Geometric Computing with CGAL and LEDA

Geometric Computing with CGAL and LEDA

Kurt Mehlhorn and Stefan Schirra

Abstract.LEDA and CGAL are platforms for combinatorial and geo-

metric computing.We discuss the use of LEDA and CGAL for geometric

computing and show that they provide a unique framework for exact,ef-

?cient and convenient geometric computing.

§1.Introduction

LEDA(Library of E?cient Data Structures and Algorithms)[16,17]and CGAL (Computational Geometry Algorithms Library)[8,26]are platforms for com-binatorial and geometric computing developed in the ESPRIT-projects AL-COM II,ALCOM-IT,CGAL,and GALIA.Concerning geometric computing, the systems provide number types,geometry kernels,geometric algorithms, and visualization.They by now provide a signi?cant fraction of the algorithms and data structures described in the computational geometry literature,where in this context computational geometry subsumes the?eld covered by the an-nual ACM Symposia on Computational Geometry.The systems are designed such that it is easy to build programs on top of them.The computations in LEDA and CGAL are exact,i.e.,behave according to their mathematical speci?cations.This is a strong point of both systems,distinguishing them form many other geometric software.

Based on the insight that algorithm design must include implementa-tion to have maximal impact,Kurt Mehlhorn and Stefan N¨a her started the development of the LEDA software library of e?cient data structures and al-gorithms in Saarbr¨u cken in’89using C++as programming language.LEDA is now developed at Max-Planck-Institut f¨u r Informatik,Saarbr¨u cken(Ger-many),and Martin-Luther-Universit¨a t Halle-Wittenberg(Germany).The idea of CGAL was conceived in fall of’94,inspired by the success of LEDA and in order to bundle forces previously put into predecessors of CGAL[2,11,20]. Development of CGAL was started in fall’96in the CGAL-project and is

Saint-Malo Proceedings1 Pierre-Jean Laurent,Paul Sablonni`e re,and Larry L.Schumaker(eds.),pp.277–286. Copyright o c2000by Vanderbilt University Press,Nashville,TN.

ISBN0-8265-1356-6.

All rights of reproduction in any form reserved.

2K.Mehlhorn and S.Schirra

now continued in the GALIA-project.GALIA is carried out by Max-Planck-Institut f¨u r Informatik,Saarbr¨u cken,ETH Z¨u rich(Switzerland),Freie Uni-versit¨a t Berlin(Germany),INRIA Sophia-Antipolis(France),Martin-Luther-Universit¨a t Halle-Wittenberg,Tel-Aviv University(Israel),and Utrecht Uni-versity(The Netherlands).The goal is to make the most important of the so-lutions and methods developed in computational geometry available to users in industry and academia in a C++library.

§2.The Need for a Geometry Software Library

Reusing code that already exists and is used and thereby tested rather than implementing everything from scratch saves development time and hence re-duces cost[5].It also eases maintenance of code.Software libraries also ease the transfer of state-of-the-art algorithmic knowledge into application areas. Since geometric computing is a wide area,many application areas can bene-?t from the availability of the re-usable code of a geometry software library. The importance of libraries of software components in subject area domains is clearly stated in a recent report of the information technology advisory committee of the president of the US[22].

In geometric computing,software libraries consisting of reliable compo-nents are particularly useful,since implementors of geometric algorithms are faced with notoriously di?cult problems[18],especially the problems of ro-bustness and degeneracies.

Robustness

Theory usually assumes exact computation with arbitrary real numbers,while the standard substitution for real numbers in scienti?c computing in practice,?oating-point arithmetic,is inherently imprecise.In practice,implementa-tions of geometric algorithms compute garbage or completely fail more or less occasionally,because rounding errors lead to wrong and contradictory deci-sions,see[14,25,27].With?oating-point arithmetic,basic laws of arithmetic, on which the correctness proof of geometric algorithms is based,of course, don’t hold anymore.We invite the reader to carry out the following simple experiment:Compute the point of intersection of the two lines with built-in ?oating point arithmetic.Then,again using built-in?oating point arithmetic, check whether the computed intersection point lies on the intersecting lines.

Figure1shows an incorrect result of a computation due to rounding errors.The task is to compute the extreme points of intersection points of a set of line segments,where a point is called extreme with respect to a set of points if its removal from the set changes the convex hull of the point set.The line segments have randomly chosen endpoints lying on a circle.In a?rst step the intersection points of the line segments are computed,then a convex hull algorithm is run on the points computed in the?rst step.With?oating-point arithmetic,some collinearities are not detected and too many extreme points are reported.Extreme points are shown as small disks in Figure1.The points surrounded by a circle are actually not extreme.In the problem considered

CGAL and LEDA3

Fig.1.Extreme points among intersection points of30line segments. here,the computed output might still be useful;for many other geometric problems,however,failures of pure?oating-point based implementations are much more drastically.They just crash.

Adding epsilons by trial and error to equality tests used to be common practice in implementations of geometric algorithms,but it in no way leads to a reliable correct implementations.Two main approaches to solving the precision-caused robustness problem can be identi?ed.The?rst is re-designing algorithms such that they can deal with imprecision,https://www.wendangku.net/doc/495333382.html,pute a good ap-proximate solution,but never crash.This approach has been applied success-fully to only very few basic problems in computational geometry so far,see [14,25,27].The second approach is exact geometric computation[28],which means computing with such a precision that an implementation behaves like its theoretical counterpart,and therefore according to its mathematical speci-?cation.This is possible for many geometric problems,at least,theoretically. Note that in practice,the input does not involve arbitrary real numbers.Of course,exact geometric computation slows down computation,but thanks to clever adaptive computation using?oating-point arithmetic whenever known to produce the correct result[10,15],it is now much closer to the speed of ?oating-point computation than it used to be a decade ago.Since libraries must be reliable in order to be usable in general,the exact geometric compu-tation approach is taken in LEDA and in CGAL.

Degeneracies

Robustness problems caused by rounding errors are closely related to degen-eracies,i.e.“exceptional”input con?gurations.Theory often neglects degen-

4K.Mehlhorn and S.Schirra eracies for the sake of a cleaner exposition of an algorithm,and also because they are rare from a theoretical point of view:they have measure zero in the set of all possible inputs over the real numbers.In practice,however,they occur frequently.Since theory papers often leave handling degenerate cases as an exercise to the reader,implementors are often left alone with the burden of investigating the details of handling degeneracies.Furthermore,this leads to treating degeneracies as an afterthought,which is,according to our expe-rience[3],not the most suitable way to think about them,since it leads to unnecessarily complicated and blown-up code.Considering degeneracies right from the beginning seems to be a much better approach.

