12.1. 导言:锥体部分

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• 选择节一. 导言:二次曲线区块

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  一. 导言:二次曲线区块

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  Roughly, conic sections are formed by the intersection of two  cones and a plane. On the surface (no pun intended), this  seems to be a rather arbitrary topic. Why would someone intersect two cones and a plane? Why is this topic an entire chapter of an algebra book? Let's address these questions one at a time.
  ::粗略地说,二次曲线系由两个锥形和一个平面的交叉点组成。在表面(不是有意的双眼),这似乎是一个相当武断的专题。为什么有人要交叉两个锥形和一个平面?为什么这个专题是代数书的完整一章?让我们一次讨论这些问题。

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  Conic sections were defined by the Greeks to address a seemingly unrelated problem. According to the story, Apollo sent a plague to the island of Delos. The citizens went to the oracle at Delphi, who told them that they needed to double the size of the altar of Apollo, which was shaped like a cube. The Delians consulted Plato, who determined that the oracle's instructions meant doubling the volume of the cube.
  ::希腊人为处理一个似乎无关的问题而定义了二次曲线段。 根据故事,阿波罗向迪洛斯岛发送了瘟疫。 公民们去了德尔菲的神器,他们告诉他们,他们需要把阿波罗祭坛的面积翻一番,这个祭坛的形状像立方体一样。 戴利安人咨询了柏拉图,他确定神器的指示意味着使立方体的体积翻一番。

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  Recall that the volume of a cube is $\begin{align*}V=s^3,\end{align*}$  where $\begin{align*}s\end{align*}$  is the length of a side. Suppose the volume of the altar of Apollo was 1 cubic unit, and the lengths of the sides were 1 unit. The new volume would have to be 2 cubic units, meaning the length of a side would be $\begin{align*}\sqrt[3]{2}\end{align*}$ , which is an irrational number. We cannot measure out  $\begin{align*}\sqrt[3]{2}\end{align*}$  units, so the Delians had to look for another solution.
  ::回顾立方体的体积是 V=3, 其中秒为侧体长度。 假设阿波罗的祭坛体积是1立方体, 而两边的体积是1立方体长度。 新体体积必须是2立方体, 意思是一面的体积是3立方体长度是3立方体, 这是不合理的数字。 我们无法测量出 3立方体, 因此德林人不得不寻找另一个解决方案 。

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  Plato gave the problem to three mathematicians, including one named Menaechmus. Menaechmus had seen some earlier work by Hippocrates of Chios, which reduced the problem to one of proportions. If there were four values where we could write equal fractions in the form   $\begin{align*}\frac{a}{x}=\frac{x}{y}=\frac{y}{b},\end{align*}$  so

  $$\begin{align*}V=s^3=\bigg(\frac{a}{x} \bigg) \bigg(\frac{x}{y} \bigg) \bigg(\frac{y}{b} \bigg)=\frac{a}{b}, \end{align*}$$
  ::柏拉图将问题交给了三位数学家,包括一位名叫Menaechmus的数学家。Menaechmus曾经见过一些希奥斯的希波克拉底早期的作品,这些作品将问题降低到一定比例。如果有四个值,我们可以以Ax=xy=yb的形式写出相等的分数,那么V=s3=(ax)(xy)(yb)=ab,

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  and $\begin{align*}a\end{align*}$  was twice  $\begin{align*}b\end{align*}$ , then the volume would be doubled ¹ .
  ::a 是b的两倍,然后数量将翻一番1。

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  Menaechmus noticed the problem yielded three relationships:
  

  $$\begin{align*}\frac{a}{x}=\frac{x}{y} \Longrightarrow x^2=ay \\ \\ \frac{x}{y}=\frac{y}{b} \Longrightarrow y^2=bx \\ \\ \frac{a}{x}=\frac{y}{b} \Longrightarrow xy=ab \\\end{align*}$$
  ::Menaechmus注意到这个问题产生了三种关系:轴=xyx2=ayxy=yby2=bxax=ybxy=ab

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  These relationships are equations for conic sections. As we will see, the 1st two are parabolas and the last is a hyperbola.  Menaechmus did not have the words parabola and hyperbola. Instead, he noticed they were "sections of a right-angled cone ² ." 
  ::这些关系是二次曲线的方程式。 正如我们所看到的, 第一次是双曲线, 最后一次是双曲线。 Menaechmus没有 parbola 和 perbola 的词。 相反, 他注意到它们是“ 右交织的锥形 ” 。

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  Conic sections are often used in lenses and mirrors. They describe the motion of planets and comets. They are also used to model the range of a radio station's signal, describe the shape of a racetrack, and determine when a business will reach its break-even point.   
  ::锥体部分通常用于镜头和镜像中,它们描述行星和彗星的运动,还用来模拟无线电台信号的范围,描述赛道的形状,并决定企业何时会达到平衡点。

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  By the end of this chapter, you should be able to: 
  ::在本章末尾,你应当能够:

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  • Find the distance between two points.
    ::查找两点之间的距离。
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  • Identify a conic section from its equation.
    ::从方程式中识别二次曲线段 。
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  • Identify properties of and graph parabolas, circles, ellipses, and hyperbolas.
    ::辨别和图示抛物体、圆圈、椭圆和双螺旋的特性。
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  • Solve systems of linear and conic sections. 
    ::解决线性和二次曲线的系统。

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  Features
  ::特征特征

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  • Section 2: It All Depends on Distance
    ::第2节:全视距离而定
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  • Section 4: I Can See Clearly
    ::第4节:我可以看得很清楚
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  • Section 6: Fix It With an Ellipse
    ::第6节:用椭圆来修补
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  • Section 7: Land Ho! 
    ::第七区: 何国!

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  Connections
  ::连接连接

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  • Our Trip Around the Sun
    ::我们环绕太阳的旅程
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  • Managing a Fishery
    ::渔业管理

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  References
  ::参考参考资料

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  1. "Doubling the Cube," last edited February 23, 2017,
  ::1. 2017年2月23日编辑的《将立方体翻倍》

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  2. Apollonius of Perga with notes by T.L. Heath, Treatise on Conic Sections. Cambridge, U.K.: Cambridge University Press, 1896. 
  ::2. 佩加的阿波罗纽斯,T.L.Heath的注解,《关于结肠部分的治疗》,剑桥,英国剑桥:剑桥大学出版社,1896年。