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河北省邯鄲一中2009屆高三上學期期末考試

英語試題

第I卷（選擇題，共95分）

第一部分英語知識運用（共三節，滿分50分）

第一節語音知識（共5小題；每小題1分，滿分5分）

從A、B、C、D四個選項中，找出其劃線部分與所給單詞的劃線部分讀音相同的選項，并在答題卡上將該項涂黑。

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

第二節語法和詞匯知識（共15小題；每小題1分，滿分15分）

從A、B、C、D四個選項中，選出可以填入空白處的最佳選項，并在答題卡上將該項涂黑。

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

第二節：完形填空(共20小題；每小題1.5分，滿分30分)

閱讀下面短文，從短文后所給各題的四個選項(A、B、c和D)中，選出可以填入空白處的最佳選項，并在答題卡上將該項涂黑。

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 (小說) 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 (職業) 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

第二部分閱讀理解（共25小題，第一節每小題2分，第二節每小題1分；滿分45分）

第一節，閱讀下面短文，從每題所給的四個選項（A、B、C和D）中，選出最佳選項，并在答題卡上將該項涂黑。

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 (樂隊)?" 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(倒車), 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(靈感) 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(冠軍), 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 (批評)"

"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 (聯系) 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(諷刺的). 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(小費) in restaurants for one reason, and one reason only: we tip to supplement (補貼) 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 (高檔的) 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 (復雜的) 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(傳感器) 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 (百頁窗) 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 (電梯) 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 (傳輸器) 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)根據對話情景和內容，從對話后所給的選項中選出能填入每一空白處的最佳選項，并在答題卡上將該選項涂黑。選項中有兩個為多余選項。

(注意：61----65填涂到答題卡71----75處)

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（非選擇題,共55分）

第三部分寫作（共三節，滿分55分）

第一節單詞拼寫（共10小題, 每小題1分，滿分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 ____ (決心的) 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______(飛)past me just now. 70._____________

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

72．What he said ____(聽起來) 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 ______(溫度) 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．________

第二節：書面表達（滿分30分）

假設2008年2月12日是你父親的生日，下面三幅圖描繪的是你給父親買完禮物后乘坐地鐵回家時經歷的一件事。請根據圖示，用英語寫一篇日記記敘這件事，并談談自己的感受。

注意：

1．詞數不少于100；

2．可適當加入細節，以使行文連貫。

參考詞匯：

地鐵（列車）----- subway train

February 12^th, 2008

________________________________________________________________________________________________________________________________________________________

_______________________________________________________________________________

試題詳情

章節復習坐標系

[教學目標]

試題詳情

《數學實驗：擺線探究》教學設計

教學目標

知識目標：

在數學實驗平臺中，觀察圓在直線上滾動時圓上定點的軌跡(平擺線)，了解平擺線的生成過程并能推導出參數方程．

了解動圓在定圓上滾動時形成的擺線（外擺線、內擺線、環擺線)的生成過程，了解三種擺線的內在聯系及相應分類，會在具體刻畫中對變幅擺線進行簡單的分類．

了解擺線在實際中應用的實例．

能力目標：

提高學生的信息技術素養，提高學生的觀察能力和研究能力

情感目標：

培養學生主動探求知識、合作交流的意識，在學生的親身操作中，感受曲線的美感受，數學的力量，改善數學學習信念．

教學過程：

一、平擺研究

1、情境創設

問題：如圖，一個人的自行車外帶上沾了一點白色油漆，當他騎車向前直行時，這個白色油漆斑點在空中會描出一條什么樣的曲線？

引導學生拖動點C，使車輪在地面上滾動，觀察點P的軌跡；學生會發現，點P的軌跡是一條以前沒有研究過的曲線．從而引入主題，即：一個圓沿著一條直線作無滑動的滾動時，圓周上的一個定點的軌跡叫做平擺線，又叫旋輪線.．

操作提示：先按【滾動】或直接拖動點C，使得圓在直線上無滑滾動，初步認識、猜想點P軌跡的形狀；繼而【追蹤】，直觀刻畫點P的軌跡形狀，

2、方程探究

通過對平擺線的直觀研究，可以發現平擺線的一些性質如：圖象是由一些呈周期性排列的拱組成，每個拱的拱高為2r，拱底長為2πr．引導學生認識到進一步研究平擺線需要研究其曲線方程．

以問題：“如何才能實現動圓在直線上無滑滾動呢？”為主線引導學生在實驗平臺中探究，學生容易發現在滾動過程中保持線段AC及弧長的長度相等，從而進一步嘗試方程的推導如下：

