IB Mathematics: Rotating 2D Coordinate Axes | 二维坐标轴旋转变换

📚 IB Mathematics: Rotating 2D Coordinate Axes | 二维坐标轴旋转变换

When a quadratic equation in two variables contains an xy-term, its graph is usually a conic section that has been tilted away from the standard axes. Rotating the coordinate axes through a suitable angle can eliminate the xy-term, revealing the true shape and orientation of the curve.

当二元二次方程中出现 xy 项时，其图象往往是一条被“旋转”过的圆锥曲线。将坐标轴旋转适当的角度，可以消去 xy 项，从而更清楚地看出曲线的形状与方向。

1. Why Rotate Axes? | 为什么需要旋转坐标轴？

The Cartesian coordinate system is convenient, but not every curve is aligned with it. An equation such as xy = 1 has a hyperbola whose axes of symmetry are the lines y = x and y = −x. If we rotate the coordinate axes by 45°, the same curve becomes (x′)² − (y′)² = 2, which is far easier to recognise and sketch.

直角坐标系固然方便，但并非所有曲线都与坐标轴对齐。例如方程 xy = 1 表示一条双曲线，其对称轴是直线 y = x 和 y = −x。如果我们将坐标轴旋转 45°，同样的曲线就变成 (x′)² − (y′)² = 2，更容易识别和作图。

A general quadratic equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0 may describe a circle, ellipse, parabola or hyperbola. The xy-term indicates that the principal axes of the conic are not parallel to the original x- and y-axes. A rotation of axes simplifies the equation, while preserving the geometric shape.

一般的二元二次方程 Ax² + Bxy + Cy² + Dx + Ey + F = 0 可能表示圆、椭圆、抛物线或双曲线。xy 项的存在意味着该圆锥曲线的主轴与原坐标轴不平行。旋转坐标轴可以使方程简化，同时保持图形的几何本质不变。

2. The Rotation Transformation | 旋转变换公式

Let a new coordinate system (x′, y′) be obtained by rotating the original axes counterclockwise through an angle θ. Then the same point, whose original coordinates are (x, y), satisfies:

设新坐标系 (x′, y′) 是将原坐标轴逆时针旋转角 θ 后得到的。则

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