河北省邯郸一中2009届高三上学期期末考试英语试题第I卷第一部分英语知识运用第一节语音知识 (共5小题,每小题1分.满分5分)从A.B.C.D四个选项中.找出其划线部分与所给单词的划线部分——青夏教育精英家教网—

河北省邯郸一中2009届高三上学期期末考试

英语试题

第I卷（选择题，共95分）

第一部分英语知识运用（共三节，满分50分）

第一节语音知识（共5小题；每小题1分，满分5分）

从A、B、C、D四个选项中，找出其划线部分与所给单词的划线部分读音相同的选项，并在答题卡上将该项涂黑。

1．stop A．lose B．woman C．shock D．rose

2．southern A．courage B．south C．youth D．trousers

3．wear A．near B．require C．cheer D．share

4．Society A．achieve B．friend C．science D．believe

5．throw A．nowadays B．tomorrow C．crowd D．town

第二节语法和词汇知识（共15小题；每小题1分，满分15分）

从A、B、C、D四个选项中，选出可以填入空白处的最佳选项，并在答题卡上将该项涂黑。

6．I don' t think we have met before. You' re taking me for ______.

A．some other B．someone else

C．other person D．one other

7．You must keep on working in the evening ______ you are sure you can finish the task in time.

A．as B．if C．when D．unless

8．-- Shall we go to the art exhibition right away?

-- ______.

A．It' s your opinion B．I don' t mind

C．It' s all up to you D．That' s your decision

9．Kathy _____ a lot of Spanish by playing with the native boys and girls.

A．picked up B．took up C．made up D．turned up

10．After dinner he gave Mr. Richardson ______ ride to ______Capital Airport.

A．the; a B．a; the C．不填; a D．不填; the

11．Helen _______ go on the trip with us, but she isn't quite sure yet.

A．shall B．must C．may D．can

12．The manager had fallen asleep where he______, without undressing.

A．was laying B．was lying

C．had laid D．had lied

13．My _______ of this weekend's activity is going out with some good friends.

A．idea B．opinion C．mind D．thought

14．That was really a splendid evening. It's years _______ I enjoyed myself so much.

A．when B．that C．before D．since

15．-- I'm terribly sorry I broke your glass.

-- ______.

A．That's right B．Bad luck C．Sorry D．You can forget it

16．John is the tallest boy in the class, ___________ according to himself.

A．five foot eight as tall as B．as tall as five foot eight

C．as five foot eight tall as D．as tall five foot eight as

17．Great changes have taken place in that school. It is no longer ______ it was 20 years ago, _______ it was so poorly equipped.

A．what; when B．that; which C．what; which D．which; that

18．He is such a man who is always ________ fault with other people.

A．putting B．seeking C．finding D．looking for

19．I really can' t understand ______her like that.

A．you treat B．you to treat C．why treat D．you treating

20．-- Maggie had a wonderful time at the party.

-- ______ and so did I.

A．So she had B．So had she C．So she did D．So did she

第二节：完形填空(共20小题；每小题1.5分，满分30分)

阅读下面短文，从短文后所给各题的四个选项(A、B、c和D)中，选出可以填入空白处的最佳选项，并在答题卡上将该项涂黑。

I did very badly at school. My headmaster thought I was 21 and when I was 14 he said, "You’re never going to be 22 but a failure. "

After five years of 23 jobs, I fell in love with a very nice middle-class girl. It was the best _24 that could have happened to me. 25 I wanted to do something positive with my life because I wanted to prove to 26 that what people said about me was 27 . Especially her mother, who had said to me, "Let's 28 it, you've failed at everything you've ever done. " So I tried hard with my 29 and went to college. My first novel (小说) 30 while I was at college.

After college I taught during the 31 in high schools and attended evening classes at London University, where I got a 32 in history. I became a lecturer at a college and was thinking of 33_ that job to write full time 34 I was offered a part-time job at Leeds University. I began to feel proud of myself -- 35 was a working-class boy who'd 36 school early, now teaching at the university.

My writing career (职业) took off when I discovered my own style. Now I' m rich and 37 , have been on TV, and met lots of film stars. 38 what does it mean? I 39 wish all the people that have put me down had _40 : "I believe in you. You'll succeed."

