📚 6D2 Mirrors and Reflections | 6D2 镜像与反射
Reflections are fundamental geometric transformations that map every point to its mirror image across a given line or plane. In linear algebra and Further Mathematics, reflections are studied as linear transformations that can be represented by matrices, making them essential for understanding symmetry, eigenvalues, and the composition of isometries. This article explores the matrices of reflection in two and three dimensions, their algebraic properties, and how they combine to produce rotations.
反射是将每一点映射到其关于给定直线或平面的镜像的基本几何变换。在线性代数和进阶数学中，反射被当作线性变换来研究，可以用矩阵表示，这使它成为理解对称性、特征值和等距变换复合的关键。本文探讨二维和三维空间中反射的矩阵、它们的代数性质，以及它们如何组合产生旋转。
When we reflect a point across a line through the origin or a plane through the origin, the transformation is linear and can be represented by an orthogonal matrix with determinant –1. Such transformations preserve distances and angles but reverse orientation. We will derive the standard reflection matrices, examine their inverses, and see how two reflections combine to give a rotation.
当我们把一个点关于通过原点的直线或平面反射时，该变换是线性的，可以用行列式为 –1 的正交矩阵表示。这些变换保持距离和角度，但使方向反转。我们将推导标准反射矩阵，考察它们的逆，并观察两次反射如何合成一个旋转。
1. Reflections as Linear Transformations | 作为线性变换的反射
In the plane, a reflection across a line through the origin is a linear map T : ℝ² → ℝ such that T(x) = Mx for every position vector x. The matrix depends only on the direction of the mirror line.
在平面中，关于通过原点的直线的反射是一个线性映射 T : ℝ² → ℝ²。因为原点不变，T(0) = 0，并且 T 保持向量加法和标量乘法。因此，它可以用一个 2 × 2 矩阵 M 表示，满足对每个位置向量 x 有 T(x) = Mx。矩阵只依赖于镜像直线的方向。
y point to its original position, so the transformation is its own inverse. In matrix language, M² = I and M⁻¹ = M. These properties distinguish reflections from rotations, which also preserve distance but have determinant +1.
反射是等距变换：它们不改变任意两点之间的距离。而且，对同一点实施两次相同的反射会让它回到原位，因此该变换是它自身的逆。用矩阵语言来说，M² = I 且 M⁻¹ = M。这些性质将反射与旋转区分开来，旋转也保持距离，但行列式为 +1。
aic rules apply: a reflection matrix in 3D is symmetric, orthogonal, and satisfies M² = I.
在进一步的学习中，我们还会考虑三维空间中的反射，此时的镜像是通过原点的平面。相同的代数法则适用：三维反射矩阵是对称的、正交的，且满足 M² = I。
2. Reflection in the x-axis | 关于x轴的反射
The simplest reflection in ℝ² sends a point (x, y) to (x, –y). This reflects across the x‑ ng how it acts on the basis vectors e₁ = (1,0)′ and e₂ = (0,1)′. Under reflection in the x‑axis, e₁ → e₁ and e₂ → –e₂, giving the matrix
ℝ² 中最简单的反射将点 (x, y) 映射到 (x, –y)。这是关于 x 轴的反射。该变换是线性的，其标准矩阵可以通过观察它如何作用于基向量 e₁ = (1,0)′ 和 e₂ = (0,1)′ 得到。在关于 x 轴的反射下，e₁ → e₁ 且 e₂ → –e₂，由此得到矩阵
| 1 | 0 |
| 0 | –1 |
Applying this matrix to the column vector (x, y)ᵀ confirms the transformation:
将该矩阵乘以列向量 (x, y)ᵀ 可验证变换：
(x′ , y′) = (x , –y)
The determinant of the matrix is (1)(–1) – (0)(0) = – tric and orthogonal. Its square is the identity matrix, as expected. The line y = 0 (the x‑axis) is the set of points left unchanged; these are eigenvectors with eigenvalue 1.
