Thread: Curve with no length

Hi!

I am asked to find some curves which don't have length. I'm not sure whether the Koch curve is appropriate as it has infinite length.

Thanks for your help!

You could also look at "space-filling curves". But really they're just other (maybe we can say "worse") examples of curves with infinite length. The term is "non-rectifiable".

Since length is defined as the integral of the magnitude of the derivative, any curve which fails to be differentiable on a set of positive measure won't have a defined length.

Originally Posted by james_bond

Hi!

I am asked to find some curves which don't have length. I'm not sure whether the Koch curve is appropriate as it has infinite length.

Thanks for your help!

Originally Posted by Tinyboss

You could also look at "space-filling curves". But really they're just other (maybe we can say "worse") examples of curves with infinite length. The term is "non-rectifiable".

Since length is defined as the integral of the magnitude of the derivative, any curve which fails to be differentiable on a set of positive measure won't have a defined length.

The usal example of a function that is nowhere differentiable is the Blancmange function

CB