📚 4. Rotation of Axes in Two Dimensions | 4. 二维坐标轴的旋转
In coordinate geometry, many second-degree equations feature an xy-term that obscures the true nature of the conic section. By rotating the coordinate axes through a carefully chosen angle, this cross-product term can be eliminated, revealing the standard form of the conic. Rotation of axes is therefore an essential technique for simplifying and classifying quadratic curves. It relies on orthogonal transformations and trigonometric identities, and it preserves geometric properties such as distances and angles.
在坐标几何中，许多二次方程含有 xy 项，使得圆锥曲线的真实性质难以辨认。通过将坐标轴旋转一个恰当选择的角度，可以消去这一交叉项，从而显示圆锥曲线的标准形式。因此，坐标轴旋转是简化和分类二次曲线的一项基本技巧。它依赖于正交变换和三角恒等式，并保持距离和角度等几何性质。
1. The Need for Axis Rotation | 为什么要旋转坐标轴
h is a rotated conic; the axes of symmetry are not aligned with the coordinate axes. Rotating the axes can align them with the axes of the conic, making it possible to identify whether the curve is an ellipse, hyperbola, or parabola without completing the square with an xy term. This rotation simplifies the equation into one with no xy term, often revealing the standard form.
含两个变量的一般二次方程可写为 Ax² + Bxy + Cy² + Dx + Ey + F = 0。当 B ≠ 0 时，图像为旋转后的圆锥曲线；其对称轴不与坐标轴对齐。旋转坐标轴可使其与圆锥曲线的轴对齐，从而无需通过配方法处理 xy 项就能识别该曲线是椭圆、双曲线还是抛物线。这种旋转将方程简化为不含 xy 项的方程，常常揭示其标准形式。
Moreover, many IB problems require you to find the angle of rotation that removes the xy-term, compute the new coefficients, and graph the resulting curve. Mastering the rotation of axes thus directly supports success in examinations and deeper understanding of conic sections.
此外，许多 IB 问题要求你找到消除 xy 项的旋转角，计算新系数并绘制所得曲线。因此，掌握坐标轴旋转直接有助于考试成功，并加深对圆锥曲线的理解。
2. The Rotation Transformation | 旋转变换
Suppose we rotate the coordinate axes counterclockwise by an angle θ. A point P with original coordinates (x, y) will have new coordinates (x', y') relative to the rotated axes. The relationship between the two coordinate systems can be derived geometrically, but the standard forward rotation formulas that express the old coordinates in terms of the new ones are:
假设我们将坐标轴逆时针旋转角度 θ。原坐标为 (x, y) 的点 P 相对于旋转后的轴将有新坐标 (x', y')。两个坐标系之间的关系可通过几何方法推导，而用新坐标表示旧坐标的标准正向旋转公式为：
x = x' cosθ − y' sinθ, y = x' sinθ + y' cosθ
In matrix form, this can be written as [x; y] = [cosθ −sinθ; sinθ cosθ] [x'; y' It is orthogonal, meaning its columns are orthonormal vectors, and its determinant is +1.
用矩阵形式表示可写为 [x; y] = [cosθ −sinθ; sinθ cosθ] [x'; y']。该 2×2 矩阵称为坐标轴旋转 θ 角的旋转矩阵。它是正交矩阵，意味着其列向量为标准正交向量，且行列式为 +1。
Conversely, the new coordinates expressed in terms of the old ones are obtained by replacing θ with −θ, which yields:
反之，用旧坐标表示新坐标可通过将 θ 替换为 −θ 得到：
x' = x cosθ + y sinθ, y' = −x sinθ + y cosθ
fixed. For axis rotation to eliminate an xy-term, however, the forward substitution x = x' cosθ − y' sinθ, y = x' sinθ + y' cosθ is more convenient.
这种逆关系通常用于在保持轴不动的情况下旋转点。然而，对于消去 xy 项的轴旋转，正向代入 x = x' cosθ − y' sinθ，y = x' sinθ + y' cosθ 更为方便。
3. Deriving the Rotation Formulas | 旋转公式的推导
the new angular coordinate of P becomes α − θ. Hence, the new Cartesian coordinates are x' = r cos(α − θ) and y' = r sin(α − θ).