21．A．bright B．useless C．simple D．hopeful

22．A．anything B．something C．everything D．nothing

23．A．low B．poor C．good D．useful

24．A．support B．happiness C．surprise D．thing

25．A．admitted B．decided C．planned D．told

26．A．me B．them C．her D．it

27．A．wrong B．right C．stupid D．faulty

28．A．see B．know C．understand D．face

29．A．experiment B．practice C．writing D．composition

30．A．came on B．came in C．came out D．came back

31．A．day B．night C．month D．year

32．A．graduation B．pass C．degree D．success

33．A．giving in B．giving back C．giving out D．giving up

34．A．while B．if C．when D．or

35．A．there B．here C．it D．that

36．A．left B．attended C．changed D．graduated

37．A．tired B．calm C．nervous D．famous

38．A．And B．But C．However D．Well

39．A．just B．exactly C．so D．very

40．A．praised B．said C．answered D．advised

第二部分阅读理解（共25小题，第一节每小题2分，第二节每小题1分；满分45分）

第一节，阅读下面短文，从每题所给的四个选项（A、B、C和D）中，选出最佳选项，并在答题卡上将该项涂黑。

A

We were on tour a few summers ago, driving through Chicago, when right outside of the city, we got pulled over. A middle-aged policeman came up to the car and was really being troublesome at first. Lecturing us, he said, "You were speeding. Where are you going in such a hurry?" Our guitarist, Tim, told him that we were on our way to Wisconsin to play a show. His way towards us totally changed. He asked, "Oh, so you boys are in a band (乐队)?" We told him that we were. He then asked all the usual broad questions about the type of music we played, and how long we had been at it. Suddenly, he stopped and said, "Tim, you want to get out of this ticket, don't you?" Tim said, "Yes." So the officer asked him to step out of the car. The rest of us, inside the car, didn't know what to think as we watched the policeman talk to Tim. Next thing we knew, the policeman was putting Tim in the back of the police car he had parked in front of us. With that, he threw the car into reverse(倒车), stopping a few feet in back of our car. Now we suddenly felt frightened. We didn't know if we were all going to prison, or if the policeman was going to sell Tim on the black market or something. All of a sudden, the policeman's voice came over in a loudspeaker. He said, "Ladies and gentlemen, for the first time ever, we have Tim here singing on Route 90.” It turned out the policeman had told Tim that the only way he was getting out of the ticket was if he sang part of one of our songs over the loudspeaker in the police car. Seconds later, Tim started screaming into the receiver. The policeman enjoyed the performance, and sent us on our way without a ticket.

41．The policeman stopped the boys to ______.

A．put them into prison B．give them a ticket

C．enjoy their performance D．ask some band questions

42．The policeman became friendly to the boys when he knew they ____

A．had long been at the band B．played the music he loved

C．were driving for a show D．promised into a performance

43．The boys probably felt ______ when they drove off.

A, joyful B．calm C．nervous D．frightened

B

Collections were the inspiration(灵感) for a project at Thomas Tallis School, which formed part of the Imagine Children's Literature Festival last autumn. Each child (aged 12-13) beatified （美化）a box and wrote a story on the subject of collections to throw inside it. The boxes were spread within the Royal Festival Hall's Ballroom. Some were left empty to encourage visitors to write their own stories.

The subject chosen by Luren was an imaginative one. "It's a sort of Cinderella (灰姑娘) story," she told me, inspired by a collection of letters from her cousin, in the story these become love letters, burned by a cruel stepmother. Lauren's best friend Charlotte is the stepmother. "I'm in Charlotte's story too," says Lauren, "and I get run over." Charlotte's tale was inspired by the girls' coin collection. "We've collected foreign coins for years ? since our families went on holiday to Tenerife." she explains. "That was before the Euro, so we put pesetas in." Lauren continues: "I find a coin in the road, go to get it and get run over. I'm in hospital and then I die." Charlotte adds: "Or she might not die. I haven't decided yet."

Millie Murray, who is a teen-novel author, thinks that setting the subject of collections was a useful inspiration to their creativity rather than a restriction(限制). "In the beginning I thought, 'Will the children be able to do it?'" she says. "But it's been fruitful. Some have their own collection, some have parents who do, and some have written complete stories. It's made them think about something they wouldn't have otherwise, which can only be a good thing."

44．What were the children asked to do in the project?

A．To meet friends at Thomas Tallis School

B．To write stories on the subject of collections.

C．To encourage visitors to write their own stories.

D．To have their friends for characters in the stories.