Symbolic perturbation schemes have been proposed as a general approach to removing degeneracies,for an overview see[24].With this approach,the input is perturbed symbolically such that no degeneracies arise anymore.The perturbed input can then be processed by an algorithm assuming general position.The computed output,however,does not correspond to the actual input,but to the perturbed input.Therefore,the complexity of the output might be much larger than the output for the actual input[3].For some problems,the symbolic perturbation approach works out?ne;for others,a postprocessing step is required to deduce the actual output from the output computed for the perturbed input.In many cases,this is a non-trivial task, as hard as dealing with degeneracies directly.

Algorithms and data structures in CGAL and LEDA handle all possi-ble degenerate case by default.So a user need not to worry about all the degenerate cases.If an algorithm or data structure should not handle a de-generate case,this is clearly stated in the documentation and this precondition is checked in the implementation.However,mainly to support rapid proto-typing,CGAL also provides tools for symbolic perturbation.A general ran-domized symbolic perturbation scheme is available for the CGAL kernels[6].

A new promising approach that has been started within the CGAL project, is controlled perturbation[23].Here the input is perturbed numerically,such that general position is guaranteed.

§3.Number Types

The lowest level in geometric computing is the arithmetic level.LEDA and CGAL provide various number types to support exact geometric computation. LEDA provides leda

rational,a number type for arbitrary precision rational arithmetic,based on leda bigfloat, a number type for?oating point arithmetic with extended precision.A user can choose the mantissa length of the leda

real[4].This number types models a subset of algebraic numbers:All integers are leda real s are closed under the operations+,?,·,/,and k√real s record the computa-tion history in an expression dag,and use adaptive evaluation to guarantee

CGAL and LEDA5

that all comparison operations give the correct results.They use bigfloat s internally.LEDA and CGAL also provide interval arithmetic.Furthermore, CGAL provides some?xed point arithmetic based on built-in float s,as well as wrappers for the gnu multiple precision integer arithmetic[12]and the number types provided by CLN[13].

§4.Geometry Kernels

The kernel of a geometry library contains the basic geometric objects and basic operations on them like points,lines,planes,sideness test,intersection and distance computations.

LEDA provides an exact geometry kernel for rational computations.In-ternally,it uses?oating-point?lters[10,15]to speed up exact computation. With?oating-point?lters,an expression whose sign has to be computed is ?rst evaluated using?oating-point arithmetic.Moreover,an upper bound on the error of the?oating-point computation is computed as well.By compari-son of the absolute computed?oating-point value with this error bound,it is checked whether the?oating-point computation is guaranteed to be reliable. If the sign can not be deduced with?oating-point arithmetic,the expression is re-evaluated with a more reliable arithmetic.In case of the rational geometry kernel in LEDA,arbitrary precision integer arithmetic is used.The rational geometry kernel of LEDA uses homogeneous coordinates and is coupled to the number types leda integer.

CGAL provides two families of geometry kernels,one based on Cartesian coordinates and one based on homogeneous coordinates[9].Both kernels are parameterized by a number type.All number types ful?lling a very small list of requirements can be used with the CGAL kernels.For example,the user might choose the Cartesian kernel with rational arithmetic or the ho-mogeneous kernel with integer arithmetic.In particular,for computations involving k-th root operations,the number type leda

real is certainly the most convenient way to get reliable computation.

LEDA also provides a kernel that uses double precision?oating-point arithmetic internally.Similarly,the CGAL kernels can be used with built-in?oating point number types as well.This might be su?cient for some problems,but since correctness can not be guaranteed,the use of these kernels is not recommended in general.

§5.Geometric Algorithms and Data Structures

CGAL and LEDA by now provide a signi?cant fraction of the algorithms and data structures described in the computational geometry literature.They pro-

6K.Mehlhorn and S.Schirra vide several algorithms to compute convex hulls in low and higher dimensions. They provide algorithms and data structures for triangulations,constrained triangulations,Delaunay triangulations,regular triangulations,and Voronoi diagrams in two-dimensional space and Delaunay triangulations and regular triangulations in three-dimensional space.Furthermore they provide several algorithms for line segment intersection and regularizing Boolean operations on polygons.CGAL and LEDA also provide a number of query data struc-tures.For example,there are range-and segment trees,kd-trees,as well as a data structure for range and nearest neighbor queries based on Delaunay triangulations.CGAL provides data structures for polyhedral surfaces based on a half-edge data structure,it provides a topological map data type and a planar map data structure,and a data structure for arrangements.The libraries also contain algorithms for curve reconstruction in the plane.

CGAL and LEDA provide algorithms for a number of geometric opti-mization problems.There are algorithms computing smallest enclosing circles, smallest enclosing ellipses,and smallest enclosing spheres,the latter in any dimension.Furthermore,there are a number of algorithms based on matrix search like computing extremal polygons and rectangular p-centers.LEDA also contains algorithms for computing smallest enclosing annuli with respect to area and width,and algorithms to compute minimum spanning trees for a set of points.

The geometric algorithms of LEDA come in two versions,one using the exact rational geometry kernel and one using the unreliable?oating-point kernel.CGAL’s algorithms and data structures are even more?exible with respect to the geometry kernel used.All algorithms and data structures of CGAL are parameterized by a template parameter called traits class.This traits class provides an algorithm or data structure with all the type infor-mation it needs.It tells the algorithm on which types it should operate and which types it should use to do that.

