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The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal curve space-filling curve first described by the German mathematician David Hilbert in 1891,D. Hilbert: Über die stetige Abbildung einer Linie auf ein Flächenstück. Mathematische Annalen 38 (1891), 459–460. as a variant of the space-filling discovered by Giuseppe Peano in 1890.G.Peano: Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36 (1890), 157–160.
Because it is space-filling, its Hausdorff dimension is 2 (precisely, its image is the unit square, whose dimension is 2 in any definition of dimension; its graph is a compact set Homeomorphism to the closed unit interval, with Hausdorff dimension 1).
The Hilbert curve is constructed as a limit of piecewise linear curves. The length of the th curve is , i.e., the length grows exponentially with , even though each curve is contained in a square with area .
==Images==
Because of this locality property, the Hilbert curve is widely used in computer science. For example, the range of used by computers can be mapped into a picture using the Hilbert curve. Code to generate the image would map from 2D to 1D to find the color of each pixel, and the Hilbert curve is sometimes used because it keeps nearby IP addresses close to each other in the picture. The locality property of the Hilbert curve has also been used to design algorithms for exploring regions with mobile robots and indexing geospatial location data.
In an algorithm called Riemersma dithering, grayscale photographs can be converted to a dithering black-and-white image using thresholding, with the leftover amount from each pixel added to the next pixel along the Hilbert curve. Code to do this would map from 1D to 2D, and the Hilbert curve is sometimes used because it does not create the distracting patterns that would be visible to the eye if the order were simply left to right across each row of pixels. Hilbert curves in higher dimensions are an instance of a generalization of , and are sometimes used for similar purposes, for similar reasons. For multidimensional databases, Hilbert order has been proposed to be used instead of Z order because it has better locality-preserving behavior. For example, Hilbert curves have been used to compress and accelerate R-tree indexesI. Kamel, C. Faloutsos, Hilbert R-tree: An improved R-tree using fractals, in: Proceedings of the 20th International Conference on Very Large Data Bases, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1994, pp. 500–509. (see Hilbert R-tree). They have also been used to help compress data warehouses. (2025). 9783540745525 ISBN 9783540745525
The linear distance of any point along the curve can be converted to coordinates in n dimensions for a given n, and vice versa, using any of several standard mathematical techniques such as Skilling's method. Programming The Hilbert Curve by John Skilling Grant Tebbin: Calculating Hilbert Curve Coordinates
It is possible to implement Hilbert curves efficiently even when the data space does not form a square. Moreover, there are several possible generalizations of Hilbert curves to higher dimensions.H. J. Haverkort, F. van Walderveen, Four-dimensional Hilbert curves for R-trees, in: Proceedings of the Eleventh Workshop on Algorithm Engineering and Experiments, 2009, pp. 63–73.
Here, "F" means "draw forward", "+" means "turn left 90°", "-" means "turn right 90°" (see turtle graphics), and "A" and "B" are ignored during drawing.
The Hilbert Curve is commonly used among rendering images or videos. Common programs such as Blender and Cinema 4D use the Hilbert Curve to trace the objects, and render the scene.
The slicer software used to convert 3D models into toolpaths for a 3D printer typically has the Hilbert curve as an option for an infill pattern.
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