Hilbert curve

Summary

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, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. 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 homeomorphic to the closed unit interval, with Hausdorff dimension 2). The Hilbert curve is constructed as a limit of piecewise linear curves. The length of the nth curve is \textstyle 2^n - {1 \over 2^n} , i.e., the length grows exponentially with n, even though each curve is contained in a square with area 1. Images File:Hilbert curve 1.svg|Hilbert curve, first order File:Hilbert curve 2.svg|Hilbert curves, first and second orders File:Hilbert curve 3.svg|Hilbert curves, first to third

Official source

https://en.wikipedia.org/wiki/Hilbert_curve

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We describe a methodology for the interactive definition of curves and motion paths using a stochastic formulation of optimal control. We demonstrate how the same optimization framework can be used in different ways to generate curves and traces that are geometrically and dynamically similar to the ones that can be seen in art forms such as calligraphy or graffiti art. The method provides a probabilistic description of trajectories that can be edited similarly to the control polygon typically used in the popular spline based methods. Furthermore, it also encapsulates movement kinematics, deformations and multivariate coordination. The user is then provided with a simple interactive interface that can generate multiple movements and traces at once, by visually defining a distribution of trajectories rather than a single one. The input to our method is a sparse sequence of targets defined as multivariate Gaussians. The output is a dynamical system generating curves that are natural looking and reflect the kinematics of a movement, similar to that produced by human drawing or writing.

2017

The chapter presents an approach for the interactive definition of curves and motion paths based on Gaussian mixture model (GMM) and optimal control. The input of our method is a mixture of multivariate Gaussians defined by the user, whose centers define a sparse sequence of keypoints, and whose covariances define the precision required to pass through these keypoints. The output is a dynamical system generating curves that are natural looking and reflect the kinematics of a movement, similar to that produced by human drawing or writing. In particular, the stochastic nature of the GMM combined with optimal control is exploited to generate paths with natural variations, which are defined by the user within a simple interactive interface. Several properties of the Gaussian mixture are exploited in this application. First, there is a direct link between multivariate Gaussian distributions and optimal control formulations based on quadratic objective functions (linear quadratic tracking), which is exploited to extend the GMM representation to a controller. We then exploit the option of tying the covariances in the GMM to modulate the style of the calligraphic trajectories. The approach is tested to generate curves and traces that are geometrically and dynamically similar to the ones that can be seen in art forms such as calligraphy or graffiti art.

Springer2019