Nov 18, 2018 · In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same ...

Since peano curves, as space filling curves, occupy all the area of space, let us say, $[0,1]\times[0,1]$, do they have area 1 square unit? They should have. However, Whether I take limit, or anything, how can I possible get the area component, all I get is length only (i dont exactly get the length but I hope you get it )?

May 6, 2018 · By this definition, $\gamma$ can easily be rectifyable without being differentiable. However, if $\gamma$ is continously differentiable, one readily sees that $(1)$ turns out to be $$\int_0^1\sqrt{1+|\gamma'(t)^2}\,\mathrm dt. $$ For the highly non-differentiable Peano curve, we'd have to go back to $(1)$ of course. It is easy to see that the ...

The construction of the Peano curve is a recursive refinement of a particular path that discretely looks one-to-one, in that it touches every coordinate point at a given scale in a bijection. In the limit there's no bijection, but at every step there is a bijection between the curve so far and the coordinates of points within $[0,1]^2 ...

So one can conclude that the topological dimension of the Peano curve must be one. But how could it be? I thought that in this context when we say curve, we mean the image of the curve. But the image of the Peano curve is $[0,1]^2$ and the topological dimension of $[0,1]^2$ is two.

Jul 14, 2018 · Peano curve is defined by iteration, while each step would split the tiny square in last step into 9, then connect the centres in a particular sequence. By doing so, one might argue, that each segment at any step , has the two ends as rational numbers.

Jul 20, 2016 · $\begingroup$ The Peano curve (more precisely, the sequence of curves converging to it) is typically described in terms of its image, rather than the actual function that yields the image. In other words, even the approximating (non-space-filling) curves don't usually come with a specific formula that tells you what the image of particular ...

It is well-known that there are continuous curves $f:I \to \mathbb R^2$ (where $I \subset \mathbb R$ is an interval) whose image have positive measure (e.g Peano curve).

Jun 5, 2018 · You can also get singular measures by composing the Cantor-Lebesgue function with Hilbert/Peano curve. A good reference for these facts is "Space filling curves" by Hans Sagan. $\endgroup$ – hot_queen

From Wikipedia, a space-filling curve is a curve (i.e. a continuous function whose domain is the unit interval $[0,1]$) whose range contains the entire 2-dimensional unit square. Many examples of space-filling curves are known, such as the Peano curve. and the Hilbert curve (both curves are the limiting curves of the iteration seen in the ...