45．The underlined word "pesetas" in Paragraph 2 is a kind of _______.

A．Charlotte hurt herself when getting a coin

B．both of them developed their imagination

C．both of them will die in each other's stories

D．Latwen's cousin posted her some love letters

46．From the stories by Lauren and Charlotte, we know that _____ .

A．Charlotte hurt herself when getting a coin

B．both of them developed their imagination

C．both of tram will die in each other's stories

D．Latwen's cousin posted her some love letters

47．Millie Murray thinks ________.

A．collections could inspire writing creativity

B．it was good for parents to have collections

C．inspirations were very useful in writing stories

D．setting collection subjects restricted inspirations

C

Paula Radcliffe, chasing (角逐) a third London marathon title(冠军), says she has become a stronger person after her terrible experience at the 2004 Athens Games.

Radcliffe, who failed to complete the Olympic marathon and the 10,000m last August, said: "Athens made me a stronger person and it made me care less about criticism (批评)"

"In the past I wanted to please everyone, but now I am going to listen even more to the people around me."

She didn't care about criticism made at the weekend by Liz McCulgan, who felt Radcliffe should have rested and let her body recover after her failure in Athens.

"Liz is someone I look up to but she hasn't spoken to me since last year and if she really cared for me, I'm sure she would have contacted (联系) me."

Instead Radcliffe won the New York City marathon just 11 weeks after Athens.

"In New York I wasn't in my best state but I did know I was good enough to win the race. Radcliffe insisted her only goal in Sunday's race would be winning a third title and not chasing world records.

However, RadcIiffe has not ruled out(排除) in the future chasing her "final" world record time and questioned sayings that marathon runners have the ability in their career to produce only four or five world-class times.

"I don't think that -- although I can't put a number on it," said RadcLiffe. "That changes from person to person."

Radcliffe is sure she can better her winning London 2003 performance at some point in the future. Following a successful three-month training period in the United States, the 31-year-old will chase a third title on Sunday after her first victory in 2002 and again 12 months later.

Radcliffe clocked a time of 2:18:56 in her first 42.2-kilometre race three years ago.

Afterwards she set a "mixed course" mark of 2:17:18 five months later in Chicago before lowering that to a time of 2:15:25 in the 2003 London event.

48．Radcliffe's failure in Athens made her ________

A．develop respect for Liz B．love people around her more

C．rest for five months D．face criticism calmly

49．Which of the following is true according to the passage?

A．Radcliffe broke the world record in the New York City marathon.

B．Radcliffe didn't fully recover before the New York City marathon.

C．Radcliffe won her first marathon title in the New York City marathon.

D．Radcliffe had a 3-month training before the New York City marathon.

50．By saying "I can't put a number on it," Radcliffe means she's not sure____.

A．if she has the ability to set a new world record

B．if she can win another race though she has won many tunes

C．how many times a marathon runner can set the world record

D．if she has the ability to produce four or five world-class times

51．According to the text, Redcliffc has won ____ London marathon title(s).

A．four B．three C．two D．one

52．What can we learn from Radcliffe's story?

A．Practice makes perfect.

B．Well begun is half done.

C．A friend in need is a fried indeed.

D．Where there is a will there is a way.

D

From Mr. Ward Hoffman.

Sir, I was halfway through Professor Raj Persaud's article “What's the tipping point"

(Financial Times Weekend, April 9-l0) when it occurred to me that what I was reading was not ironic(讽刺的). If Prof Persaud wants to know why Americans tip in restaurants, he needs only to ask the first American he meets in London.

Americans tip(小费) in restaurants for one reason, and one reason only: we tip to supplement (补贴) the salary of restaurant workers. Quality of service does not enter into it, beyond the fact that one may tip a bit less for poor service, or a little more for good service.

Not tipping at all in a non-fast-food restaurant is not a choice. In the US, one used to tip about 15 per cent for dining in a family-style restaurant or in an up-market (高档的) restaurant. Here, in San Francisco Bay area restaurants, we are encouraged to tip 20 per cent or more, to help restaurant workers live in this very expensive area.

After eating at an Italian restaurant in my city, I left a tip of 20 per cent on the non-tax part of our dinner bill. It was expecteD．There is nothing more complicated (复杂的) than that about Americans tipping in restaurants.

Ward Hoffman,

Palo Alto, CA 94306, US

* * *

From Mr. Philip McBride Johnson.