该矩阵的行列式为 (1)(–1) – (0)(0) = –1，且矩阵是对称正交的。它的平方是单位矩阵，正如预期。直线 y = 0（x 轴）是保持不变的点集，这些点就是特征值为 1 的特征向量。
3. Reflection in the y-axis | 关于y轴的反射
Reflection across the y‑axis maps (x, y) to (–x, y). The basis vectors transform as e₁ → –e₁ and e₂ → e₂, giving the matrix
关于 y 轴的反射将 (x, y) 映射到 (–x, y)。基向量变换为 e₁ → –e₁、e₂ → e₂，得到矩阵
| –1 | 0 |
| 0 | 1 |
Again, the determinant is – e matrix is its own inverse. The invariant line is x = 0. Multiplying this matrix by the x‑axis reflection matrix (in either order) yields
同样，行列式为 –1，矩阵是自己的逆。不变直线是 x = 0。将该矩阵与 x 轴反射矩阵相乘（无论哪个顺序）都会得到
| –1 | 0 |
| 0 | –1 |
which is a rotation by 180° (or a point reflection through the origin). This illustrates that two perpendicular reflections compose to a half‑turn.
这是一个 180° 的旋转（或关于原点的点反射）。这说明两次垂直反射的复合是一个半周旋转。
4. Reflection in the line y = x | 关于直线 y = x 的反射
he line y = x. The map (x, y) → (y, x) exchanges the basis vectors: e₁ ↔ e₂. The matrix is
交换坐标相当于关于直线 y = x 的反射。映射 (x, y) → (y, x) 交换了基向量：e₁ ↔ e₂。矩阵为
| 0 | 1 |
| 1 | 0 |
It is symmetric and has determinant – f points (a, a) that are fixed, so (1,1) is an eigenvector with eigenvalue 1. The line y = –x is mapped to its opposite, giving eigenvalue –1.
它是对称的，行列式为 –1。直线 y = x 由所有坐标相等的点 (a, a) 组成，这些点不变，因此 (1,1) 是特征值为 1 的特征向量。直线 y = –x 被映射到其反向，给出特征值 –1。
This reflection is particularly useful for studying the relationship between a function and its inverse. If a graph undergoes this reflection, the roles of x and y are swapped.
这个反射在研究函数与其反函数的关系时特别有用。图像经过该反射后，x 和 y 的角色互换。
5. Reflection in the line y = –x | 关于直线 y = –x 的反射
Reflection in the line y = –x sends (x, y) to (–y, –x). The matrix representing this transformation is
关于直线 y = –x 的反射将 (x, y) 映射到 (–y, –x)。表示此变换的矩阵为
| 0 | –1 |
| –1 | 0 |
Check: applying the matrix to (x, y)ᵀ gives ( –y , –x )ᵀ. The determinant is (0)(0) – (–1)(–1) = –1, and the matrix is symmetric. Points of the form (a, – y 180° followed by a reflection across x = y, or equivalently as a reflection across a perpendicular mirror.
验证：将矩阵乘到 (x, y)ᵀ 得到 ( –y , –x )ᵀ。行列式为 (0)(0) – (–1)(–1) = –1，且矩阵对称。形如 (a, –a) 的点位于镜像直线上并保持不变。该变换在交换坐标的同时改变两者的符号，可视为先旋转 180° 再进行 y = x 反射，或者等效为关于垂直镜面的反射。
6. General Reflection in a line through the origin | 一般通过原点的直线反射
To find the matrix that reflects across an arbitrary line through the origin making an angle θ with the positive x‑axis, we can decompose the transformation into a rotation, a reflection in the x‑axis, and a rotation back. Specifically, rotate by –θ, reflect across the x‑axis, then rotate by θ:
为了找出关于任意一条通过原点且与 x 轴正方向夹角为 θ 的直线的反射矩阵，我们可以将该变换分解为一个旋转、一个关于 x 轴的反射和一次反向旋转。具体说，先旋转 –θ，再关于 x 轴反射，最后旋转 θ：
M(θ) = R(θ) · Mₓ · R(–θ)
Working through the matrix multiplications using the standard rotation matrix
| cos θ | –sin θ |
| sin θ | cos θ |
and the x‑axis reflection matrix, we obtain the general reflection matrix
利用标准旋转矩阵和 x 轴反射矩阵进行乘法运算，我们得到一般反射矩阵
| cos 2θ | sin 2θ |
| sin 2θ | –cos 2θ |
This matrix is symmetric, has determinant – th angle θ is the eigenspace for eigenvalue 1, while the perpendicular line (angle θ + ½π) gives eigenvalue –1. By varying θ, we can represent the reflection in any line through the origin.