The parameterization and the resulting genericity of CGAL’s algorithms and data structures is best illustrated by a simple,but instructive example. Computing the convex hull of a set of points in the plane is an intensively studied problems in computational geometry.The input is a set of points in the plane,the output is the counterclockwise sequence of extreme points. Andrew’s variant[1]of the Graham scan algorithm can be formulated in such a way that it needs only two primitive operations on the points,namely a primitive to compare two points in order to sort the points lexicographically by their Cartesian coordinates,and a primitive to check the order type of a triple of points,more precisely,to check whether a sequence of three points forms a left turn.The CGAL implementation of Andrew’s algorithm is parameterized by a point type and the two required primitive operations.The latter two are passed as function object types and need to correspond to the point type. We call the parameter types of an algorithm or data structure the traits types. To avoid long lists of template parameters,the traits types are collected in the traits class.Note that the parameterization is on the level of data types, not on the level of objects.In order to use the CGAL implementation of

CGAL and LEDA7 Andrew’s algorithm with a point type from a CGAL kernel,no traits class needs to be speci?ed.CGAL adds an appropriate one.If a user wants to run the algorithm on a di?erent point type,for example,a point type from some other C++library or system,for example from LEDA or from some Geographical Information System,an appropriate traits class for this point type must be passed to the function in order to tell it which operations it should use.That’s it.Given such a traits class,the algorithm now works with non-CGAL types as well.CGAL provides traits classes for both LEDA kernels.

The parameterization by a traits class can be used to avoid explicit trans-formation of the data.Assume that we have points in three dimensional space. Using a traits class that provides a comparison primitive and an order type predicate that both operate on x and y coordinates of the points only,the CGAL implementation of Andrew’s algorithm can be used to compute the se-quence of three-dimensional points whose projections onto the xy-plane form the convex hull of the projections of all points onto that plane.There is no need to explicitly transform the points into two-dimensional points.With an appropriate traits class,the algorithm can directly operate on the three-dimensional points.This saves time and space.

This feature is most likely even more interesting for Delaunay triangula-tions.Assume we have a triangulated irregular network(TIN),and we want to make a TIN with the same set of vertices without long and skinny triangles. This is usually accomplished by computing the two-dimensional Delaunay tri-angulation of the projections of the vertices of the TIN and lifting the vertices and triangles again.CGAL allows you to do this without explicit projection using an appropriate traits class.There are further examples where traits classes can be used nicely in the context of geometric transformations.

§6.Visualization

In LEDA,there is a data type leda

scene maintains a container with geometric objects.The GeoWin data type can be used for the construction and display of geometric objects and data structures,the visualization of geometric algorithms,writing interactive demos for geometric algorithms and debugging geometric algorithms.

8K.Mehlhorn and S.Schirra

§7.Conclusions

Reliability means that software behaves as speci?ed.Unfortunately,there are many exceptions to this rule for geometric software,mainly due to the issues discussed in Section2.Correctness and reliability are even more important for the components of a software library.You might be willing to accept shortcomings of a program designed for a special purpose,if problematic input instances never arise in your context.Since library component need to be generally applicable,any such shortcomings are not acceptable.CGAL(if used with a number type for exact geometric computation)and LEDA(with the rational kernel)provide geometric software that behaves according to its mathematical speci?cation.That makes it easy to combine components form these libraries,and to build larger entities out of these components.

The use of exact computation alone can not guarantee correctness.CGAL and LEDA also use program checking[19]to increase reliability of its compo-nents.A program checker need not compute the output for a given input.It already gets both input and output,and then has to verify that the output is the correct output for the given input.While a program gets x and has to compute f(x),a checker gets x and y and must only check whether y=f(x). The latter step should be computationally simpler,such that it is less likely that its implementation is buggy.

At present,LEDA and CGAL consists of more than100,000lines of C++ code each.Neither library provides class libraries in the sense of Smalltalk,but both provide fairly small class hierarchies if any.CGAL uses the generic pro-gramming paradigm that became known with the Standard Template Library (STL),which is now part of Standard C++.This makes CGAL very?exible, more?exible than LEDA.On the other hand,LEDA is a more complete, closed programming framework that also contains very useful components for combinatorial computing.Due to its generic design,CGAL is more open.It often relies on other sources for basic non-geometric data structures,mainly on the STL.Due to its generic design,it works well together with LEDA. Since CGAL has a more modern design and is developed by a larger group of people,the future will certainly be with CGAL.However,LEDA’s com-ponents for geometric computing will continue to be useful,especially within CGAL.For more information and to download LEDA,see

http://www.mpi-sb.mpg.de/LEDA

For more information and to download CGAL,see

http://www.cs.uu.nl/CGAL Acknowledgments.Work on this paper has been supported by ESPRIT-IV LTR project28155(GALIA).

CGAL and LEDA9

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Kurt Mehlhorn and Stefan Schirra

Max-Planck-Institut f¨u r Informatik

66123Saarbr¨u cken

Germany

mehlhorn@mpi-sb.mpg.de

stschirr@mpi-sb.mpg.de

With的用法全解

With的用法全解 with结构是许多英语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 一、 with结构的构成 它是由介词with或without+复合结构构成，复合结构作介词with或without的复合宾语，复合宾语中第一部分宾语由名词或代词充当，第二部分补足语由形容词、副词、介词短语、动词不定式或分词充当，分词可以是现在分词，也可以是过去分词。With结构构成方式如下： 1. with或without-名词/代词+形容词； 2. with或without-名词/代词+副词； 3. with或without-名词/代词+介词短语； 4. with或without-名词/代词 +动词不定式； 5. with或without-名词/代词 +分词。 下面分别举例： 1、 She came into the room，with her nose red because of cold.（with+名词+形容词，作伴随状语）

2、 With the meal over ， we all went home.（with+名词+副词，作时间状语） 3、The master was walking up and down with the ruler under his arm。（with+名词+介词短语，作伴随状语。） The teacher entered the classroom with a book in his hand. 4、He lay in the dark empty house，with not a man ，woman or child to say he was kind to me.（with+名词+不定式，作伴随状语）He could not finish it without me to help him.（without+代词 +不定式，作条件状语） 5、She fell asleep with the light burning.（with+名词+现在分词，作伴随状语） Without anything left in the with结构是许多英 语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 二、with结构的用法 with是介词，其意义颇多，一时难掌握。为帮助大家理清头绪，以教材中的句子为例，进行分类，并配以简单的解释。在句子中with结构多数充当状语，表示行为方式，伴随情况、时间、原因或条件（详见上述例句）。 1.带着，牵着…… (表动作特征)。如： Run with the kite like this.