設是軌跡上任一點，

則

那么，從而得出相應的軌跡參數方程．

操作提示：按【比較】會顯示線段AC及弧長的動態度量值，從而能幫助學生認識其中的規律．

3、平擺變幅

在問題：“P點的軌跡是平擺線，若直線OP上另有一點Q，那么圓在直線上作無滑滾動時，Q點的軌跡如何呢？” 的指引下，引導學生探究相應的軌跡，從而得出下列結論：一個圓沿著一條直線作無滑動的滾動時，圓周上的一個定點的軌跡叫做平擺線，圓內一個定點的軌跡叫做短幅平擺線；圓外一個定點的軌跡叫做長幅平擺線．

操作提示：按紐【軌跡】會給出點Q的軌跡，拖動點Q能幫助學生初步對變幅擺線形成動態印象；按紐【比較】會給出兩條典型的短幅擺線和長幅擺線，再次拖動點Q可以幫助學生形成整體動態認識，從而能認識到可以根據點Q相對于圓的位置對變幅擺線加以分類．

二、拓展研究

1、外擺線探究

有了平擺線的研究經歷和基礎，學生對類似情境“動圓在定圓外滾動會得到什么樣的曲線呢”的研究會產生極大的興趣．學生在如圖所示的實驗平臺中容易發現，軌跡是一條與平擺線相似的曲線，從而引出探究主題，即動圓在定圓外無滑滾動時，動圓圓周上的一個定點的軌跡是外擺線．

操作提示：與平擺線的研究相類似，學生在外擺線的探究上也須經歷一個由感性認識到理性升華的過程，先讓學生在【滾動】中猜想外擺線的形狀，通過【追蹤】驗證直覺判斷，繼而通過【比較】度量值的大小，從而對動圓在定圓外無滑滾動有所感悟，從而對外擺線有個全面而深刻的認識．

2、擺線拓展

在前面探究的基礎上，改變相應的R、r的值，可以得到不同類型的擺線模型，初步認識到擺線的形狀取決于R、r的比例．

操作提示：選中參數R或r的度量值，按動小鍵盤上的“+”或“-”改變相應參數的值，可以發現擺線的形狀發生變化，當r值為負時表示動圓在定圓內滾動；拖動半徑r的兩端點可以改變圖形的大小，從而能在屏幕上得到合適的顯示．

3、擺線全景

在實驗平臺中同時顯示四個圖，目的是讓學生對擺線形成整體認識，即擺線包括三種類型：外擺線、內擺線、環擺線，并在比較中歸納出擺線的分類（如下表，根據R、r的相對比例）：

操作提示：按動按紐【選中R】將選中全部四個圖中參數R，這樣再按小鍵盤上的“+”或“-”就可以同時改變四個圖中的參數了，從而可以地實現對擺線的整體研究，按動按紐【選中r】也有相類似的效果．

在R或r的值過大的情況下，對應的圖形也會超出相應的范圍，這樣不利于觀察圖形的特征，可以拖動綠色點（與紅色點接近圖形縮小，反之則圖形放大）后再按按紐【統一長度】即可解決問題．

4、擺線變幅

在順利研究了擺線的分類后，與平擺線的變幅相類似，學生在問題“P點的軌跡是擺線，若直線OP上另有一點Q，那么動圓P在定圓O外（或內）作無滑滾動時，Q點的軌跡如何呢”的引領下，很自然地會想到研究擺線的變幅．

在相應數學實驗平臺的支持下，學生容易認識到：當Q點在圓P上時，Q點的軌跡是擺線；當Q點在圓P內時，Q點的軌跡是短幅擺線；當Q點在圓P外時，Q點的軌跡是長幅擺線．

操作提示：在顯示【軌跡】的基礎上，拖動點Q，隨著點Q相對于動圓的位置不同，將產生不同的變幅擺線；同時也可改變參數R或r，觀察不同類型的擺線的變幅，感受其異同之處，對擺線的變幅形成整體認識．

5、哥白尼定理

在擺線全景中，學生會發現特殊情形下的擺線會成為一條直線，從而產生探究興趣．動圓P在定圓O內無滑滾動且R=2|r|時，動圓上一個定點的軌跡是定圓的一條直徑（哥白尼定理）．根據哥白尼定理，可以把旋轉運動變成往返的直線運動，這一定點機械設計上是很有用的．上述條件下，動圓所在平面內與動圓固定地連接在一起的圓內（或圓外）一點的軌跡是橢圓（卡丹轉盤）．