Courbe de Hilbert

Résumé

La courbe de Hilbert est une courbe continue remplissant un carré. Elle a été décrite pour la première fois par le mathématicien allemand David Hilbert en 1891. Comme elle couvre un carré, sa dimension de Hausdorff et sa dimension topologique sont égales à 2. On la considère cependant comme faisant partie des fractales. La longueur euclidienne de H (la courbe approchée continue obtenue à la n-ième itération) est 2^n - {1 \over 2^n} ; elle croit donc exponentiellement avec n. Pour le parcours des bases de données multi-dimensionnelles, la courbe de Hilbert a été proposée à la place de la courbe de Lebesgue parce qu'elle a un comportement préservant mieux la localité. Construction géométrique Hilbert définit la fonction f:[0,1]\to[0,1]^2 comme limite de fonctions f_n:[0,1]\to[0,1]^2 donnant les approximations successives de la courbe de Hilbert.

À l'étape 0, le graphe H se limite à un seul point, disposé a

Source officielle

https://fr.wikipedia.org/wiki/Courbe_de_Hilbert

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Generating Calligraphic Trajectories with Model Predictive Control

Sylvain Calinon

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

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Interactive Generation of Calligraphic Trajectories from Gaussian Mixtures

Sylvain Calinon

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

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