Normal (geometry)

Summary

In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (in which case it is a unit normal vector) or its length may represent the curvature of the object (a ); its algebraic sign may indicate sides (interior or exterior). In three-dimensional space, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word normal is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point

Official source

https://en.wikipedia.org/wiki/Normal_(geometry)

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Examples of normal, not geometrically normal, projective Gorenstein del Pezzo surfaces with at most Du Val singularities

Ursina Schweizer

While over fields of characteristic at least 5, a normal, projective and Gorenstein del Pezzo surface is geometrically normal, this does not hold for characteristic 2 and 3. There is no characterization of all such non-geometrically normal surfaces, but there is a complete characterization of all possible surfaces that can arise as the normalization of the base change to the algebraic closure of the base field. In four of these instances, for a given normalization, we describe the construction of such a non-geometrically normal surface.

EPFL2021

Electromagnetic modelling of planar circuits in bounded layered media

Pedro Crespo Valero

Printed circuits in bounded media encompass a wide range of practical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even two-dimensional printed periodic structures. The Electromagnetic (EM) modeling of printed circuits in layered bounded media is performed via an Integral Equation (IE) technique. Green's functions (GFs) are specially defined to satisfy both the Boundary Conditions (BCs) imposed by the layered media and by the transverse boundary enclosing the entire structure. Finally, a system of IEs on the equivalent sources can be solved numerically by means of the Method of Moments (MoM). Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Nevertheless, our formulation introduces a unified approach applicable to all the aforementioned problems. Due to the symmetry presented by a bounded layered media, the GF problem can be reduced into a two-dimensional transverse boundary problem and a one-dimensional transmission line problem in the normal direction. Both problems can be treated independently. This thesis proposes and fully develops an efficient technique that encompasses different laterally bounded multilayered problems with a seamless transition between them. The method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by zero-curl & constant-charge Basis Functions (BFs). It offers a unified and versatile approach that, on one hand eliminates redundancy in the formulation and on the other hand simplifies each particular problem to the evaluation of constant coefficients or basic line integrals. Analytical solutions can be found for the combination of linear subsectional basis functions in rectangular and circular Perfect Electric Conductor (PEC) boundaries as well as for periodic lattices. This thesis then solves the problem of transmission line model in the longitudinal direction by proposing an efficient algorithm that guarantees numerical stability under a variety of known critical conditions where other already known formulations fail. In addition, it introduces alternate equivalent expressions of this formulation that allow new interpretations of the problem. Due to its practical interest, the method is applied for the EM modeling of multilayered boxed printed circuits. This motivated the implementation of a dedicated software tool for the efficient analysis of these topologies including losses. Extensive numerical experiments have been carried out to assess the validity of the aforementioned theory and some properties of test-structures (losses, mesh, etc).

EPFL2007

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Many manipulation tasks require coordinated motions for arm and fingers. Complexity increases when the task requires to control for the force at contact against a non-flat surface; This becomes even more challenging when this contact is done on a human. All these challenges are regrouped when one, for instance, massages a human limb. When massaging, the robotic arm is required to continuously adapt its orientation and distance to the limb while the robot fingers exert desired patterns of forces and motion on the skin surface. To address these challenges, we adopt a Dynamical System (DS) approach that offers a unified motion-force control approach and enables to easily coordinate multiple degrees of freedom. As each human limb may slightly differ, we learn a model of the surface using support vector regression (SVR) which enable us to obtain a distance-to-surface mapping. The gradient of this mapping, along with the DS, generates the desired motions for the interaction with the surface. A DS-based impedance control for the robotic fingers allows to control separately for force along the normal direction of the surface while moving in the tangential plane. We validate our approach using the KUKA IIWA robotic arm and Allegro robotic hand for massaging a mannequin arm covered with a skin-like material.We show that our approach allows for 1) reactive motion planning to reach for an unknown surface, 2) following desired motion patterns on the surface, and 3) exerting desired interaction forces profiles. Our results show the effectiveness of our approach; especially the robustness toward uncertainties for shape and the given location of the surface.

2020

Electromagnetic modelling of planar circuits in bounded layered media

Pedro Crespo Valero

EPFL2007