这个矩阵是对称的，行列式为 –1，且满足 M(θ)² = I。夹角为 θ 的直线是特征值 1 的特征空间，而与之垂直的直线（夹角 θ + ½π）对应特征值 –1。通过改变 θ，我们可以表示任何通过原点的直线反射。
uation y = mx (so tan θ = m), we can use trigonometric identities to rewrite the matrix as
(1/(1+m²)) · [ 1–m² , 2m ; 2m , m²–1 ]
Although the angle form is usually cleaner, this rational form is occasionally required in exam contexts. Both forms embody the same underlying geometry.
尽管角度形式通常更简洁，但在考试中偶尔也需要这种有理式形式。两种形式都体现了相同的几何本质。
7. Properties of Reflection Matrices | 反射矩阵的性质
All reflection matrices for lines through the origin in ℝ² share key properties. They are orthogonal, meaning Mᵀ = M⁻¹. Because a reflection is its own inverse, we also have Mᵀ = M, so reflection matrices are symmetric. Their determinants are always –1. The eigenvalues are 1 (for the mirror line) and –1 (for the perpendicular direction).
ℝ² 中所有关于通过原点的直线的反射矩阵都具有一些关键性质。它们都是正交的，即 Mᵀ = M⁻¹。因为反射是自身的逆，我们也有 Mᵀ = M，所以反射矩阵是对称的。它们的行列式总是 –1。特征值为 1（对应镜像直线）和 –1（对应垂直方向）。
Geometrically, being orthogonal combined with determinant –1 gives a clear classification: any 2 × 2 orthogonal matrix with det = – malises the intuitive picture that doing a mirror reflection twice returns the original figure.
从几何上看，正交矩阵加上行列式 –1 提供了一个清晰的分类：任何 2 × 2 行列式为 –1 的正交矩阵都表示一个反射（或反射后再旋转，而该组合本身是一个反射）。矩阵的平方是单位矩阵 I。这形式化了一个直观图像：两次镜像反射使图形恢复原样。
These properties are not coincidental; they stem from the fact that reflections preserve the dot product up to a change in sign of orientation. As linear isometries, they form part of the orthogonal group O(2), with the reflections making up the coset of determinant –1.
这些性质并非偶然；它们源于反射保持点积但改变定向符号这一事实。作为线性等距变换，它们构成正交群 O(2) 的一部分，而行列式为 –1 的反射则构成其一个陪集。
8. Determinant and Inverse | 行列式和逆
For any reflection matrix M, det(M) = – ticlockwise) will appear clockwise after reflection. The inverse of M is M itself, as M² = I. If we compute the adjugate or simply recall that for an orthogonal matrix M⁻¹ = Mᵀ, symmetry makes M⁻¹ = M.
对于任何反射矩阵 M，det(M) = –1。这立即告诉我们该变换反转定向：一个正向形状（例如顶点按逆时针排列的三角形）在反射后会呈现顺时针。M 的逆是它自身，因为 M² = I。如果我们计算伴随矩阵或直接回忆正交矩阵有 M⁻¹ = Mᵀ，对称性使得 M⁻¹ = M。
f the determinants. Thus, composing two reflections gives a transformation with determinant (–1)(–1) = +1, which must be a rotation (or identity). We will explore this next.