by with and in区别

WITH/BY/IN的用法区别："The mountain is covered with/in/by snow". The meanings are so similar that the three can be used almost interchangeably, but some subtle nuances may apply. When referring to a substance(物体) that sticks to another, use in or with, but not by: ?The actress was covered in blood, or ?The actress was covered with blood, but not ?The actress was covered by blood. When referring something that physically protects something else, use with or by, but not in: ?The field(田地) was covered with a tarp(油布), or ?The field was covered by a tarp, but not ?The field was covered in a tarp. Use covered with to indicate an unusual amount of something on top of something else; use covered by to connote a covering so dense that the object being covered is completely obscured from view: ?The mountain was covered with fog. ?The mountain was covered by fog. Another example: ?Our grass(草坪) was covered with butterflies. ?Our grass was covered by butterflies. Somehow, the latter (covered by) paints a picture where the butterflies are so close together that I can hardly see the grass at all, but in the former (covered with), I picture a lot of butterflies, just not necessarily so many that I can't see the grass. When talking about metaphorical coverage, use covered by: ?The roof damage was covered by insurance(上保险), but not ?The roof damage was covered with insurance, or ?The roof damage was covered in insurance. Another example: ?The city council meeting was covered by the news station, but not ?The city council meeting was covered with the news station, or ?The city council meeting was covered in the news station.

compare 的两个重要词组区别

compare to 和compare with 的区别是什么 Compare to 是“把……比作”的意思。例如： We compare him to a little tiger. 我们把他比作小老虎。 The last days before liberation are often compared to the darkness before the dawn. 将要解放的那些日子常常被比作黎明前的黑暗。 Compare ... with 是“把……和……比较”的意思。例如： We must compare the present with the past. 我们要把现在和过去比较一下。 We compared the translation with the original. 我们把译文和原文比较了下。 从上面比较可以看出，compare with 侧重一个仔细的比较过程。有时，两者都可以互相代替。例如： He compared London to (with) Paris. 他把伦敦比作巴黎。 London is large, compared to (with) Paris. 同巴黎比较而言，伦敦大些。 在表示“比不上”、“不能比”的意思时，用compare with 和compare to 都可以。例如： My spoken English can't be compared with yours. 我的口语比不上你的。 The pen is not compared to that one. 这笔比不上那支。 1、c ompare…to…意为“把…比作”，即把两件事物相比较的同时，发现某些方面相似的地方。这两件被比较的事物 或人在本质方面往往是截然不同的事物。如： He compared the girl to the moon in the poem. 他在诗中把那姑娘比作月亮。 2、compare…with…“与…相比，把两件事情相比较，从中找出异同”，这两件事又往往是同类的， 如：I'm afraid my English compares poorly with hers. 恐怕我的英语同她的英语相比要差得多。 compare to和compare with有何区别,当说打比方时和做比较是分别用哪个？ compare…to…比喻.例如: The poets often compare life to a river. 诗人们经常把生活比喻成长河． compare…with…相比．例如： My English can't compare with his. 我的英文水平不如他．

五种计算机语言的特点与区别

php语言，PHP（PHP: Hypertext Preprocessor的缩写，中文名：“PHP：超文本预处理器”）是一种通用开源脚本语言。语法吸收了C语言、Java和Perl的特点，入门门槛较低，易于学习，使用广泛，主要适用于Web开发领域。 特性：PHP 独特的语法混合了C、Java、Perl 以及PHP 自创新的语法；PHP可以比CGI 或者Perl更快速的执行动态网页——动态页面方面，与其他的编程语言相比，PHP是将程序嵌入到HTML文档中去执行，执行效率比完全生成htmL标记的CGI要高许多，PHP具有非常强大的功能，所有的CGI的功能PHP都能实现；PHP支持几乎所有流行的数据库以及操作系统；最重要的是PHP可以用C、C++进行程序的扩展。 Java语言，Java是一种可以撰写跨平台应用软件的面向对象的程序设计语言，是由Sun Microsystems公司于1995年5月推出的Java程序设计语言和Java平台（即JavaSE, JavaEE, JavaME）的总称。 Java 技术具有卓越的通用性、高效性、平台移植性和安全性，广泛应用于个人PC、数据中心、游戏控制台、科学超级计算机、移动电话和互联网，同时拥有全球最大的开发者专业社群。在全球云计算和移动互联网的产业环境下，Java更具备了显著优势和广阔前景。 Java的优势，与传统程序不同，Sun 公司在推出Java 之际就将其作为一种开放的技术。全球数以万计的Java 开发公司被要求所设计的Java软件必须相互兼容。“Java 语言靠群体的力量而非公司的力量”是Sun公司的口号之一，并获得了广大软件开发商的认同。这与微软公司所倡导的注重精英和封闭式的模式完全不同。 Sun 公司对Java 编程语言的解释是：Java 编程语言是个简单、面向对象、分布式、解释性、健壮、安全与系统无关、可移植、高性能、多线程和动态的语言。 python语言，是一种面向对象、直译式计算机程序设计语言，Python语法简洁而清晰，具有丰富和强大的类库。它常被昵称为胶水语言，它能够很轻松的把用其他语言制作的各种模块（尤其是C/C++）轻松地联结在一起。 常见的一种应用情形是，使用python快速生成程序的原型（有时甚至是程序的最终界面），然后对其中有特别要求的部分，用更合适的语言改写。 Python是完全面向对象的语言。函数、模块、数字、字符串都是对象。并且完全支持继承、重载、派生、多继承，有益于增强源代码的复用性。 Python支持重载运算符和动态类型。相对于Lisp这种传统的函数式编程语言，Python对函数式设计只提供了有限的支持。有两个标准库(functools, itertools)提供了Haskell和Standard

with用法归纳

with用法归纳 （1）“用……”表示使用工具，手段等。例如： ①We can walk with our legs and feet. 我们用腿脚行走。 ②He writes with a pencil. 他用铅笔写。 （2）“和……在一起”，表示伴随。例如： ①Can you go to a movie with me? 你能和我一起去看电影'>电影吗？ ②He often goes to the library with Jenny. 他常和詹妮一起去图书馆。 （3）“与……”。例如： I’d like to have a talk with you. 我很想和你说句话。 （4）“关于，对于”，表示一种关系或适应范围。例如： What’s wrong with your watch? 你的手表怎么了？ （5）“带有，具有”。例如： ①He’s a tall kid with short hair. 他是个长着一头短发的高个子小孩。 ②They have no money with them. 他们没带钱。 （6）“在……方面”。例如： Kate helps me with my English. 凯特帮我学英语。 （7）“随着，与……同时”。例如： With these words, he left the room. 说完这些话，他离开了房间。 [解题过程] with结构也称为with复合结构。是由with+复合宾语组成。常在句中做状语，表示谓语动作发生的伴随情况、时间、原因、方式等。其构成有下列几种情形： 1.with+名词(或代词)+现在分词 此时，现在分词和前面的名词或代词是逻辑上的主谓关系。 例如:1)With prices going up so fast, we can't afford luxuries. 由于物价上涨很快，我们买不起高档商品。（原因状语） 2)With the crowds cheering, they drove to the palace. 在人群的欢呼声中，他们驱车来到皇宫。（伴随情况） 2.with+名词(或代词)+过去分词 此时，过去分词和前面的名词或代词是逻辑上的动宾关系。