设点 P 相对于原 x 轴的极坐标为 (r, α)，则 x = r cosα，y = r sinα。若将坐标轴逆时针旋转 θ，点 P 的新角坐标变为 α − θ。因此，新直角坐标为 x' = r cos(α − θ)，y' = r sin(α − θ)。
Using the cosine and sine difference identities, we obtain:
运用余弦和正弦的差角公式可得：
x' = r cosα cosθ + r sinα sinθ = x cosθ + y sinθ
y' = r sinα cosθ − r cosα sinθ = y cosθ − x sinθ = −x sinθ + y cosθ
This shows the inverse rotation directly. Solving these two linear equations for x and y gives the forward formulas mentioned in Section 2. The derivation confirms that no area or length is distorted because the transformation is a rotation.
这直接给出了逆旋转公式。解这两个关于 x 和 y 的线性方程组即可得到第 2 节中的正向公式。该推导证实了变换不会扭曲面积或长度，因为它是旋转变换。
4. Inverse Rotation and Orthogonality | 逆旋转与正交性
The rotation matrix R = [cosθ −sinθ; sinθ cosθ] has the property that its inverse equals its transpose: R⁻¹ = Rᵀ. Indeed, replacing θ by −θ gives [cosθ sinθ; −sinθ which is exactly the transpose. This reflects the orthogonal nature of the transformation.
旋转矩阵 R = [cosθ −sinθ; sinθ cosθ] 的特性是其逆矩阵等于其转置：R⁻¹ = Rᵀ。事实上，将 θ 替换为 −θ 得到 [cosθ sinθ; −sinθ cosθ]，这正是其转置。这反映了变换的正交性。
he transformation preserves orientation. Lengths are invariant: for any point, x² + y² = x'² + y'². This is why the discriminant B² − 4AC stays the same after rotation.
由于 R 的行列式为 1，该变换保持定向。长度不变：对任意点有 x² + y² = x'² + y'²。这正是为什么旋转后判别式 B² − 4AC 保持不变的原因。
These invariant properties are extremely useful when analysing conics, because they allow classification without completing the entire rotation algebra.
这些不变性质在分析圆锥曲线时非常有用，因为它们允许我们在不完成整个旋转代数的情况下进行分类。
5. Rotating a Point – Worked Example | 旋转点——计算示例
th θ = −30°. However, it is often simpler to think in terms of rotating the point: using x' = x cosθ + y sinθ, y' = −x sinθ + y cosθ with θ = +30° gives the new coordinates relative to the original axes.
假设我们希望求出点 (3, 4) 在坐标轴顺时针旋转 30° 后的坐标（等价于点逆时针旋转以实现相同的相对位置）。若坐标轴顺时针旋转 30°，我们使用 θ = −30° 的逆公式。然而，通常考虑旋转点更为简单：使用 x' = x cosθ + y sinθ，y' = −x sinθ + y cosθ 并取 θ = +30°，可得到点相对于原坐标轴的新坐标。
Taking cos30° = √3/2, sin30° = 1/2, we get:
代入 cos30° = √3/2，sin30° = 1/2，得：
x' = 3(√3/2) + 4(1/2) = (3√3 + 4)/2, y' = −3(1/2) + 4(√3/2) = (4√3 − 3)/2
This is useful for checking transformations. For axis rotation to simplify equations, however, we always substitute the rotation formulas into the original equation.
这对于验证变换很有用。然而，在通过轴旋转简化方程时，我们总是将旋转公式代入原方程。
6. General Conic Equation and the xy-Term | 一般二次曲线方程与 xy 项
olely responsible for the tilt of the conic. When B ≠ 0, the principal axes of the conic are not parallel to the x- or y-axis. We can eliminate the Bxy term by rotating the axes through an appropriate angle θ.
x 和 y 的最一般二次方程为 Ax² + Bxy + Cy² + Dx + Ey + F = 0。交叉项系数 B 是导致圆锥曲线倾斜的唯一原因。当 B ≠ 0 时，圆锥曲线的主轴不平行于 x 轴或 y 轴。我们可以通过将坐标轴旋转一个适当的角度 θ 来消除 Bxy 项。
Upon substituting x = x' cosθ − y' sinθ and y = x' sinθ + y' cosθ into the equation, the new coefficient of x'y' mes (C − A) sin 2θ + B cos 2θ. Setting this to zero yields the equation that determines θ.
将 x = x' cosθ − y' sinθ 和 y = x' sinθ + y' cosθ 代入方程后，新的 x'y' 项系数变为 (C − A) sin 2θ + B cos 2θ。令其为零即可得到确定 θ 的方程。
7. Choosing θ to Eliminate the xy-Term | 选择 θ 以消去 xy
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