知道行列式在组合问题中很有用。当两个变换复合时，乘积的行列式等于行列式的乘积。因此，两次反射的复合给出行列式为 (+1) 的变换，那必定是一个旋转（或恒等变换）。接下来我们将探讨这一点。
9. Composition of Two Reflections | 两次反射的复合
Let M₁ and M₂ be the matrices of two reflections across lines L₁ and L₂ that pass through the origin and have angles θ₁ and θ₂ with the positive x‑axis. Then the product M₂M₁ represents the composition “reflect first across L₁, then across L₂”. Using the general form from Section 6, a straightforward multiplication yields
设 M₁ 和 M₂ 分别是关于通过原点且与 x 轴正方向夹角为 θ₁ 和 θ₂ 的直线 L₁ 和 L₂ 的反射矩阵。那么乘积 M₂M₁ 表示“先关于 L₁ 反射，再关于 L₂ 反射”的复合。利用第6节的一般形式，直接相乘得到
M₂M₁ = R(2(θ₂ – θ₁))
which is the rotation matrix for an angle 2(θ₂ – me, θ₂ = θ₁, the rotation angle is 0 and we get the identity, consistent with M² = I. If the mirrors are perpendicular, the rotation angle is 180°, a half‑turn.
这是一个旋转角度为 2(θ₂ – θ₁) 的旋转矩阵。这个美妙的结果表明，两次反射的复合是一个旋转，转过的角度是两镜面夹角的两倍。如果两条直线重合，θ₂ = θ₁，旋转角为 0，得到恒等变换，这与 M² = I 一致。若两镜面垂直，旋转角为 180°，即半周旋转。
Conversely, any rotation can be decomposed into two reflections. This fact is fundamental in geometry and is used to analyse symmetry groups. In tests, you may be asked to find the single transformation equivalent to a given sequence of reflections.
反之，任何旋转都可以分解为两次反射。这一事实是几何的基础，并用于分析对称群。在考试中，你可能会被要求找出与给定反射序列等效的单一变换。
10. Reflections in Three Dimensions | 三维空间中的反射
nsformation is linear and can be represented by a 3 × 3 orthogonal matrix with determinant –1. The simplest examples are reflections across the coordinate planes:
在三维空间中，反射是关于通过原点的一个平面（镜面）进行的。该变换是线性的，可以用一个 3 × 3、行列式为 –1 的正交矩阵表示。最简单的例子是关于坐标平面的反射：
- Reflection in the xy‑plane (z = 0): maps (x, y, z) → (x, y, –z).
- 关于 xy 平面 (z = 0) 的反射：将 (x, y, z) 映射到 (x, y, –z)。
The matrix is diagonal with entries (1, 1, –1):
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | –1 |
- Reflection in the xz‑plane (y = 0): (x, y, z) → (x, –y, z), matrix diag(1, –1, 1).
- 关于 xz 平面 (y = 0) 的反射：(x, y, z) → (x, –y, z)，矩阵 diag(1, –1, 1)。
- Reflection in the yz‑plane (x = 0): (x, y, z) → (–x, y, z), matrix diag(–1, 1, 1).
- 关于 yz 平面 (x = 0) 的反射：(x, y, z) → (–x, y, z)，矩阵 diag(–1, 1, 1)。
in two dimensions, these matrices are symmetric, their own inverses, and have determinant – reflection is the eigenspace for λ = 1, and the line perpendicular to the plane gives λ = –1. More general reflection planes can be constructed using Householder matrices, but for A‑level Further Mathematics, the coordinate‑plane reflections are the main focus.
如同二维情形，这些矩阵对称、自逆、行列式为 –1。反射平面是特征值 λ = 1 的特征空间，垂直于平面的直线给出 λ = –1。更一般的反射平面可以用 Householder 矩阵构建，但在 A‑level 进阶数学中，坐标平面反射是重点。
The composition of two reflections in 3D across intersecting planes yields a rotation about the line of intersection, with angle equal to twice the dihedral angle between the planes. This generalises the 2D
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