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Concept# Cylinder

Summary

A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the right circular cylinder.
The definitions and results in this section are taken from the 1913 text Plane and Solid Geometry by George Wentworth and David Eugene Smith .
A is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line. Any line in this family of parallel lines is called an element of the cylindrical surface. From a kinematics point of view, given a plane curve, called the directrix, a cylindrical surface is that surface traced out by a line, called the generatrix, not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of the generatrix is an element of the cylindrical surface.
A solid bounded by a cylindrical surface and two parallel planes is called a (solid) . The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a , otherwise it is called an .

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Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Cylinder

A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces.

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points.

Related lectures (329)

This thesis is devoted to the understanding of topological graphs. We consider the following four problems: 1. A \emph{simple topological graph} is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once, at endpoint or at a crossing. Let $G$ be a complete simple topological graph on $n$ vertices. The three edges induced by any triplet of vertices in $G$ form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an \emph{empty triangle}. In 1998, Harborth proved that $G$ has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least $2n/3$. We settle Harborth's conjecture in the affirmative. 2. A \emph{monotone cylindrical} graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called \emph{simple} if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are \emph{disjoint} if they do not intersect. We show that every simple complete monotone cylindrical graph on $n$ vertices contains $\Omega(n^{1-\epsilon})$ pairwise disjoint edges for any $\epsilon>0$. As a consequence, we show that every simple complete topological graph (drawn in the plane) with $n$ vertices contains $\Omega(n^{\frac 12-\epsilon})$ pairwise disjoint edges for any $\epsilon>0$. By extending some of the ideas here we are then able to get rid of the $\epsilon$ term in the exponent, showing that in fact we can always guarantee a set with $\Omega(n^{\frac 12})$ pairwise disjoint edges. This improves the previous lower bound of $\Omega(n^\frac 13)$ by Suk and independently by Fulek. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges. 3. A {\em tangle} is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a {\em degenerate tangle}. We settle a problem of Pach, Radoi\v ci'c, and T'oth \cite{TTpaper}, by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles. 4. We show that for a constant $t\in \NN$, every simple topological graph on $n$ vertices has $O(n)$ edges if the graph has no two sets of $t$ edges each such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i'c, and T'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges.

Fil Winding: Mechanics AnalysisPHYS-101(g): General physics : mechanics

Analyzes the winding of a thread around a fixed cylinder and the motion of a point mass attached to it.

Descriptive Geometry: Cones and Cylinders

Explains tangent planes on cones and cylinders, contours apparent, and shadows cast by surfaces.

MATH-126: Geometry for architects II

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

MATH-213: Differential geometry

Ce cours est une introduction à la géométrie différentielle classique des courbes et des surfaces, principalement dans le plan et l'espace euclidien.

PHYS-101(f): General physics : mechanics

Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de pr

We present a new second-order method, based on the MAC scheme on cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. In this approach, the solid boundary is embedded in the cartesian computational mesh. Discretizations of the viscous and convective terms are formulated in the context of finite volume methods ensuring local conservation properties of the scheme. Classical second-order centered schemes are applied in mesh cells which are sufficiently far from the obstacle. In the mesh cells cut by the obstacle, first-order approximations are proposed. The resulting linear system is nonsymmetric but the stencil remains local as in the classical MAC scheme on cartesian grids. The linear systems are solved by a fast direct method based on the capacitance matrix method. The time integration is achieved with a second-order projection scheme. While in cut-cells the scheme is locally first-order, a global second-order accuracy is recovered. This property is assessed by computing analytical solutions for a Taylor-Couette problem. The efficiency and robustness of the method is supported by numerical simulations of 2D flows past a circular cylinder at Reynolds number up to 9500. Good agreement with experimental and published numerical results are obtained. (c) 2012 Elsevier Ltd. All rights reserved.

Lattice Boltzmann method has become a common tool for computational fluid dynamics. Thanks to the weak numerical dissipation of the numerical scheme, it seems very well fitted for aeroacoustics studies. However there exists only few validations in literature. In this semester project we propose to evaluate on a simple and well known case (laminar flow past one or two cylinders) the ability of the method to reproduce the aeroacoustic fields in a reliable manner. Firstly, we present the Lattice Boltzmann Method (LBM) and, in order to get familiar with LBM, a simple Matlab code is written and the flow past a 2D cylinder is simulated. Then, the ability of LBM to simulate aeroacoustics is tested. With this aim, the Palabos library (open-source and based on C++) is used to run the simulation in a more efficient way. The wake cylinder flow is one of the most studied and well known flow and it is therefore natural to validate the numerical scheme with this flow case. The LBM has to be validated for such particular case where numerical simulations tries to catch tiny variations of pressure. The simulations of the flow obtained with the LBM are compared to the results obtained in literature.

2015