Search results
Results From The WOW.Com Content Network
Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. [1] 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 ...
First six iterations of the Hilbert curve. The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, [1] as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890.
A fourth-stage Gosper curve The line from the red to the green point shows a single step of the Gosper curve construction. The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve [1] and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7.
Peano curve: And a family of curves built in a similar way, such as the Wunderlich curves. 2: Moore curve: Can be extended in 3 dimensions. 2: Lebesgue curve or z-order curve: Unlike the previous ones this space-filling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the Hilbert Curve it can be extended in 3D.
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [5] [6] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [7] or they can be used to build a pointer based quadtree.
Mandelbrot shows how to calculate the Hausdorff dimension of each of these curves, each of which has a dimension D between 1 and 2 (he also mentions but does not give a construction for the space-filling Peano curve, which has a dimension exactly 2).
Variant of the Peano curve with the middle line erased creates a Sierpiński carpet. The area of the carpet is zero (in standard Lebesgue measure). Proof: Denote as a i the area of iteration i. Then a i + 1 = 8 / 9 a i. So a i = ( 8 / 9 ) i, which tends to 0 as i goes to infinity. The interior of the carpet is empty.