Polar Coordinates · 极坐标
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| polar coordinates/ˈpəʊlə kəʊˈɔːdɪnəts/ | 极坐标 | jí zuò biāo |
| radians/ˈreɪdɪənz/ | 弧度 | hú dù |
| polar grid/ˈpəʊlə ɡrɪd/ | 极坐标网格 | jí zuò biāo wǎng gé |
A different way to say "where"
- Usually we locate a point by "go right $x$, go up $y$" — rectangular coordinates.
- But a radar screen or a compass thinks differently: "how far, and in which direction?"
- That is the polar way: a distance and an angle.
- For anything that spins or radiates, polar coordinates are far more natural.
说明"在哪里"的另一种方式
- 通常我们用"向右 $x$,向上 $y$"来定位一个点——直角坐标。
- 但雷达屏幕或指南针的想法不同:"多远,朝哪个方向?"
- 那就是极坐标的方式:一个距离和一个角。
- 对任何旋转或辐射的事物,极坐标要自然得多。
A point as (r, θ)
- Polar coordinates 极坐标 name a point by $(r, \theta)$.
- $r$ is the distance from the origin (the pole); $\theta$ is the angle from the positive x-axis.
- The angle is measured in radians 弧度 (or degrees), counter-clockwise.
- So $(2, \tfrac{\pi}{4})$ means "go out $2$ units at a $45°$ direction".
一个点作为 (r, θ)
- 极坐标(polar coordinates)用 $(r, \theta)$ 来命名一个点。
- $r$ 是离原点(极点)的距离;$\theta$ 是与正 x 轴的夹角。
- 角用弧度(radians)(或度)度量,逆时针方向。
- 所以 $(2, \tfrac{\pi}{4})$ 意味着"朝 $45°$ 方向走出 $2$ 个单位"。
In polar coordinates $(r, \theta)$, the two numbers give… · 在极坐标 $(r, \theta)$ 中,这两个数给出……
Polar coordinates · 极坐标 use · 使用 $r$ (distance from the origin) and $\theta$ (angle), unlike the $(x, y)$ of rectangular coordinates. · 极坐标用 $r$(离原点的距离)和 $\theta$(角),与直角坐标的 $(x, y)$ 不同。
The polar grid
- A polar grid 极坐标网格 is a set of concentric circles (constant $r$) and radial lines (constant $\theta$).
- Reading a point is easy: find its circle, then its radial line.
- The origin is $r = 0$, the single centre of all the circles.
- This grid replaces the familiar square graph paper of rectangular coordinates.
极坐标网格
- 极坐标网格(polar grid)是一组同心圆($r$ 恒定)和放射状直线($\theta$ 恒定)。
- 读一个点很容易:找到它的圆,再找它的放射线。
- 原点是 $r = 0$,是所有圆唯一的中心。
- 这个网格取代了直角坐标里熟悉的方格纸。
Locate a point by radius and angle · 用半径和角度定位一个点
Set the angle and the radius to place a point. Polar coordinates (r, theta) describe location by direction and distance. · 设定角度和半径来放置一个点。极坐标 (r, theta) 用方向和距离来描述位置。
To convert $(r, \theta)$ to rectangular $(x, y)$, use… · 要把 $(r, \theta)$ 转成直角坐标 $(x, y)$,用……
The unit-circle relationship scales up by $r$: $x = r\cos\theta$ and $y = r\sin\theta$. · 单位圆的关系按 $r$ 放大:$x = r\cos\theta$,$y = r\sin\theta$。
A point has $r = 2$ and $\theta = 0$. What is its $x$-coordinate? · 一个点有 $r = 2$,$\theta = 0$。它的 $x$ 坐标是多少?
$x = r\cos\theta = 2\cos 0 = 2 \cdot 1 = 2$; and $y = 2\sin 0 = 0$. · $x = r\cos\theta = 2\cos 0 = 2 \cdot 1 = 2$;而 $y = 2\sin 0 = 0$。
Select all · 所有 true statements about polar coordinates. · 选出关于极坐标的所有正确说法。
Polar forms are not · 不 unique — many $(r, \theta)$ name the same point. The other three are correct. · 极坐标形式不唯一——许多 $(r, \theta)$ 命名同一个点。其余三条正确。
Converting between systems
- Polar to rectangular: $x = r\cos\theta$ and $y = r\sin\theta$.
- Rectangular to polar: $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\!\tfrac{y}{x}$ (mind the quadrant).
- These come straight from the unit-circle definitions, scaled by $r$.
- Switch to whichever system makes the problem simpler.
在两种系统之间转换
- 极坐标转直角:$x = r\cos\theta$,$y = r\sin\theta$。
- 直角转极坐标:$r = \sqrt{x^2 + y^2}$,$\theta = \arctan\!\tfrac{y}{x}$(注意象限)。
- 这些直接来自单位圆的定义,按 $r$ 放大。
- 切换到让问题更简单的那个系统。
The same point can have more than one set of polar coordinates (e.g. adding $2\pi$ to the angle). · 同一个点可以有不止一组极坐标(例如给角加上 $2\pi$)。
Adding a full turn to $\theta$ lands on the same point, so polar coordinates are not unique. · 给 $\theta$ 加上一整圈会落到同一个点,所以极坐标不是唯一的。
Not a unique address
- Unlike $(x, y)$, a polar point has many names.
- Adding $2\pi$ to $\theta$ lands on the very same point.
- A negative $r$ points in the opposite direction, giving yet another name.
- So $(2, 0)$, $(2, 2\pi)$, and $(-2, \pi)$ are all the same point.
不是唯一的地址
- 与 $(x, y)$ 不同,一个极坐标点有许多名字。
- 给 $\theta$ 加上 $2\pi$ 会落到完全相同的点。
- 负的 $r$ 指向相反方向,又给出另一个名字。
- 所以 $(2, 0)$、$(2, 2\pi)$ 和 $(-2, \pi)$ 都是同一个点。
Polar coordinates are not unique: one point has infinitely many $(r, \theta)$ names. When converting or solving, watch the quadrant and remember that $\arctan$ alone can land you in the wrong half of the plane.
极坐标不唯一:一个点有无穷多个 $(r, \theta)$ 名字。转换或求解时,要注意象限,并记住单靠 $\arctan$ 可能把你送到平面错误的一半。
Convert the polar point $(4, \tfrac{\pi}{2})$ to rectangular coordinates.
- $x = r\cos\theta = 4\cos\tfrac{\pi}{2} = 4 \cdot 0 = 0$.
- $y = r\sin\theta = 4\sin\tfrac{\pi}{2} = 4 \cdot 1 = 4$.
- So the point is $(0, 4)$ — straight up the y-axis.
把极坐标点 $(4, \tfrac{\pi}{2})$ 转成直角坐标。
- $x = r\cos\theta = 4\cos\tfrac{\pi}{2} = 4 \cdot 0 = 0$。
- $y = r\sin\theta = 4\sin\tfrac{\pi}{2} = 4 \cdot 1 = 4$。
- 所以点是 $(0, 4)$——沿 y 轴正上方。
Polar coordinates locate a point as $(r, \theta)$: a distance $r$ from the origin at angle $\theta$, plotted on a polar grid. Convert with $x = r\cos\theta$, $y = r\sin\theta$. Polar names are not unique — adding a full turn gives the same point.
极坐标把一个点定位为 $(r, \theta)$:在角 $\theta$ 方向上离原点距离 $r$,画在极坐标网格上。用 $x = r\cos\theta$、$y = r\sin\theta$ 转换。极坐标名字不唯一——加上一整圈得到同一个点。