📚 Transformations: Reflections, Rotations and Translations | 平面图形的变换：反射、旋转与平移
Imagine sliding a shape across a grid, flipping it over a line, or spinning it around a point – these movements are called transformations. In KS3 Cambridge Mathematics, you will learn to describe and perform reflections, rotations, and translations with precision using coordinates, vectors, and angles. Mastering these ideas helps you understand symmetry, maps, and even computer graphics.
想象一下，在方格纸上移动一个图形，将它沿着一条线翻转，或者绕着一个点旋转——这些操作就叫做变换。在KS3剑桥数学课程中，你将学习用坐标、向量和角度来精确描述并完成反射、旋转和平移。掌握这些概念能帮助你理解对称、地图甚至计算机图形学。
1. What Are Transformations? | 什么是变换？
A transformation changes the position or orientation of a shape without altering its size or shape. In KS3, the three principle transformations are translation (sliding), reflection (flipping), and rotation (turning). An object is the original shape, and the image is the shape after transformation.
变换会改变图形的位置或方向，但不改变其大小和形状。在KS3阶段，三种主要的变换是平移（滑动）、反射（翻转）和旋转（转动）。原始图形称为原像，变换后的图形称为像。
2. Coordinates and the Cartesian Plane | 坐标与直角坐标平面
We describe positions using coordinates (x, y) on a grid. The x-axis runs horizontally, and the y-axis runs vertically. Knowing how to plot points accurately is essential before applying transformations. For instance, the point A(2, 3) is 2 units right and 3 units up from the origin (0,0).
我们使用坐标 (x, y) 在网格上描述位置。x 轴为水平方向，y 轴为垂直方向。在应用变换之前，准确地标记点是至关重要的。例如，点 A(2, 3) 意味着从原点 (0,0) 向右2个单位、向上3个单位。
3. Translation: Sliding a Shape | 平移：滑动图形
A translation moves every point of a figure the same distance in the same direction. We often describe a translation using a column vector, for example (4, −2), which means move 4 units to the right and 2 units down. The shape stays congruent; it does not turn or flip.
平移是将图形上的每一个点沿相同方向移动相同距离。我们通常使用列向量来描述平移，例如 (4, −2) 表示向右移动4个单位、向下移动2个单位。平移后的图形与原图形全等，既没有转动也没有翻转。
4. Describing Translations with Vectors | 用向量描述平移
The vector (a, b) tells us the horizontal shift (a) and the vertical shift (b). If a is positive, the movement is to the right; if negative, to the left. If b is positive, the shape moves up; if negative, it moves down. This compact notation makes it easy to specify translations without lengthy sentences.
向量 (a, b) 告诉我们水平位移量 (a) 和垂直位移量 (b)。如果 a 为正，则向右移动；如果为负，则向左移动。如果 b 为正，则向上移动；如果为负，则向下移动。这种简洁的表示法可以轻松指定平移，无需冗长的句子。
5. Reflection: Flipping Over a Mirror Line | 反射：沿镜面线翻转
A reflection creates a mirror image of a shape across a line called the line of reflection. Every point on the original shape is mapped to a point on the opposite side of the line, at an equal perpendicular distance. The image is congruent to the original but reversed.
反射是沿着一条叫做反射线的直线创建图形的镜像。原图形上的每一个点都映射到反射线另一侧等垂直距离的点上。反射像与原图形全等，但方向相反。
6. Common Lines of Reflection | 常见的反射线
Typical lines of reflection include the x-axis (y = 0), the y-axis (x = 0), and lines such as y = x or y = −x. When reflecting in the x-axis, the y-coordinate changes sign: (x, y) → (x, −y). Reflecting in the y-axis changes the sign of x: (x, y) → (−x, y). For y = x, coordinates swap: (x, y) → (y, x).
常见的反射线包括 x 轴 (y = 0)、y 轴 (x = 0)，以及直线 y = x 或 y = −x 等。关于 x 轴反射时，y 坐标变号：(x, y) → (x, −y)。关于 y 轴反射时，x 坐标变号：(x, y) → (−x, y)。关于 y = x 反射时，坐标互换：(x, y) → (y, x)。
7. Rotation: Turning Around a Point | 旋转：绕一个点转动
A rotation turns a shape around a fixed point called the centre of rotation. To fully describe a rotation we need three pieces of information: the centre of rotation, the angle of rotation (e.g., 90°, 180°), and the direction (clockwise or anticlockwise). The distance from the centre to each point stays the same, and the shape remains congruent.
旋转是让图形绕着一个固定点（旋转中心）转动。要完整描述一个旋转，我们需要三个要素：旋转中心、旋转角度（如 90°、180°）和旋转方向（顺时针或逆时针）。每个点到旋转中心的距离保持不变，图形仍全等。
8. Rules for Common Rotations | 常见旋转的规则
When the centre is the origin (0,0), there are helpful coordinate patterns. For a rotation of 90° clockwise about the origin, (x, y) becomes (y, −x). For 90° anticlockwise, (x, y) becomes (−y, x). A rotation of 180° (half turn) maps (x, y) to (−x, −y). These rules allow you to quickly find the image of any point without drawing.
当旋转中心为原点 (0,0) 时，有一些有用的坐标模式。绕原点顺时针旋转 90°，(x, y) 变为 (y, −x)。逆时针旋转 90°，(x, y) 变为 (−y, x)。旋转 180°（半圈）将 (x, y) 映射为 (−x, −y)。这些规则使你无需绘图就能快速找到任何点的像。
9. Combining Transformations | 组合变换
Sometimes a shape undergoes more than one transformation, such as a reflection followed by a translation. The order matters: performing translation then rotation can give a different final image than rotation then translation. You must apply each step carefully, recording the intermediate images or using algebraic methods.
有时一个图形会经历不止一次变换，例如先反射再平移。顺序很重要：先平移后旋转与先旋转后平移可能得到不同的最终像。你必须仔细完成每一步，记录中间图像或使用代数方法。
10. Checking Your Work and Common Mistakes | 检查作业与常见错误
Always verify that the image is congruent to the object. In reflections, measure perpendicular distances to the mirror line. In rotations, use tracing paper or check that distances to the centre match. A common mistake is miscounting squares or confusing clockwise with anticlockwise – double-check your direction and angle.
务必检查像和原图形是否全等。在反射中，测量到反射线的垂直距离。在旋转中，使用描图纸或检查到旋转中心的距离是否一致。常见的错误是数错格子或混淆顺时针与逆时针——请再次确认你的方向和角度。
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