Sir, I agree with most of Raj Persaud's opinion about the doubtful value of tipping, but with one exception(例外). Tips can be very useful when one is a repeat customer or diner.

It is only when the tipper is a stranger and likely to remain so that the system does not work to his or her advantage. But frequent a hotel or a restaurant, always tip a bit more, and the difference in service and treatment will be easily felt.

Phfiip McBnde Johnson,

Great Falls, VA 22066, US

53．What can we learn from Hoffrnan's letter?

A．Quality of service determines tipping in the US.

B．Americans don't tip in non fast-food restaurants.

C．Tipping in US market restaurants is unnecessary.

D．How to tip in the United States is not complicated.

54．Johnson's letter shows ________.

A．a stranger in a restaurant is likely to tip a bit more

B．diners receive better service if they frequent a restaurant

C．repeat diners may get good service if they tip a bit more

D．the tipping system works to the advantage of new customers

55．From the two letters, we can learn Professor Raj Persaud ______ .

A．feels doubtful about the value of tipping

B．believes tipping improves quality of service

C．wants to ask Hoffman about tipping m the US

D．thinks tipping a bit mom one can get good service

56．The two letters most probably appears in a ______.

A．notice B．handbook

C．book review D．newspaper

E

At Dallm/Fort Worth Airport, the lights are controlled by sensors(传感器) that measure sunlight. They dim immediately when it's sunny and brighten when a passing cloud blocks the sun.

A wall of windows at a University of Pennsylvania engineering budding has built-in blinds (百页窗) controlled by a computer program that follows the sun's path.

Buildings are getting smarter -- and the next generation of building materials expected to do even more.

Windows could catch the sun's energy to heat water. Sensors that measure the carbon dioxide breathed out by people in a room could determine whether the air conditioning needs to be turned up.

Many new materials and technology have been designed in the last 15 years. They are now being used in a wave of buildings designed to save as much energy as possible. They include old ideas, like "green roofs," where a belt of plants on a roof helps the building keep heat in winter and stay cool in summer, and new ideas, like special coating for windows that lets light in, but keeps heat out.

As technologies such as sensors become cheaper, their uses spread.

The elevators (电梯) at Seven World Trade Center, which is under construction in New York, use a system that groups people traveling to nearby floor into the same elevator, thus saving elevator stops. People who work in the building will enter it by swiping (刷) ID cards that will tell the elevators their floor; readouts will then tell them which elevator to use. The building also has windows with a coating that blocks heat while letting in light.

More new building materials and technology are in development. A Philadelphia building firm is now working on "smart wrap" that uses tiny solar collectors to catch the sun's energy and transmitters (传输器) the width of a human hair to move it. They are expected to change the face of the construction industry in the next ten years or so.

57．_______ will be developed and used in the construction industry.

A．"Green mops" that cool or heat buildings

B．"Smart wrap" that catches the sun's energy

C．Sunlight-measuring sensors that control lights

D．Window coating that lets light in, but keeps heat out

58．The elevators at Seven World Trade Center are special because they can ______.

A．send people to floors with fewer stops

B．teach people how to use their ID cards

C．make people stay very cool in summer

D．help people go traveling in the building

59．The underlined word "it' in the last paragraph refers to _____.

A．a human being B．smart wrap

C．the sun's energy D．a transmitter

60．What might be the most suitable title for the text?

A．Buildings Are Becoming Smarter

B．Buildings Are Getting More Sunlight

C．Buildings Are Lacking in Much Energy

D．Buildings Are Using Cheaper Materials

(二)根据对话情景和内容，从对话后所给的选项中选出能填入每一空白处的最佳选项，并在答题卡上将该选项涂黑。选项中有两个为多余选项。

(注意：61----65填涂到答题卡71----75处)

Sandy is at home. She is now calling Lisa, her classmate.

Sandy: Hi, Lisa. Have you finished helping your parents?

Lisa: I finished cleaning the living room a little while ago, but I haven’t cleaned my bedroom yet. 61

Sandy: I know what you mean. 62 Do you think you’ll be finished soon?

Lisa: I should be done in about an hour.

Sandy: 63

Lisa: I did the English homework last night. 64 I don’t understand it.

Sandy: Me neither. 65_ We can help each other work it out.

Lisa: OK. Then we can go and have our hair cut.

Sandy: Great. See you in a little while.

