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Bézier curve

A Bézier curve is a parametric curve used in computer graphics and related fields. The curve, related to the Bernstein polynomial, is named after Pierre Bézier, who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces; the Bézier triangle is a special case of the latter. In vector graphics, Bézier curves are used to model smooth curves. "Paths", as they are referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are intuitive to modify. Bézier curves are used in the time domain in animation, user interface design and smoothing cursor trajectory in eye gaze controlled interfaces. For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than moving at a fixed number of pixels per step; when animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

This applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear. The mathematical basis for Bézier curves—the Bernstein polynomials—had been known since 1912, but the polynomials were not applied to graphics until some 50 years when they were publicised by the French engineer Pierre Bézier, who used them to design automobile bodies at Renault. However, the study of these curves was first developed in 1959 by mathematician Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves at Citroën, another French automaker. A Bézier curve is defined by a set of control points P0 through Pn; the first and last control points are always the end points of the curve. The sums in the following sections are to be understood as affine combinations, the coefficients sum to 1. Given distinct points P0 and P1, a linear Bézier curve is a straight line between those two points; the curve is given by B = P 0 + t = P 0 + t P 1, 0 ≤ t ≤ 1 and is equivalent to linear interpolation.

A quadratic Bézier curve is the path traced by the function B, given points P0, P1, P2, B = + t, 0 ≤ t ≤ 1,which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from P0 to P1 and from P1 to P2 respectively. Rearranging the preceding equation yields: B = 2 P 0 + 2 t P 1 + t 2 P 2, 0 ≤ t ≤ 1; this can be written in a way that highlights the symmetry with respect to P1: B = P 1 + 2 + t 2, 0 ≤ t ≤ 1. Which gives the derivative of the Bézier curve with respect to t: B ′ = 2 + 2 t. from which it can be concluded that the tangents to the curve at P0 and P2 intersect at P1. As t increases from 0 to 1, the curve departs from P0 in the direction of P1 bends to arrive at P2 from the direction of P1; the sec

Rheinisch-Bergischer Kreis

The Rheinisch-Bergische Kreis is a Kreis in the Cologne Bonn Region of North Rhine-Westphalia, Germany. Neighboring districts are Kreis Mettman, Oberbergischer Kreis and Rhein-Sieg, the district-free cities Cologne, Leverkusen and Remscheid; the area of the Bergisches Land belonged to the earldom Berg for most of medieval times, still gives the district its name. In 1816 after the whole Rhineland area did come to Prussia the districts of Wipperfürth, Mülheim, Lennep and Solingen were created on the area now covered by the district. In 1819 Opladen and Solingen were merged into a bigger Solingen district. In 1929 a new Rhein-Wupper district was created, while several municipalities were incorporated into the cities Wuppertal and Solingen. 1932 the districts Mülheim and Wipperfürth were merged to form the old Rheinisch-Bergische Kreis. In 1975 most area of the two districts Rhein-Wupper and Rheinisch-Bergisch was merged to form the current district; the district form the western part of the Bergisches Land, where the hills of the Sauerland descend into the Rhine valley.

Media related to Rheinisch-Bergischer Kreis at Wikimedia Commons Official website

Paesaggio Urbano

Paesaggio Urbano - Urban Design is a bimonthly magazine focusing on architecture and urban design, founded in 1989, published by Gruppo Maggioli. The magazine offers a multi-disciplinary approach on urban phenomena like sociology, urban typology, economics and local and international cultural trends; the main focus of the magazine is the urban transformation and the analysis of influencing factors that impact on contemporary architecture. Since 2010 the magazine has started its internationalization process offering contents in English as part of its major restyle in 2011; every issue include a monographic dossier focusing on specific topics, like colour, building renovation and urban design related themes. Editor in Chief: Amalia Maggioli Director: Marcello Balzani Vice Director: Nicola Marzot Paolo Baldeschi Lorenzo Berna Marco Bini Ricky Burdett Giovanni Carbonara Manuel Gausa Pierluigi Giordani Giuseppe Guerrera Thomas Herzog Winy Maas Francesco Moschini Attilio Petruccioli Franco Purini Carlo Quintelli Alfred Rütten Livio Sacchi Pino Scaglione Giuseppe Strappa Kimmo Suomi Francesco Taormina The magazine went online and created a web site to make its archive available to a wider audience.

The website is updated with the latest issues. Issues released before the complete table of contents. Issues published from 2011 presents English abstracts of all articles and an English version of the main articles. Since 2012 the magazine launched a Vimeo channel offering a number of videos and documentaries which complete the èrinted articles and provide more insights on the leading topics. Gruppo Maggioli Paesaggio Urbano boards and info Direzione Generale per i Beni Storici e Paesaggistici progettarepertutti.org The official website