🔄 坐标旋转计算器（绕原点或任意点）Rotation Calculator (Rotate Points)
把一个或多个点 (x, y) 绕原点或任意中心点旋转指定角度（逆时针或顺时针，角度或弧度），给出旋转后的坐标和旋转矩阵，适合几何作业、图形和 CAD 计算。Rotate one or more points (x, y) about the origin or any centre by a given angle — counter-clockwise or clockwise, in degrees or radians — and get the new coordinates and the rotation matrix. Handy for geometry homework, graphics and CAD.
使用说明与常见问题How-to & FAQ
「坐标旋转计算器（绕原点或任意点）」是什么？What is the Rotation Calculator (Rotate Points) tool?
把一个或多个点 (x, y) 绕原点或任意中心点旋转指定角度（逆时针或顺时针，角度或弧度），给出旋转后的坐标和旋转矩阵，适合几何作业、图形和 CAD 计算。Rotate one or more points (x, y) about the origin or any centre by a given angle — counter-clockwise or clockwise, in degrees or radians — and get the new coordinates and the rotation matrix. Handy for geometry homework, graphics and CAD.
如何使用「坐标旋转计算器（绕原点或任意点）」？How do I use the Rotation Calculator (Rotate Points) tool?
在「坐标旋转计算器（绕原点或任意点）」中，填写「旋转角度」、「单位」、「方向」、「中心 x」、「中心 y」等参数，结果会随输入实时计算并显示在下方。In the Rotation Calculator (Rotate Points): Fill in "Rotation angle", "Unit", "Direction", "Centre x", "Centre y"; results are recalculated instantly as you type.
旋转公式是什么？What is the rotation formula?
绕中心 (a, b) 逆时针旋转 θ：x′ = a + (x − a)·cosθ − (y − b)·sinθ，y′ = b + (x − a)·sinθ + (y − b)·cosθ。顺时针旋转等于逆时针旋转 −θ。绕原点时 a = b = 0，对应的旋转矩阵是 [cosθ −sinθ; sinθ cosθ]。Rotating counter-clockwise by θ about (a, b): x′ = a + (x − a)·cosθ − (y − b)·sinθ and y′ = b + (x − a)·sinθ + (y − b)·cosθ. Clockwise is the same with −θ. About the origin a = b = 0, and the rotation matrix is [cosθ −sinθ; sinθ cosθ].
为什么 90° 的结果是精确的整数？Why are 90° results exact whole numbers?
当角度是 90° 的整数倍时，工具直接使用精确的 cos、sin 值（0、1、−1），避免浮点数产生 6.1×10⁻¹⁷ 这样的噪声；其它角度绝对值小于 10⁻¹² 的结果也显示为 0。For multiples of 90° the exact cosine and sine (0, 1, −1) are used, avoiding floating-point noise like 6.1×10⁻¹⁷; for other angles results smaller than 10⁻¹² in absolute value are shown as 0.
坐标系的方向是怎样的？Which way do the axes point?
按数学惯例：x 轴向右、y 轴向上，逆时针为正方向。屏幕和很多绘图软件的 y 轴向下，那时这里的“逆时针”在屏幕上看起来是顺时针。The math convention: x to the right, y up, counter-clockwise positive. On screens and in many drawing programs y points down, so 'counter-clockwise' here looks clockwise on screen.
计算精度如何？How precise are the results?
「坐标旋转计算器（绕原点或任意点）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Rotation Calculator (Rotate Points) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「坐标旋转计算器（绕原点或任意点）」免费吗？需要注册吗？Is the Rotation Calculator (Rotate Points) free? Do I need an account?
「坐标旋转计算器（绕原点或任意点）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Rotation Calculator (Rotate Points) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.