战略与策略的主要区别

战略与策略的主要区别 一，什么是战略营销？ 必须首先明确，什么是战略。 1，战略的本质是一个企业的选择。为什么要做选择？因为任何一个企业都不是全能的。不可能做所有的事情，也不是所有的事情都能做好！任何企业的资源和能力都是有限的。战略就是要把有限的资源和能力，用到产出最大的地方。战略就是一个选择的过程，选择什么？如何选择？这是企业战略规划所要研究的课题。 2，战略首先意味着放弃。在中国目前的经济环境下，战略对于企业家的意义，更为重要的是“放弃”。中国的经济处在快速发展期，有太多的市场机会可供选择。但选择意味着放弃，而放弃是一件很痛苦的事情。 综上所述，战略选择的核心是对企业目标客户群的选择。而战略营销就是从战略的高度思考和规划企业的营销过程，是聚焦最有价值客户群的营销模式。 我们都知道80/20原理，20%的客户创造了企业80%的利润。战略营销要做的就是找到适合企业的目标客户群，并锁定他们进行精确打击，使企业的资源和能力发挥最大的效益，并实现企业能力的持续提升。 因此，战略营销的三个关键要素就是：1）客户细分；2）聚焦客户价值；3）为股东和客户增值。 二，什么是策略营销？ 策略营销主要指的是在市场营销中，将企业的市场策略运用到营销中的过程。 比如： 1，低成本策略 通过降低产品生产和销售成本,在保证产品和服务质量的前提下,使自己的产品价格低于竞争对手的价格,以迅速扩大的销售量提高市场占有率的竞争策略。 2.差别化策略 通过发展企业别具一格的营销活动,争取在产品或服务等方面具有独特性,使消费者产生兴趣而消除价格的可比性,以差异优势产生竞争力的竞争策略。 3.聚焦策略 通过集中企业力量为某一个或几个细分市场提供有效的服务,充分满足一部分消费者的特殊需求,以争取局部竞争优势的竞争策略。 一个企业的市场营销策略必须是在企业的战略营销策略下确定的，可以简单把策略营销理解成企业在市场的战术营销。这就是两者的区别！

真理的定义和特点以及谬误的区别

、真理的定义和特点以及谬误的区别 定义：真理是人们对客观事物及其规律的正确反映。 特点：1、真理具有客观性。真理的内容是客观的;检验真理的标准是客观的。 2、真理具有价值性。真理的价值性是指真理对人类实践活动的功能性，它揭示了客观真理具有能满足主体需要、对主体有用的属性。 9．资本循环和资本周转（资本循环的三个阶段三大职能，两大前提条件；资本周转的定义，影响周转的因素） 资本循环指产品资本从一定的形式出发，经过一系列形式的变化，又回到原来出发点的运动。产品资本在循环过程中要经历三个不同的阶段，于此相联系的是资本依次执行三种不同的职能： 第一个阶段是购买阶段，即生产资料与劳动力的购买阶段。它属于商品的流通过程，在这一阶段，产业资本执行的是货币资本的职能。 第二个阶段是生产阶段，即生产资料与劳动者相结合生产物质财富并使生产资本得以增值，执行的是生产资本的职能。 第三个阶段是售卖阶段，即商品资本向货币资本的转化阶段。在此阶段产业资本所执行的是商品资本的职能，通过商品买卖实现商品的价值，满足人们的需要。 资本循环必须具备两个基本前提条件： 一是产业资本的三种职能形式必须在空间上同时并存，也就是说，产业资本必须按照一定比例同时并存于货币资本、生产资本和商品资本三种形式中。 二是产业资本的三种职能形式必须在时间上继起，也就是说，产业资本循环的三种职能形式必须保持时间上的依次连续性。 资本周转是资本反复不断的循环运动所形成的周期性运动。 影响资本周转最重要的两个要素是：一是资本周转的时间；二是生产资本的固定资本和流动资本的构成。要加快资本周转的时间，获得更多的剩余价值，就要缩短资本周转时间，加快流动资本周转速度。 第五章 2.垄断条件下竞争的特点 竞争目的上，垄断竞争是获取高额利润，并不断巩固和扩大自己的垄断地位和统治权力；竞争手段上，垄断组织的竞争，除采取各种形式的经济手段外，还采取非经济手段，使经济变得更加复杂、更加激烈； 在竞争范围上，国际市场的竞争越来越激烈，不仅经济领域的竞争多种多样，而且还扩大到经济领域范围以外进行竞争。 总之，垄断条件下的竞争，不仅规模大、时间长、手段残酷、程度更加激烈，而且具有更大的破坏性。 3.金融寡头如何握有话语权 金融寡头在经济领域中的统治主要通过“参与制”实现。所谓参与制，即金融寡头通过掌握

独立主格with用法小全

独立主格篇 独立主格，首先它是一个“格”，而不是一个“句子”。在英语中任何一个句子都要有主谓结构，而在这个结构中，没有真正的主语和谓语动词，但又在逻辑上构成主谓或主表关系。独立主格结构主要用于描绘性文字中，其作用相当于一个状语从句，常用来表示时间、原因、条件、行为方式或伴随情况等。除名词/代词+名词、形容词、副词、非谓语动词及介词短语外，另有with或without短语可做独立主格，其中with可省略而without不可以。*注：独立主格结构一般放在句首，表示原因时还可放在句末；表伴随状况或补充说明时，相当于一个并列句，通常放于句末。 一、独立主格结构： 1. 名词/代词+形容词 He sat in the front row, his mouth half open. Close to the bank I saw deep pools, the water blue like the sky. 靠近岸时，我看见几汪深池塘，池水碧似蓝天。 2. 名词/代词+现在分词 Winter coming, it gets colder and colder. The rain having stopped, he went out for a walk.