A．I don’t like math at all.

B．But I haven’t started my math yet.

C．Housework is tiring, and I’m tired

D．What do you think we should do then?

E. Why don’t we do it together this afternoon?

F. Have you looked at Monday’s homework yet

G. I also hate doing the cleaning around the house.

第Ⅱ卷（非选择题,共55分）

第三部分写作（共三节，满分55分）

第一节单词拼写（共10小题, 每小题1分，满分10分）

66．______(十二月) is the twelfth month of the year. 66._____________

67．Now we have a growing population of senior______(公民). 67._____________

68．They are ____ (决心的) to solve problems of common people. 68._____________

69．My brother often comes to see me,_____(尤其) when he is free. 68.___________

70．A lovely bird______(飞)past me just now. 70._____________

71．Operated on in time, the ______(病人) is likely to recover soon. 71.___________

72．What he said ____(听起来) so strange that I couldn’t believe it! 72.___________

73．Well, you are the______ (二十), that is , the last to come. 73._____________

74．I must ______(道歉) for not having written to you sooner. 74._____________

75．In summer, a sudden change in ______(温度) could bring rain. 75.___________

When I was four years old , I got ill. I took medicine twice

a day. The medicine had so a bitter taste that I took it mixed in 76．________

orange juice . The problem was that I can still taste the medicine. 77．________

In a particular afternoon my mother brought the drink in. As 78．________

she gave it to me, when the phone rang and she went to answer 79．________

it. I looked at the orange drink and decided I couldn’t face them, 80．________

so I hid the glass behind a pile magazines . Of course, when 81．________

my mother asked, “ Have you already taken your medicine ? ” 82．_______

I said, “Yes. ” I felt very pleased with me. The next day, my 83．________

mother found drink when she was doing the housework. She 84．________

looked at me serious and said, “ It’s bad to tell lies ! ” 85．________

第二节：书面表达（满分30分）

假设2008年2月12日是你父亲的生日，下面三幅图描绘的是你给父亲买完礼物后乘坐地铁回家时经历的一件事。请根据图示，用英语写一篇日记记叙这件事，并谈谈自己的感受。

注意：

1．词数不少于100；

2．可适当加入细节，以使行文连贯。

参考词汇：

地铁（列车）----- subway train

February 12^th, 2008

________________________________________________________________________________________________________________________________________________________

_______________________________________________________________________________

试题详情

章节复习坐标系

[教学目标]

试题详情

《数学实验：摆线探究》教学设计

教学目标

知识目标：

在数学实验平台中，观察圆在直线上滚动时圆上定点的轨迹(平摆线)，了解平摆线的生成过程并能推导出参数方程．

了解动圆在定圆上滚动时形成的摆线（外摆线、内摆线、环摆线)的生成过程，了解三种摆线的内在联系及相应分类，会在具体刻画中对变幅摆线进行简单的分类．

了解摆线在实际中应用的实例．

能力目标：

提高学生的信息技术素养，提高学生的观察能力和研究能力

情感目标：

培养学生主动探求知识、合作交流的意识，在学生的亲身操作中，感受曲线的美感受，数学的力量，改善数学学习信念．

教学过程：

一、平摆研究

1、情境创设

问题：如图，一个人的自行车外带上沾了一点白色油漆，当他骑车向前直行时，这个白色油漆斑点在空中会描出一条什么样的曲线？

引导学生拖动点C，使车轮在地面上滚动，观察点P的轨迹；学生会发现，点P的轨迹是一条以前没有研究过的曲线．从而引入主题，即：一个圆沿着一条直线作无滑动的滚动时，圆周上的一个定点的轨迹叫做平摆线，又叫旋轮线.．

操作提示：先按【滚动】或直接拖动点C，使得圆在直线上无滑滚动，初步认识、猜想点P轨迹的形状；继而【追踪】，直观刻画点P的轨迹形状，

2、方程探究

通过对平摆线的直观研究，可以发现平摆线的一些性质如：图象是由一些呈周期性排列的拱组成，每个拱的拱高为2r，拱底长为2πr．引导学生认识到进一步研究平摆线需要研究其曲线方程．

以问题：“如何才能实现动圆在直线上无滑滚动呢？”为主线引导学生在实验平台中探究，学生容易发现在滚动过程中保持线段AC及弧长的长度相等，从而进一步尝试方程的推导如下：

