhere, not only every orator but every man should let out all the length of all the reins; should find or make a frank and hearty expression of what force and meaning is in him.”
—Ralph Waldo Emerson (1803–1882)