The question having been settled, we wound up the meeting. 也可以The question settled, we wound up the meeting. 但含义稍有差异。前者强调了动作的先后。 We redoubled our efforts, each man working like two. 我们加倍努力，一个人干两个人的活。 3. 名词/代词+过去分词 The job finished, we went home. More time given, we should have done the job much better. *当表人体部位的词做逻辑主语时，不及物动词用现在分词，及物动词用过去分词。 He lay there, his teeth set, his hands clenched, his eyes looking straight up. 他躺在那儿，牙关紧闭，双拳紧握，两眼直视上方。 4. 名词/代词+不定式 We shall assemble at ten forty-five, the procession to start moving at precisely eleven. We divided the work, he to clean the windows and I to sweep the floor.

in,on ,at, by ,of ,with 介词区别与用法

in，on ，at，by ，of ，with 介词区别与用法 in用在年月季节前，还有上午、下午等固定习语里 at用于传统的节日前，如at Christmas等；还有固定词组：at noon, at night;在点时间前用at 如at 7.15 on 用于具体的日期前，星期几，几号，包括那天的上午下午晚上等，如on Friday afternoon with: 一、with表拥有某物 Mary married a man with a lot of money . 马莉嫁给了一个有着很多钱的男人。 I often dream of a big house with a nice garden . 我经常梦想有一个带花园的大房子。 The old man lived with a little dog on the lonely island . 这个老人和一条小狗住在荒岛上。 二、with表用某种工具或手段 I cut the apple with a sharp knife . 我用一把锋利的刀削平果。 Tom drew the picture with a pencil . 汤母用铅笔画画。 三、with表人与人之间的协同关系 make friends with sb talk with sb quarrel with sb struggle with sb fight with sb play with sb work with sb cooperate with sb I have been friends with Tom for ten years since we worked with each other , and I have never quarreled with him . 自从我们一起工作以来，我和汤母已经是十年的朋友了，但我们从没有吵过架。 四、with 表原因或理由 John was in bed with high fever . 约翰因发烧卧床。 He jumped up with joy . 他因高兴跳起来。 Father is often excited with wine . 父亲常因白酒变的兴奋。 五、with 表“带来”，或“带有,具有”，在…身上，在…身边之意 The girl with golden hair looks beautiful . 那个金头发的女孩看起来漂亮。 The famous director will come to the meeting with the leading actor and actress .

功能和特点的区别Excel的主要功能和特点

功能和特点的区别Excel的主要功能和特点 Excel的主要功能和特点 Excel电子表格是office系列办公软的-种，实现对日常生活、工作中的表格的数据处理。它通过友好的人机界面，方便易学的智能化操作方式，使用户轻松拥有实用美观个性十足的实时表格，是工作、生活中的得力助手。 一、Excel功能概述; 1、功能全面:几乎可以处理各种数据 2、操作方便:菜单、窗口、对话框、工具栏 3、丰富的数据处理函数 4、丰富的绘制图表功能:自动创建各种统计图表 5、丰富的自动化功能:自动更正、自动排序、自动筛选等 6、运算快速淮确: 7、方便的数据交换能力 8、新增的Web工具 二、电子数据表的特点Excel 电子数据表软工作于Windows平台，具有Windows环境软的所有优点。而在图形用户界面、表格处理、数据分析、图表制作和网络信息共享等方面具有更突出的特色。工.图形用户界面Excel 的图形用户界面是标准的Windows的窗口形式，有控制菜单、最大化、最小化按钮、标题栏、菜单栏等内容。其中的

菜单栏和工具栏使用尤为方便。菜单栏中列出了电子数据表软的众多功能，工具栏则进一步将常用命令分组，以工具按钮的形式列在菜单栏的下方。而且用户可以根据需要，重组菜单栏和工具栏。在它们之间进行复制或移动操作，向菜单栏添加工具栏按钮或是在工具栏上添加菜单命令，甚至定义用户自己专用的菜单和工具栏。当用户操作将鼠标指针停留在菜单或工具按钮时,菜单或按钮会以立体效果突出显示，并显示出有关的提示。而当用户操作为单击鼠标右键时,会根据用户指示的操作对象不同，自动弹出有关的快捷菜单，提供相应的最常用命令。为了方便用户使用工作表和建立公式，Excel 的图形用户界面还有编辑栏和工作表标签。. 2.表格处理 Excel的另-个突出的特点是采用表格方式管理数据，所有的数据、信息都以二维表格形式(工作表)管理，单元格中数据间的相互关系一目了然。从而使数据的处理和管理更直观、更方便、更易于理解。对于曰常工作中常用的表格处理操作，例如，增加行、删除列、合并单元格、表格转置等操作，在Excel中均只需询单地通过菜单或工具按钮即可完成。此外Excel还提供了数据和公式的自动填充，表格格式的自动套用，自动求和，自动计算，记忆式输入，选择列表，自动更正，拼写检查，审核，排序和筛选等众多功能，可以帮助用户快速高效地建立、编辑、编排和管理各种表格。

with用法小结

with用法小结 一、with表拥有某物 Mary married a man with a lot of money . 马莉嫁给了一个有着很多钱的男人。 I often dream of a big house with a nice garden . 我经常梦想有一个带花园的大房子。 The old man lived with a little dog on the lonely island . 这个老人和一条小狗住在荒岛上。 二、with表用某种工具或手段 I cut the apple with a sharp knife . 我用一把锋利的刀削平果。 Tom drew the picture with a pencil . 汤母用铅笔画画。 三、with表人与人之间的协同关系 make friends with sb talk with sb quarrel with sb struggle with sb fight with sb play with sb work with sb cooperate with sb I have been friends with Tom for ten years since we worked with each other, and I have never quarreled with him . 自从我们一起工作以来，我和汤姆已经是十年的朋友了，我们从没有吵过架。 四、with 表原因或理由 John was in bed with high fever . 约翰因发烧卧床。 He jumped up with joy . 他因高兴跳起来。 Father is often excited with wine . 父亲常因白酒变的兴奋。 五、with 表“带来”，或“带有,具有”，在…身上，在…身边之意