设是轨迹上任一点，

则

那么，从而得出相应的轨迹参数方程．

操作提示：按【比较】会显示线段AC及弧长的动态度量值，从而能帮助学生认识其中的规律．

3、平摆变幅

在问题：“P点的轨迹是平摆线，若直线OP上另有一点Q，那么圆在直线上作无滑滚动时，Q点的轨迹如何呢？” 的指引下，引导学生探究相应的轨迹，从而得出下列结论：一个圆沿着一条直线作无滑动的滚动时，圆周上的一个定点的轨迹叫做平摆线，圆内一个定点的轨迹叫做短幅平摆线；圆外一个定点的轨迹叫做长幅平摆线．

操作提示：按纽【轨迹】会给出点Q的轨迹，拖动点Q能帮助学生初步对变幅摆线形成动态印象；按纽【比较】会给出两条典型的短幅摆线和长幅摆线，再次拖动点Q可以帮助学生形成整体动态认识，从而能认识到可以根据点Q相对于圆的位置对变幅摆线加以分类．

二、拓展研究

1、外摆线探究

有了平摆线的研究经历和基础，学生对类似情境“动圆在定圆外滚动会得到什么样的曲线呢”的研究会产生极大的兴趣．学生在如图所示的实验平台中容易发现，轨迹是一条与平摆线相似的曲线，从而引出探究主题，即动圆在定圆外无滑滚动时，动圆圆周上的一个定点的轨迹是外摆线．

操作提示：与平摆线的研究相类似，学生在外摆线的探究上也须经历一个由感性认识到理性升华的过程，先让学生在【滚动】中猜想外摆线的形状，通过【追踪】验证直觉判断，继而通过【比较】度量值的大小，从而对动圆在定圆外无滑滚动有所感悟，从而对外摆线有个全面而深刻的认识．

2、摆线拓展

在前面探究的基础上，改变相应的R、r的值，可以得到不同类型的摆线模型，初步认识到摆线的形状取决于R、r的比例．

操作提示：选中参数R或r的度量值，按动小键盘上的“+”或“-”改变相应参数的值，可以发现摆线的形状发生变化，当r值为负时表示动圆在定圆内滚动；拖动半径r的两端点可以改变图形的大小，从而能在屏幕上得到合适的显示．

3、摆线全景

在实验平台中同时显示四个图，目的是让学生对摆线形成整体认识，即摆线包括三种类型：外摆线、内摆线、环摆线，并在比较中归纳出摆线的分类（如下表，根据R、r的相对比例）：

操作提示：按动按纽【选中R】将选中全部四个图中参数R，这样再按小键盘上的“+”或“-”就可以同时改变四个图中的参数了，从而可以地实现对摆线的整体研究，按动按纽【选中r】也有相类似的效果．

在R或r的值过大的情况下，对应的图形也会超出相应的范围，这样不利于观察图形的特征，可以拖动绿色点（与红色点接近图形缩小，反之则图形放大）后再按按纽【统一长度】即可解决问题．

4、摆线变幅

在顺利研究了摆线的分类后，与平摆线的变幅相类似，学生在问题“P点的轨迹是摆线，若直线OP上另有一点Q，那么动圆P在定圆O外（或内）作无滑滚动时，Q点的轨迹如何呢”的引领下，很自然地会想到研究摆线的变幅．

在相应数学实验平台的支持下，学生容易认识到：当Q点在圆P上时，Q点的轨迹是摆线；当Q点在圆P内时，Q点的轨迹是短幅摆线；当Q点在圆P外时，Q点的轨迹是长幅摆线．

操作提示：在显示【轨迹】的基础上，拖动点Q，随着点Q相对于动圆的位置不同，将产生不同的变幅摆线；同时也可改变参数R或r，观察不同类型的摆线的变幅，感受其异同之处，对摆线的变幅形成整体认识．

5、哥白尼定理

在摆线全景中，学生会发现特殊情形下的摆线会成为一条直线，从而产生探究兴趣．动圆P在定圆O内无滑滚动且R=2|r|时，动圆上一个定点的轨迹是定圆的一条直径（哥白尼定理）．根据哥白尼定理，可以把旋转运动变成往返的直线运动，这一定点机械设计上是很有用的．上述条件下，动圆所在平面内与动圆固定地连接在一起的圆内（或圆外）一点的轨迹是椭圆（卡丹转盘）．