Orthogonalité

Résumé

En géométrie classique, l'orthogonalité est une propriété liée à l'existence d'un angle droit (orthos = droit, gônia = angle). Dans l'espace, deux droites sont orthogonales si elles sont chacune parallèles à des droites se coupant en angle droit ; deux perpendiculaires étant deux droites orthogonales et sécantes. Une droite est orthogonale à un plan si elle est orthogonale aux droites du plan. On parle de vecteurs orthogonaux pour des vecteurs directeurs de droites orthogonales et de segments orthogonaux pour des segments portés par des droites orthogonales. Cette notion d'orthogonalité se généralise dans un premier temps à des espaces euclidiens, c'est-à-dire des espaces vectoriels de dimension finie sur lesquels on peut parler de distance et d'angle grâce à la définition d'un produit scalaire : dans les espaces euclidiens, deux vecteurs sont orthogonaux si leur produit scalaire est nul. Deux sous-ensembles d'un espace euclidien sont orthogonaux si tout vecteur de l'un est orthogo

Source officielle

https://fr.wikipedia.org/wiki/Orthogonalit%C3%A9

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Fixed-wing micro UAV open data with digicam and raw INS/GNSS

Davide Antonio Cucci, Kenneth Joseph Paul, Jan Skaloud

We have recently released in the open domain data originating from a series of flights conducted with a fixed-wing micro UAV carrying high-quality small camera and navigation sensors. This data was previously used in several peer-reviewed publications. However, the data that we describe in the following is part of a larger series that will be released gradually after incorporating user feedback (e.g., on formats, description, etc.) from the first (three) released open data-sets. In the first part of this work we provide a thorough description of the common elements of these data sets, notably the UAV, its sensors, methods of time-stamping and synchronizing data streams, reference geometrical relations among them (system calibration) as well as time-invariant sensor parameters (e.g., lens distortion, non-orthogonality of inertial sensors) together with ground control points that are valid over the whole series. In the second part we describe the individual missions and provide the links to the released data sets.

2021

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MATHICSE Technical Report : On the dynamically orthogonal approximation of time dependent random PDEs

Eleonora Musharbash, Fabio Nobile

In this work we discuss the Dynamically Orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion. We analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the S-terms DO solution by the corresponding S-terms best approximation, i.e. the truncated S-terms Karhunen-Loève expansion at each time instant, under the assumption that the latter is continuously differentiable in time. Properties of the DO approximations are analyzed on simple cases of deterministic equations with random initial data. Numerical tests are presented that confirm the theoretical bound and show potentials and limitations of the proposed approach.

MATHICSE2014

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In this work we discuss the dynamically orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion. We analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the S-terms DO solution by the corresponding S-terms best approximation, i.e., the truncated S-terms Karhunen-Loeve expansion at each time instant. The bound is applicable on the largest time interval in which the best S-terms approximation is continuously time differentiable. Properties of the DO approximations are analyzed on simple cases of deterministic equations with random initial data. Numerical tests are presented that confirm the theoretical bound and show potentials and limitations of the proposed approach.

2015

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