as和with区别

一、with （介词）的复合结构：一做状语，二做定语。 作状语： 1:with +名词/代词，表示随着....， Times change and we must change with them. 时代变了,我们也要跟着变。 The risk of developing heart disease increases with the number of cigarettes smoked. 吸食香烟的数量越多，患心脏病的风险就越大。 Blood pressure decreases with exercise. 血压随着锻炼而降低。 With all his abilities, he failed completely. 尽管很有能力,他还是一败涂地了。 2：with +名词/代词+非谓语动词（现在分词/过去分词/不定式）作伴随状语/原因状语 (1 )with + 名词/代词+V-ing（现在分词） With prices going up so fast, we can't afford luxuries. 由于物价上涨很快，我们买不起高档商品。(原因状语) With the crowds cheering, they drove to the palace. 在人群的欢呼声中，他们驱车来到皇宫。(伴随情况) 注：现在分词和前面的名词或代词是逻辑上的主谓关系，也就是说动作由with 后的名词或代词发出，是主动者 (2)with + 名词/代词+V-ed（过去分词） I sat in my room for a few minutes with my eyes fixed on the ceiling. 我在房间坐了一会儿，眼睛盯着天花板。(伴随情况) She had to walk home with her bike stolen. 自行车被偷，她只好步行回家。(原因状语) 注：过去分词和前面的名词或代词是逻辑上的动宾关系，也就是说with后面的宾语是动作的承受者，是被动的 (3)with或without+名词/代词+ to do (动词不定式) With no one to talk to, John felt miserable.

各类格式的特点区分

在用各类软件设计时相信大家肯定存在着这样的问题，各种各样的格式让大家很是迷惑。没关系，福利来了，这里就给大家介绍了各种格式的特点应用。 TIFF格式 标签图像文件格式(Tagged Image File Format，简写为TIFF) 是一种主要用来存储包括照片和艺术图在内的图像的文件格式。它最初由Aldus公司与微软公司一起为PostScript 打印开发.TIFF文件格式适用于在应用程序之间和计算机平台之间的交换文件，它的出现使得图像数据交换变得简单。 TIFF是最复杂的一种位图文件格式。TIFF是基于标记的文件格式，它广泛地应用于对图像质量要求较高的图像的存储与转换。由于它的结构灵活和包容性大，它已成为图像文件格式的一种标准，绝大多数图像系统都支持这种格式。用Photoshop 编辑的TIFF文件可以保存路径和图层。 应用广泛 (1)TIFF可以描述多种类型的图像；(2)TIFF拥有一系列的压缩方案可供选择；(3)TIFF 不依赖于具体的硬件；(4)TIFF是一种可移植的文件格式。 可扩展性 在TIFF 6.0中定义了许多扩展，它们允许TIFF提供以下通用功能：(1)几种主要的压缩方法；(2)多种色彩表示方法；(3)图像质量增强；(4)特殊图像效果；(5)文档的存储和检索帮助。 格式复杂 TIFF文件的复杂性给它的应用带来了一些问题。一方面，要写一种能够识别所有不同标记的软件非常困难。另一方面，一个TIFF文件可以包含多个图像，每个图像都有自己的IFD 和一系列标记，并且采用了多种压缩算法。这样也增加了程序设计的复杂度。 文档图像中的TIFF TIFF格式是文档图像和文档管理系统中的标准格式。在这种环境中它通常使用支持黑白（也称为二值或者单色）图像的CCITT Group IV 2D压缩。在大量生产的环境中，文档通常扫描成黑白图像（而不是彩色或者灰阶图像）以节约存储空间。A4大小200dpi（每英寸点数分辨率）扫描结果平均大小是30KB，而300dpi的扫描结果是50KB。300dpi比200dpi更

