Conic Sections · 圆锥曲线
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| ellipse/ɪˈlɪps/ | 椭圆 | tuǒ yuán |
| parabola/pəˈræbələ/ | 抛物线 | pāo wù xiàn |
| hyperbola/haɪˈpɜːbələ/ | 双曲线 | shuāng qū xiàn |
| conic section/ˈkɒnɪk ˈsekʃn/ | 圆锥曲线 | yuán zhuī qū xiàn |
Slicing a cone
- Take a double cone and slice it with a flat plane at different tilts.
- Cut straight across and you get a circle; tilt a little and it stretches to an ellipse.
- Tilt more and it opens into a parabola; steeper still and it splits into a hyperbola.
- These four curves — the conic sections — appear all over science and design.
切割一个圆锥
- 取一个对顶双锥,用一个平面以不同的倾斜角切割它。
- 直着横切得到一个圆;稍微一斜,它就拉长成一个椭圆。
- 再斜一些,它张开成一条抛物线;更陡时,它裂成一条双曲线。
- 这四条曲线——圆锥曲线——在科学和设计中随处可见。
The family of conics
- A conic section 圆锥曲线 is a curve where a plane cuts a cone.
- The four types are the circle, the ellipse, the parabola, and the hyperbola.
- Each has a clean implicit equation and a geometric "distance" definition.
- Planets orbit on ellipses; thrown balls and satellite dishes trace parabolas.
圆锥曲线家族
- 圆锥曲线(conic section)是平面切割圆锥所得的曲线。
- 四种类型是:圆、椭圆、抛物线和双曲线。
- 每一种都有一个干净的隐式方程和一个几何的"距离"定义。
- 行星沿椭圆轨道运行;抛出的球和卫星天线描出抛物线。
The conic sections are the curves formed by… · 圆锥曲线是由……形成的曲线
A conic section is the intersection of a plane and a cone; the slice angle gives a circle, ellipse, parabola, or hyperbola. · 圆锥曲线是平面与圆锥体的交线；切片角度决定了它是圆、椭圆、抛物线还是双曲线。
A circle is a special conic section (an ellipse whose two foci coincide). · 圆是一种特殊的圆锥曲线（两个焦点重合的椭圆）。
When the two foci merge into one centre, the ellipse becomes a perfectly round circle. · 当两个焦点合并为一个中心时，椭圆变成一个完美的圆。
The ellipse
- An ellipse 椭圆 is the set of points whose distances to two fixed foci add to a constant.
- Its equation is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, a stretched circle.
- A circle is just the special case where the two foci merge into one centre.
- The longer axis is the major axis; the shorter is the minor axis.
椭圆
- 椭圆(ellipse)是到两个固定焦点的距离之和为常数的点的集合。
- 它的方程是 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$,一个被拉伸的圆。
- 圆只是两个焦点合并成一个中心的特殊情形。
- 较长的轴是长轴;较短的轴是短轴。
An ellipse · 椭圆 is the set of points where the sum of distances to two foci is… · 椭圆是到两个焦点的距离之和为……的所有点的集合
Every point on an ellipse · 椭圆 has the same · 相同 total distance to the two foci — that constant defines the shape. · 椭圆上的每一点到两个焦点的总距离都是相同的——该常数定义了形状。
Select all · 所有 the conic sections. · 选择所有圆锥曲线。
A triangle is not a conic. The circle, ellipse, parabola, and hyperbola are the conic sections. · 三角形不是圆锥曲线。圆、椭圆、抛物线和双曲线才是圆锥曲线。
Parabola and hyperbola
- A parabola 抛物线 is the set of points equidistant from a focus and a fixed line (the directrix).
- Its equation looks like $y = ax^2$ — the graph of a quadratic.
- A hyperbola 双曲线 is where the difference of distances to two foci is constant.
- Its equation is $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, opening into two branches.
抛物线与双曲线
- 抛物线(parabola)是到一个焦点和一条固定直线(准线)距离相等的点的集合。
- 它的方程形如 $y = ax^2$——一个二次函数的图像。
- 双曲线(hyperbola)是到两个焦点距离之差为常数的地方。
- 它的方程是 $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$,张开成两支。
Which conic section? · 哪种圆锥曲线？
The squared terms tell you the conic. Sort each equation. · 平方项告诉你这是哪种圆锥曲线。对每个方程进行分类。
A ____ is the conic where each point is equidistant from a focus and a directrix line. · ____是圆锥曲线，其中每一点到一个焦点和一条准线的距离相等。
A parabola · 抛物线 balances distance to a focus against distance to a fixed line (the directrix). · 抛物线平衡了到焦点的距离与到固定直线（准线）的距离。
Recognising a conic
- Read the equation's form: an $x^2 + y^2$ with equal coefficients is a circle.
- Different positive coefficients on $x^2$ and $y^2$ give an ellipse.
- Only one squared term gives a parabola; a minus sign between them gives a hyperbola.
- The sign pattern of the squared terms tells you which conic you have.
辨认一条圆锥曲线
- 读方程的形式:$x^2 + y^2$ 且系数相等是一个圆。
- $x^2$ 和 $y^2$ 上不同的正系数给出一个椭圆。
- 只有一个平方项给出抛物线;它们之间有减号给出双曲线。
- 平方项的符号模式告诉你这是哪一种圆锥曲线。
Not every conic is a function of $x$. Circles, ellipses, and sideways parabolas fail the vertical-line test. Describe them with implicit equations or parametric forms, not by trying to force a single $y = f(x)$.
并非每条圆锥曲线都是 $x$ 的函数。圆、椭圆和横向的抛物线都通不过垂直线检验。用隐式方程或参数形式来描述它们,而不要试图硬凑成单个 $y = f(x)$。
Identify the conic $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$.
- Both terms are squared, positive, and different coefficients.
- That is the equation of an ellipse.
- It stretches to $x = \pm 3$ (major axis) and $y = \pm 2$ (minor axis).
辨认圆锥曲线 $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$。
- 两个项都是平方、为正,且系数不同。
- 那是一个椭圆的方程。
- 它伸展到 $x = \pm 3$(长轴)和 $y = \pm 2$(短轴)。
The conic sections — circle, ellipse, parabola, and hyperbola — are the curves formed by slicing a cone. Each has a distance definition (an ellipse sums distances to two foci) and a signature implicit equation, and most are not functions of $x$.
圆锥曲线——圆、椭圆、抛物线和双曲线——是切割圆锥所形成的曲线。每一种都有一个距离定义(椭圆是到两焦点距离之和),以及一个标志性的隐式方程,而且大多数都不是 $x$ 的函数。