📚 The Relationship Between Cartesian and Polar Coordinates | 笛卡尔坐标与极坐标的关系
Coordinate systems allow us to describe the position of a point in a plane using numbers. The Cartesian system uses perpendicular axes, while the polar system uses distance and angle from a fixed origin. Understanding the relationship between these two systems is essential for solving many problems in A-Level mathematics, particularly in calculus and curve sketching.
坐标系让我们能用数字描述平面中一个点的位置。笛卡尔坐标系使用互相垂直的坐标轴，而极坐标系则使用到固定原点的距离和与正方向的夹角。理解这两种坐标系之间的关系，对于解决A-Level数学中的许多问题，尤其是微积分和曲线作图问题，至关重要。
1. Introduction to Cartesian Coordinates | 笛卡尔坐标系简介
The Cartesian coordinate system was introduced by René Descartes. It locates a point \(P\) by an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the \(y\)-axis and \(y\) is the vertical distance from the \(x\)-axis. The axes intersect at the origin \(O = (0,0)\).
笛卡尔坐标系由勒内·笛卡尔引入。它通过有序数对 \((x, y)\) 来定位点 \(P\)，其中 \(x\) 是到 \(y\) 轴的水平距离，\(y\) 是到 \(x\) 轴的垂直距离。两条坐标轴相交于原点 \(O = (0,0)\)。
2. Defining Polar Coordinates | 极坐标的定义
In polar coordinates, a point is described by \((r, \theta)\), where \(r\) is the radial distance from the origin (pole), and \(\theta\) is the angle measured counter-clockwise from the positive \(x\)-axis (polar axis). By convention, \(r \geq 0\) unless stated otherwise, and \(\theta\) is often given in radians.
在极坐标系中，一个点用 \((r, \theta)\) 描述，其中 \(r\) 是从极点（原点）到点的径向距离，\(\theta\) 是从极轴（正 \(x\) 轴）按逆时针方向测量得到的角度。按惯例，除非另有说明，通常取 \(r \geq 0\)，且 \(\theta\) 通常以弧度表示。
3. Conversion from Polar to Cartesian | 极坐标转换为笛卡尔坐标
Given a point in polar form \((r, \theta)\), its Cartesian coordinates are obtained by projecting the radius onto the axes. The conversion formulas are:
给定极坐标形式 \((r, \theta)\) 的点，通过将半径分别投影到两个坐标轴上，即可得到其笛卡尔坐标。转换公式为：
x = r cos θ, y = r sin θ
For example, the polar point \((4, \pi/3)\) has Cartesian coordinates \(x = 4\cos(\pi/3) = 2\) and \(y = 4\sin(\pi/3) = 2\sqrt{3}\).
例如，极坐标点 \((4, \pi/3)\) 的笛卡尔坐标为 \(x = 4\cos(\pi/3) = 2\)，\(y = 4\sin(\pi/3) = 2\sqrt{3}\)。
4. Conversion from Cartesian to Polar | 笛卡尔坐标转换为极坐标
To convert from Cartesian \((x, y)\) to polar \((r, \theta)\), we first find the radial distance using the Pythagorean theorem, then find the angle using an inverse trigonometric function.
要将笛卡尔坐标 \((x, y)\) 转换为极坐标 \((r, \theta)\)，我们先利用勾股定理求出径向距离，再利用反三角函数求出角度。
r = √(x² + y²), θ = arctan(y / x)
The formula \(r = \sqrt{x^2 + y^2}\) always gives the correct length from the origin. However, ‘θ = arctan(y/x)’ requires careful attention to the quadrant of the point, since the tangent function is periodic with period \(\pi\).
公式 \(r = \sqrt{x^2 + y^2}\) 始终给出从原点出发的正确长度。然而，公式 \(θ = \arctan(y/x)\) 需要仔细考虑点所在的象限，因为正切函数以 \(\pi\) 为周期。
5. Handling the Quadrant Problem | 象限与 θ 的正确判断
The basic arctangent only returns values between \(-\pi/2\) and \(\pi/2\). For points in the second or third quadrants, simply using \(\arctan(y/x)\) gives the wrong angle. We must adjust by adding or subtracting \(\pi\).
基本的反正切函数只能返回 \(-\pi/2\) 到 \(\pi/2\) 之间的值。对于第二或第三象限中的点，直接使用 \(\arctan(y/x)\) 会得到错误的角度。我们必须通过加减 \(\pi\) 来校正。
- If \(x > 0\) and \(y > 0\) (first quadrant): \(\theta = \arctan(y/x)\).
- If \(x < 0\) (second or third quadrant): \(\theta = \arctan(y/x) + \pi\).
- If \(x > 0\) and \(y < 0\) (fourth quadrant): \(\theta = \arctan(y/x)\) or \(\theta = \arctan(y/x) + 2\pi\) if you need a positive angle.
- 若 \(x > 0\) 且 \(y > 0\)（第一象限）：\(\theta = \arctan(y/x)\)。
- 若 \(x < 0\)（第二或第三象限）：\(\theta = \arctan(y/x) + \pi\)。
- 若 \(x > 0\) 且 \(y < 0\)（第四象限）：\(\theta = \arctan(y/x)\)；如果需要正角，则取 \(\theta = \arctan(y/x) + 2\pi\)。
Many calculators provide a two-argument function usually named ‘atan2(y, x)’ which automatically returns the correct angle in the range \((-\pi, \pi]\).