with的用法

with[wIT] prep.1.与…(在)一起，带着：Come with me. 跟我一起来吧。/ I went on holiday with my friend. 我跟我朋友一起去度假。/ Do you want to walk home with me? 你愿意和我一道走回家吗 2.(表带有或拥有)有…的，持有，随身带着：I have no money with me. 我没有带钱。/ He is a man with a hot temper. 他是一个脾气暴躁的人。/ We bought a house with a garden. 我们买了一座带花园的房子。/ China is a very large country with a long history. 中国是一个具有历史悠久的大国。3.(表方式、手段或工具)以，用：He caught the ball with his left hand. 他用左手接球。/ She wrote the letter with a pencil. 她用铅笔写那封信。4.(表材料或内容)以，用：Fill the glass with wine. 把杯子装满酒。/ The road is paved with stones. 这条路用石头铺砌。5.(表状态)在…的情况下，…地：He can read French with ease. 他能轻易地读法文。/ I finished my homework though with difficulty. 虽然有困难，我还是做完了功课。6.(表让步)尽管，虽然：With all his money, he is unhappy. 尽管他有钱，他并不快乐。/ With all his efforts, he lost the match. 虽然尽了全力，他还是输了那场比赛。7.(表条件)若是，如果：With your permission, I’ll go. 如蒙你同意我就去。8.(表原因或理由)因为，由于：He is tired with work. 他工作做累了。/ At the news we all jumped with joy. 听到这消息我们都高兴得跳了起来。9.(表时间)当…的时候，在…之后：With that remark, he left. 他说了那话就离开了。/ With daylight I hurried there to see what had happened. 天一亮我就去那儿看发生了什么事。10. (表同时或随同)与…一起，随着：The girl seemed to be growing prettier with each day. 那女孩好像长得一天比一天漂亮。11.(表伴随或附带情况)同时：I slept with the window open. 我开着窗户睡觉。/ Don’t speak with your mouth full. 不要满嘴巴食物说话。12.赞成，同意：I am with you there. 在那点上我同你意见一致。13.由…照看，交…管理，把…放在某处：I left a message for you with your secretary. 我给你留了个信儿交给你的秘书了。/ The keys are with reception. 钥匙放在接待处。14 (表连同或包含)连用，包含：The meal with wine came to ￡8 each. 那顿饭连酒每人8英镑。/ With preparation and marking a teacher works 12 hours a day. 一位老师连备课带批改作业每天工作12小时。15. (表对象或关系)对，关于，就…而言，对…来说：He is pleased with his new house. 他对他的新房子很满意。/ The teacher was very angry with him. 老师对他很生气。/ It’s the same with us students. 我们学生也是这样。16.(表对立或敌对)跟，以…为对手：The dog was fighting with the cat. 狗在同猫打架。/ He’s always arguing with his brother. 他老是跟他弟弟争论。17.(在祈使句中与副词连用)：Away with him! 带他走！/ Off with your clothes! 脱掉衣服！/ Down with your money! 交出钱来！ 【用法】1.表示方式、手段或工具等时(=以，用)，注意不要受汉语意思的影响而用错搭配，如“用英语”习惯上用in English，而不是with English。2.与某些抽象名词连用时，其作用相当于一个副词：with care=carefully 认真地/ with kindness=kindly 亲切地/ with joy=joyfully 高兴地/ with anger=angrily 生气地/ with sorrow=sorrowfully 悲伤地/ with ease=easily 容易地/ with delight=delightedly 高兴地/ with great fluency =very fluently 很流利地3.表示条件时，根据情况可与虚拟语气连用：With more money I would be able to buy it. 要是钱多一点，我就买得起了。/ With better equipment, we could have finished the job even sooner. 要是设备好些，我们完成这项工作还要快些。4.比较with 和as：两者均可表示“随着”，但前者是介词，后者是连词：He will improve as he grows older. 随着年龄的增长，他会进步的。/ People’s ideas change with the change of the times. 时代变了，人们的观念也会变化。5.介词with和to 均可表示“对”，但各自的搭配不同，注意不要受汉语意思的影响而用错，如在kind, polite, rude, good, married等形容词后通常不接介词with而接to。6.复合结构“with+宾语+宾语补足语”是一个很有用的结构，它在句中主要用作状语，表示伴随、原因、时间、条件、方式等；其中的宾语补足语可以是名词、形容词、副词、现在分词、过去分词、不定式、介词短语等：I went out with the windows open. 我外出时没有关窗户。/ He stood before his teacher with his head down. 他低着头站在老师面前。/ He was lying on the bed with all his clothes on. 他和衣躺在床上。/ He died with his daughter yet a schoolgirl. 他去世时，女儿还是个小学生。/ The old man sat there with a basket beside her. 老人坐在那儿，身边放着一个篮子。/ He fell asleep with the lamp burning. 他没熄灯就睡着了。/ He sat there with his eyes closed. 他闭目坐在那儿。/ I can’t go out with all these clothes to wash. 要洗这些衣服，我无法出去了。这类结构也常用于名词后作定语：The boy with nothing on is her son. 没穿衣服的这个男孩子是她儿子。 （摘自《英语常用词多用途词典》金盾出版社） - 1 -

认清维也纳华尔兹中的重要区别

认清维也纳华尔兹中的重要区别 维也纳华尔兹中的重要区别： 1、左转步与右转步不相同。左转步反身，右转步摆荡； 2、男士步法与女士步法不相同。男士前进摆荡，女士前进无摆荡； 3、前进小节与后退小节不相同。男士前进小节大步向前，后退小节小步调整； 4、节拍长短不相同。每一拍时间值长短不相同，不是平均占一拍。具体来说： 1、维也纳华尔兹左转步与右转步不相同。 在维也纳华尔兹中，右转和左转的跳法是不对称的，右旋转是横并式结构，右转步强调向前流动，强调摆荡，有倾斜，有起伏，步幅大，以单侧拉腰为主；左旋转是锁式结构，左转步强调拧腰胯，反身，无摆荡，无升降，锁步，步幅小，要不停地反身。 2、维也纳华尔兹男士步法与女士步法不相同。

在维也纳华尔兹中，男士与女士步法不相同，男士的前进转身小节是女士的后退转身小节，男士前进右转摆荡，女士后退右转也摆荡；男士后退右转无摆荡，女士前进右转也无摆荡。 3、维也纳华尔兹前进小节与后退小节跳法不相同。 在维也纳华尔兹中，前进与后退小节跳法不相同，男士前进小节大步向前，后退小节小步调整。右转男士后退（女士前进）那个小节不摆荡，步子也较小，相当于休息。 4、维也纳华尔兹中节拍长短不相同。 在维也纳华尔兹中，每两小节六步为一组，每一节拍时间值长短不相同，不都是平均占一拍。六个节拍时间值分别是：1.5、0.75、0.75、1.5、0.75、0.75，第一、四拍最长，第三、六拍最短，口令：慢、快、快、慢、快、快 跳快三的要领 (2011-11-14 10:18:09) 转载▼ 标签：

杂谈

维也纳华尔兹俗称快三，它是舞中之王，跳快三是很难跳得好的，我虽然跳舞多年，长期以来被错误的观点支配，也是最近才掌握到跳快三的要领。 快三看似简单，只有四种基本步法，左转、右转、左换步、右换步，但如果不掌握要领，光靠看视频，听舞友指点，不容易领会关键的要领，舞就跳不好。 很多人以为快三就是比慢三转快一些，这就错了，这也是跳不好快三的原因。我以前也是用这种思维去跳的，结果一直转不好，转起来不畅顺，不能绕舞池转。开始还以为对方没跳好，但与多个人跳过也不好，最近才发现，是自己没跳对，不会带舞伴，跳和带的方法不对。 不久前，在网上无意间找到了2句跳好快三的要决，原来快三的转与慢三的转完全不一样，慢三是转园圈，像车轮那样。而快三的转是折转或翻转，像蛇爬行时一样。其次，快三不是以3拍为一小节，而是以6拍为一小节。一小节中跳半个大圈和半个小圈，不是两个半圈相同的，跳时男女互相错开，男跳大半圈时女跳小半圈，只有跳大半圈时才发力。这就是对快三的新认识，是跳快三的要领。 从以上认识入手，还需要学会用力的方法，以前我和很多人一样用手发力来带对方转，这显然不能到位。其实，关键是要从腰发力，用侧腰的力去带动身体前进，以前进带动转动。 快三的左转和右转也很不一样，很多人右转不错，但左转就不妥，这也是上面说的原因，没有认识快三的转的实质。在跳快三过