许多计算器提供通常名为 ‘atan2(y, x)’ 的双参数函数，它会自动返回 \((-\pi, \pi]\) 范围内的正确角度。
6. Worked Examples | 例题解析
Example 1: Convert the point with Cartesian coordinates \((-3, 4)\) to polar form.
例1：将笛卡尔坐标 \((-3, 4)\) 的点转换为极坐标形式。
Using \(r = \sqrt{(-3)^2 + 4^2} = 5\). Since \(x < 0\), \(\theta = \arctan(4/(-3)) + \pi \approx 2.214\) rad. Thus the polar form is approximately \((5, 2.214)\).
利用 \(r = \sqrt{(-3)^2 + 4^2} = 5\)。由于 \(x < 0\)，\(\theta = \arctan(4/(-3)) + \pi \approx 2.214\) 弧度。因此极坐标形式约为 \((5, 2.214)\)。
Example 2: Convert the polar point \((6, 5\pi/6)\) to Cartesian coordinates.
例2：将极坐标点 \((6, 5\pi/6)\) 转换为笛卡尔坐标。
\(x = 6\cos(5\pi/6) = 6 \times (-\sqrt{3}/2) = -3\sqrt{3}\). \(y = 6\sin(5\pi/6) = 6 \times (1/2) = 3\). So the Cartesian coordinates are \((-3\sqrt{3}, 3)\).
\(x = 6\cos(5\pi/6) = 6 \times (-\sqrt{3}/2) = -3\sqrt{3}\)。\(y = 6\sin(5\pi/6) = 6 \times (1/2) = 3\)。所以笛卡尔坐标为 \((-3\sqrt{3}, 3)\)。
7. Equations of Curves: Lines and Circles | 曲线方程：直线与圆
Some curves have much simpler equations in polar coordinates. For example, a circle of radius \(a\) centered at the origin has equation \(r = a\) in polar form, whereas its Cartesian equation is \(x^2 + y^2 = a^2\).
有些曲线在极坐标中的方程要简单得多。例如，圆心在原点、半径为 \(a\) 的圆，在极坐标中的方程为 \(r = a\)，而它的笛卡尔方程为 \(x^2 + y^2 = a^2\)。
A line through the origin making an angle \(\alpha\) with the positive \(x\)-axis is described simply by \(\theta = \alpha\) in polar form. In Cartesian form this is \(y = (\tan\alpha) x\).
过原点且与正 \(x\) 轴成角 \(\alpha\) 的直线，在极坐标中只需用 \(\theta = \alpha\) 描述。在笛卡尔形式下，它是 \(y = (\tan\alpha)x\)。
8. Roses and Spirals in Polar Form | 玫瑰曲线与螺线
Polar coordinates are especially powerful for describing curves that are complicated in Cartesian form. The rose curve \(r = a\cos(k\theta)\) produces \(k\) petals if \(k\) is odd, or \(2k\) petals if \(k\) is even.
极坐标在描述笛卡尔形式下很复杂的曲线时尤其强大。玫瑰曲线 \(r = a\cos(k\theta)\) 在 \(k\) 为奇数时产生 \(k\) 个花瓣，在 \(k\) 为偶数时产生 \(2k\) 个花瓣。
Another classic shape is the Archimedean spiral \(r = a\theta\). Its Cartesian coordinates are given parametrically by \(x = a\theta\cos\theta\), \(y = a\theta\sin\theta\), which is far less elegant than the polar expression.
另一个经典图形是阿基米德螺线 \(r = a\theta\)。它的笛卡尔坐标可用参数式表达为 \(x = a\theta\cos\theta\)，\(y = a\theta\sin\theta\)，远不如极坐标表达式简洁。
9. Choosing the Right Coordinate System | 选择合适的坐标系
The choice between Cartesian and polar coordinates often depends on the symmetry of the problem. If a curve has circular symmetry, such as a disk or a sector, polar coordinates simplify integration and differentiation.
在笛卡尔坐标与极坐标之间进行选择，通常取决于问题的对称性。如果曲线具有圆形对称性，例如圆盘或扇形，极坐标可以简化积分和微分运算。
On the other hand, motion along straight lines or rectangular regions is usually easier in Cartesian coordinates. In physics, forces acting radially or tangentially are naturally expressed using polar coordinates.
另一方面，沿直线运动或矩形区域的问题通常用笛卡尔坐标更容易处理。在物理学中，沿径向或切向作用的力自然适合用极坐标表示。
10. Relationship Summary Table | 关系总结表
The table below summarizes the key conversion formulas and their domain considerations.
下表总结了关键的转换公式及其定义域注意事项。
| Conversion | Formula | Notes |
| Polar → Cartesian | x = r cos θ, y = r sin θ | Valid for all r, θ |
| Cartesian → Polar (r) | r = √(x² + y²) | Always non-negative |
| Cartesian → Polar (θ) | θ = arctan(y/x) adjusted for quadrant | Use atan2 in calculators |
Mastering the relationship between Cartesian and polar coordinates gives you flexibility in tackling problems in pure mathematics, mechanics, and further calculus. Practice converting both ways with points and equations until the process becomes automatic.
掌握笛卡尔坐标与极坐标之间的关系，能让你在纯数学、力学和进阶微积分问题中拥有更大的灵活性。通过点和方程的双向转换多加练习，直至过程变得自动熟练。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply