Transformations and Symmetry — KS3 Cambridge Mathematics 变换与对称性

Introduction to Transformations | 变换入门

In mathematics, a transformation is a way of changing the position, size, or orientation of a shape. Transformations are a fundamental topic in geometry and form a key part of the KS3 Cambridge Mathematics curriculum. Understanding transformations helps students develop spatial reasoning skills and lays the foundation for more advanced topics such as vectors, matrices, and coordinate geometry at GCSE and A-Level.

在数学中，变换是改变图形位置、大小或方向的一种方式。变换是几何学中的一个基础主题，也是 KS3 剑桥数学课程的重要组成部分。理解变换有助于学生培养空间推理能力，并为 GCSE 和 A-Level 中更高级的主题（如向量、矩阵和坐标几何）奠定基础。

There are four main types of transformations that you need to know: translation, reflection, rotation, and enlargement. Each type of transformation changes a shape in a different way, and some transformations produce images that are congruent to the original shape (same size and shape), while others produce images that are similar (same shape but different size). Understanding the difference between congruence and similarity is essential for mastering transformations.

你需要了解四种主要的变换类型：平移、反射、旋转和放大。每种变换以不同的方式改变图形，有些变换产生的图形与原图形全等（大小和形状相同），而另一些则产生相似的图形（形状相同但大小不同）。理解全等和相似的区别对于掌握变换至关重要。

Transformations can be described using mathematical language and notation. For example, a translation can be described using a column vector, a reflection can be described by naming the mirror line, a rotation requires a centre of rotation, an angle, and a direction, and an enlargement requires a centre of enlargement and a scale factor. Let us explore each transformation in detail.

变换可以用数学语言和符号来描述。例如，平移可以用列向量描述，反射可以通过命名镜线来描述，旋转需要旋转中心、角度和方向，而放大则需要放大中心和比例因子。让我们逐一详细探讨每种变换。

What Are Transformations? | 什么是变换？

A transformation is a rule that maps each point of a shape to a new position. The original shape is called the object, and the shape after the transformation is called the image. When we perform a transformation, we often label the vertices of the object with letters (such as A, B, C) and the corresponding vertices of the image with the same letters followed by a prime symbol (such as A’, B’, C’). This notation helps us keep track of which point went where during the transformation process.

变换是将图形上的每个点映射到新位置的规则。原始图形称为原图，变换后的图形称为像。当我们进行变换时，我们通常用字母（如 A、B、C）标记原图的顶点，并用相同字母加撇号（如 A’、B’、C’）标记原像的对应顶点。这种记法帮助我们在变换过程中跟踪每个点的去向。

Translation | 平移

Translation is the simplest type of transformation. A translation moves every point of a shape by the same distance in the same direction. The shape does not change its size, orientation, or appearance – it simply slides from one position to another. Translations produce congruent images, meaning the object and image are identical in shape and size.

平移是最简单的变换类型。平移将图形上的每个点沿相同方向移动相同的距离。图形的大小、方向或外观不会改变 – 它只是从一个位置滑动到另一个位置。平移产生全等的像，这意味着原图和像在形状和大小上完全相同。

In coordinate geometry, a translation is described using a column vector. A column vector is written as a pair of numbers stacked vertically inside brackets. The top number tells you how far to move in the x-direction (positive means right, negative means left), and the bottom number tells you how far to move in the y-direction (positive means up, negative means down). For example, the column vector (3, -2) means move 3 units to the right and 2 units down.

在坐标几何中，平移用列向量来描述。列向量写为括号内垂直堆叠的一对数字。上面的数字告诉你在 x 方向移动多远（正数表示向右，负数表示向左），下面的数字告诉你在 y 方向移动多远（正数表示向上，负数表示向下）。例如，列向量 (3, -2) 表示向右移动 3 个单位，向下移动 2 个单位。

To perform a translation on a grid, simply take each vertex of the shape and add the column vector components to its coordinates. If a vertex is at (x, y) and the translation vector is (a, b), then the new position is (x + a, y + b). The translated shape maintains exactly the same side lengths and angles as the original.

在网格上进行平移时，只需取图形的每个顶点，将列向量的分量加到其坐标上。如果顶点位于 (x, y)，平移向量为 (a, b)，则新位置为 (x + a, y + b)。平移后的图形保持与原图形完全相同的边长和角度。

Example: Translate triangle ABC with vertices A(1, 2), B(3, 5), and C(4, 1) by the vector (2, 3). The new vertices are A'(3, 5), B'(5, 8), and C'(6, 4). Notice that each x-coordinate increased by 2 and each y-coordinate increased by 3. The triangle’s size, shape, and orientation remain unchanged.

例子：将顶点为 A(1, 2)、B(3, 5) 和 C(4, 1) 的三角形 ABC 按向量 (2, 3) 平移。新顶点为 A'(3, 5)、B'(5, 8) 和 C'(6, 4)。注意每个 x 坐标增加了 2，每个 y 坐标增加了 3。三角形的大小、形状和方向保持不变。

Reflection | 反射（对称）

Reflection is a transformation that flips a shape over a line called the mirror line or line of reflection. The reflected image is the mirror image of the object – every point on the object is mapped to a point on the opposite side of the mirror line, at the same perpendicular distance from the line. Reflections produce congruent images, meaning the size and shape of the object are preserved.

反射是一种将图形翻折到称为镜线或反射线的直线另一侧的变换。反射后的像是原图的镜像 – 原图上的每个点映射到镜线另一侧与镜线垂直距离相等的点。反射产生全等的像，意味着原图的大小和形状保持不变。

In the KS3 curriculum, you will encounter reflections across various mirror lines: the x-axis (y = 0), the y-axis (x = 0), the line y = x, the line y = -x, and any horizontal or vertical line such as x = 2 or y = -1. When reflecting across a vertical line, the y-coordinate stays the same and the x-coordinate changes. When reflecting across a horizontal line, the x-coordinate stays the same and the y-coordinate changes.

在 KS3 课程中，你会遇到各种镜线的反射：x 轴 (y = 0)、y 轴 (x = 0)、直线 y = x、直线 y = -x，以及任何水平或垂直线如 x = 2 或 y = -1。当跨垂直线反射时，y 坐标保持不变而 x 坐标改变。当跨水平线反射时，x 坐标保持不变而 y 坐标改变。

When you reflect a shape, you should also think about what happens to its orientation. A reflection changes the orientation of the shape – if the vertices of the original shape were labelled in clockwise order, the vertices of the reflected image will appear in anticlockwise order. This is an important property that distinguishes reflections from translations and rotations.

当你反射一个图形时，你还应该考虑它的方向会发生什么变化。反射会改变图形的方向 – 如果原图的顶点按顺时针顺序标记，反射后的像的顶点将按逆时针顺序出现。这是区分反射与平移和旋转的重要性质。

Example: Reflect the point P(3, 4) in the y-axis (x = 0). The y-coordinate stays the same (4) and the x-coordinate changes sign: P'(-3, 4). Reflect P(3, 4) in the line y = x. The x and y coordinates swap: P'(4, 3). These simple rules make reflections straightforward to perform once you have identified the mirror line correctly.

例子：将点 P(3, 4) 在 y 轴 (x = 0) 上反射。y 坐标保持不变 (4)，x 坐标变号：P'(-3, 4)。将 P(3, 4) 在直线 y = x 上反射。x 和 y 坐标互换：P'(4, 3)。一旦正确识别了镜线，这些简单的规则使反射变得容易执行。

Rotation | 旋转

Rotation is a transformation that turns a shape around a fixed point called the centre of rotation. Every point on the shape moves along a circular path around the centre. The amount of turning is called the angle of rotation, which is measured in degrees. Rotations produce congruent images – the shape keeps the same size and shape but its orientation changes.

旋转是一种将图形绕一个称为旋转中心的固定点转动的变换。图形上的每个点沿圆弧路径绕中心移动。转动的量称为旋转角度，以度为单位测量。旋转产生全等的像 – 图形保持相同的大小和形状，但其方向改变了。

To fully describe a rotation, you need three pieces of information: the centre of rotation, the angle of rotation (such as 90°, 180°, or 270°), and the direction of rotation (clockwise or anticlockwise). For example, “rotate the triangle 90 degrees clockwise about the point (2, 1)” is a complete description. At KS3, the most common rotation angles are 90°, 180°, and 270°, and the centre of rotation is often at the origin (0, 0) or another clearly marked point on the grid.

要完整描述一次旋转，你需要三部分信息：旋转中心、旋转角度（如 90°、180° 或 270°）以及旋转方向（顺时针或逆时针）。例如，”将三角形绕点 (2, 1) 顺时针旋转 90 度”是一个完整的描述。在 KS3 阶段，最常见的旋转角度是 90°、180° 和 270°，旋转中心通常位于原点 (0, 0) 或网格上另一个明确标记的点。

There are some standard results that are worth remembering. A rotation of 180° about the origin maps (x, y) to (-x, -y). A rotation of 90° anticlockwise about the origin maps (x, y) to (-y, x). A rotation of 90° clockwise about the origin maps (x, y) to (y, -x). Knowing these patterns can help you quickly determine the coordinates of a rotated shape without needing to trace it on paper every time.

有一些值得记住的标准结果。绕原点旋转 180° 将 (x, y) 映射为 (-x, -y)。绕原点逆时针旋转 90° 将 (x, y) 映射为 (-y, x)。绕原点顺时针旋转 90° 将 (x, y) 映射为 (y, -x)。了解这些模式可以帮助你快速确定旋转后图形的坐标，而无需每次都在纸上描摹。

When performing a rotation using tracing paper (a common KS3 exam technique), place the tracing paper over the grid, trace the shape and the centre of rotation, hold the centre of rotation fixed with a pencil point, and rotate the tracing paper by the required angle. Then mark the new positions of the vertices on the grid beneath. This practical method is very reliable and is recommended for students who find it difficult to visualise rotations mentally.

使用描图纸进行旋转时（一种常见的 KS3 考试技巧），将描图纸放在网格上，描出图形和旋转中心，用铅笔尖固定旋转中心，将描图纸旋转所需的角度。然后在下面的网格上标记顶点的新位置。这种实用方法非常可靠，推荐给难以在脑海中想象旋转的学生使用。

Enlargement | 放大（缩放）

Enlargement is a transformation that changes the size of a shape. Unlike translations, reflections, and rotations, an enlargement does not always produce a congruent image. Instead, it produces a similar image – the shape is the same but the size is different. An enlargement is defined by two things: a centre of enlargement and a scale factor.

放大是一种改变图形大小的变换。与平移、反射和旋转不同，放大并不总是产生全等的像。相反，它产生相似的像 – 形状相同但大小不同。放大由两个要素定义：放大中心和比例因子。

The scale factor tells you how much bigger or smaller the image is compared to the object. If the scale factor is greater than 1, the image is larger. If the scale factor is between 0 and 1, the image is smaller (this is sometimes called a reduction or a fractional enlargement). If the scale factor is exactly 1, the image is the same size as the object and the transformation has no visible effect. If the scale factor is negative, the image appears on the opposite side of the centre of enlargement – this is a more advanced concept typically introduced at GCSE level.

比例因子告诉你像与原图相比有多大或多小。如果比例因子大于 1，像更大。如果比例因子介于 0 和 1 之间，像更小（有时称为缩小或分数放大）。如果比例因子恰好为 1，像与原图大小相同，变换没有可见效果。如果比例因子为负，像出现在放大中心的另一侧 – 这是通常在 GCSE 级别引入的更高级概念。

To perform an enlargement from a given centre, measure the distance from the centre to each vertex of the object, multiply each distance by the scale factor, and then mark the new vertex positions along the same ray extending from the centre through each original vertex. The image will be similar to the object, meaning that all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).

要从给定的中心进行放大，测量从中心到原图每个顶点的距离，将每个距离乘以比例因子，然后沿从中心穿过每个原顶点的相同射线上标记新顶点位置。像与原图相似，意味着所有对应角相等，所有对应边成相同比例（比例因子）。

Example: Enlarge triangle ABC with vertices A(1, 1), B(3, 1), and C(2, 4) from the centre (0, 0) with scale factor 2. The new vertices are A'(2, 2), B'(6, 2), and C'(4, 8). Each coordinate is simply multiplied by 2. The side lengths are doubled, but the angles remain the same. The area of the enlarged triangle is 4 times the area of the original, because area scales by the square of the scale factor.

例子：以 (0, 0) 为中心，比例因子为 2，放大顶点为 A(1, 1)、B(3, 1) 和 C(2, 4) 的三角形 ABC。新顶点为 A'(2, 2)、B'(6, 2) 和 C'(4, 8)。每个坐标都简单地乘以 2。边长加倍，但角度保持不变。放大后三角形的面积是原面积的 4 倍，因为面积按比例因子的平方缩放。

Symmetry | 对称性

Symmetry is closely related to transformations, particularly to reflection and rotation. A shape has symmetry if there is a transformation that maps the shape onto itself. There are two main types of symmetry studied at KS3: line symmetry (also called reflection symmetry or mirror symmetry) and rotational symmetry.

对称性与变换密切相关，特别是与反射和旋转相关。如果存在将图形映射到自身的变换，则该图形具有对称性。在 KS3 阶段学习两种主要类型的对称性：线对称（也称为反射对称或镜面对称）和旋转对称。

Line Symmetry | 线对称

A shape has line symmetry if it can be folded along a line so that one half fits exactly on top of the other half. This line is called a line of symmetry or an axis of symmetry. The number of lines of symmetry a shape has depends on its properties. For example, a square has 4 lines of symmetry, an equilateral triangle has 3, a rectangle has 2, a rhombus has 2, and a circle has infinitely many lines of symmetry.

如果一个图形可以沿一条线折叠，使其中一半恰好与另一半重合，则该图形具有线对称。这条线称为对称线或对称轴。一个图形有多少条对称线取决于其性质。例如，正方形有 4 条对称线，等边三角形有 3 条，矩形有 2 条，菱形有 2 条，而圆有无限多条对称线。

To find the lines of symmetry in a shape, try visualising a fold through the shape. If the two halves match exactly (including any patterns or colours), then you have found a line of symmetry. Regular polygons have a special property: the number of lines of symmetry equals the number of sides. A regular pentagon has 5 lines of symmetry, a regular hexagon has 6, and so on.

要找到图形中的对称线，尝试想象一条穿过图形的折线。如果两半完全匹配（包括任何图案或颜色），那么你就找到了一条对称线。正多边形有一个特殊性质：对称线的数量等于边的数量。正五边形有 5 条对称线，正六边形有 6 条，以此类推。

Rotational Symmetry | 旋转对称

A shape has rotational symmetry if it can be rotated about its centre by an angle less than 360 degrees and still look exactly the same as it did before the rotation. The order of rotational symmetry is the number of different positions in which the shape looks the same during one complete turn. For example, a square has rotational symmetry of order 4 because it looks the same after rotations of 90°, 180°, 270°, and 360°.

如果一个图形绕其中心旋转小于 360 度的角度后，看起来与旋转前完全相同，则该图形具有旋转对称。旋转对称的阶数是在一次完整旋转中图形看起来相同的不同位置的个数。例如，正方形具有 4 阶旋转对称，因为它在 90°、180°、270° 和 360° 旋转后看起来相同。

The order of rotational symmetry can be found by counting how many times a shape matches itself during a full 360-degree rotation. An equilateral triangle has rotational symmetry of order 3, a rectangle has order 2, a rhombus has order 2, and a parallelogram has order 2. A shape with no rotational symmetry (it only matches itself at 360 degrees) is said to have rotational symmetry of order 1.

旋转对称的阶数可以通过计算在一次完整的 360 度旋转中图形与自身重合的次数来找到。等边三角形具有 3 阶旋转对称，矩形具有 2 阶，菱形具有 2 阶，平行四边形具有 2 阶。没有旋转对称的图形（仅在 360 度时与自身重合）被称为具有 1 阶旋转对称。

It is important not to confuse the order of rotational symmetry with the angle of rotation. If a shape has rotational symmetry of order n, then the smallest angle of rotation that maps the shape onto itself is 360° divided by n. For example, an equilateral triangle has order 3, so its smallest rotation angle is 120°. This relationship helps you check your work and develop a deeper understanding of how rotational symmetry works.

重要的是不要混淆旋转对称的阶数和旋转角度。如果一个图形具有 n 阶旋转对称，那么将图形映射到自身的最小旋转角度是 360° 除以 n。例如，等边三角形具有 3 阶，因此其最小旋转角度是 120°。这种关系有助于你检查作业并加深对旋转对称工作原理的理解。

Combining Transformations | 组合变换

It is possible to apply more than one transformation to a shape, one after the other. This is called a combination of transformations or a composition of transformations. The final image after applying multiple transformations depends on the order in which the transformations are applied – changing the order can produce a different result. This is an important concept that bridges KS3 work with more advanced topics at GCSE and beyond.

可以对一个图形连续应用多个变换。这称为变换的组合或变换的复合。应用多个变换后的最终像取决于应用变换的顺序 – 改变顺序可能产生不同的结果。这是一个重要的概念，将 KS3 的学习与 GCSE 及以后更高级的主题连接起来。

When describing a combination of transformations, work step by step. First apply transformation A to the object to get image A’. Then apply transformation B to image A’ to get the final image B’. Make sure you label each intermediate image clearly to avoid confusion. Using different numbers of prime marks (A’, A”, A”’) can help track which stage you are at in the transformation sequence.

在描述组合变换时，要逐步进行。首先将变换 A 应用于原图得到像 A’。然后将变换 B 应用于像 A’ 得到最终像 B’。确保清楚地标记每个中间像以避免混淆。使用不同数量的撇号 (A’, A”, A”’) 可以帮助追踪你在变换序列中的哪个阶段。

Example: Reflect triangle P(1, 1), Q(3, 1), R(2, 4) in the y-axis, then translate the result by the vector (2, -1). Step 1: Reflect in y-axis gives P'(-1, 1), Q'(-3, 1), R'(-2, 4). Step 2: Translate by (2, -1) gives P”(1, 0), Q”(-1, 0), R”(0, 3). The order matters – if the translation was done first, the result would be different.

例子：将三角形 P(1, 1)、Q(3, 1)、R(2, 4) 在 y 轴上反射，然后将结果按向量 (2, -1) 平移。步骤 1：在 y 轴上反射得到 P'(-1, 1)、Q'(-3, 1)、R'(-2, 4)。步骤 2：按 (2, -1) 平移得到 P”(1, 0)、Q”(-1, 0)、R”(0, 3)。顺序很重要 – 如果先进行平移，结果会不同。

Real-World Applications of Transformations | 变换的实际应用

Transformations are not just abstract mathematical concepts – they appear everywhere in the real world. Architects use transformations when designing buildings with repeating patterns and symmetrical facades. Computer graphics and video games rely heavily on transformations to move, rotate, and scale objects on the screen. Artists like M.C. Escher used transformations to create famous tessellations and impossible constructions that captivate viewers to this day.

变换不仅仅是抽象的数学概念 – 它们在现实世界中无处不在。建筑师在设计具有重复图案和对称立面的建筑时使用变换。计算机图形和视频游戏严重依赖变换来在屏幕上移动、旋转和缩放物体。像 M.C. 埃舍尔这样的艺术家使用变换创作了著名的镶嵌图案和不可能结构，至今仍吸引着观众。

In nature, symmetry and transformation patterns are abundant. Snowflakes exhibit six-fold rotational symmetry, butterfly wings display reflection symmetry, and sunflower seed arrangements follow spiral patterns related to the golden ratio and rotational transformations. Understanding these mathematical principles enriches our appreciation of the natural world and helps scientists model and predict natural phenomena.

在自然界中，对称和变换模式比比皆是。雪花呈现六重旋转对称，蝴蝶翅膀展示反射对称，向日葵种子的排列遵循与黄金比例和旋转变换相关的螺旋模式。理解这些数学原理丰富了我们对自然界的欣赏，并帮助科学家建模和预测自然现象。

In engineering and manufacturing, transformations are essential for designing parts that fit together, creating patterns for textiles and wallpaper, and programming robotic arms to perform precise movements. The mathematical foundations of transformations that you learn at KS3 are the same principles used by engineers to design cars, planes, bridges, and countless other structures that shape our modern world.

在工程和制造领域，变换对于设计相互配合的零件、为纺织品和壁纸创建图案以及编程机器人手臂执行精确运动至关重要。你在 KS3 学习的变换数学基础与工程师设计汽车、飞机、桥梁和无数其他塑造现代世界的结构所使用的原理是相同的。

Summary | 总结

Transformations are a cornerstone of geometry that describe how shapes can be moved, flipped, turned, and resized. The four main types – translation, reflection, rotation, and enlargement – each have distinct properties and rules. Translation moves a shape without changing its orientation; reflection flips a shape over a mirror line; rotation turns a shape around a fixed centre; and enlargement changes the size of a shape by a scale factor. Translations, reflections, and rotations produce congruent images, while enlargements produce similar images.

变换是几何学的基石，描述了图形如何被移动、翻转、转动和缩放。四种主要类型 – 平移、反射、旋转和放大 – 各有不同的性质和规则。平移在不改变方向的情况下移动图形；反射将图形翻折到镜线另一侧；旋转将图形绕固定中心转动；放大通过比例因子改变图形的大小。平移、反射和旋转产生全等的像，而放大产生相似的像。

Symmetry is deeply connected to transformations. Line symmetry relates to reflection, while rotational symmetry relates to rotation. Recognising symmetry in shapes helps you understand their properties and classify them correctly. The skills you develop in this topic – visualising movements, working with coordinates, and solving multi-step problems – will serve you well throughout your mathematics education and beyond.

对称性与变换密切相关。线对称与反射相关，而旋转对称与旋转相关。识别图形中的对称性有助于你理解它们的性质并正确分类。你在这个主题中培养的技能 – 可视化运动、处理坐标以及解决多步骤问题 – 将在你的整个数学教育及以后的学习中为你提供帮助。

Remember that practice is essential for mastering transformations. Work through examples systematically, always clearly identifying the type of transformation, its defining parameters, and the effect on coordinates. Use tracing paper for rotations, count squares carefully for reflections, and always check your answers by verifying that the image satisfies the given transformation rules. With consistent practice, transformations will become second nature.

记住，练习对于掌握变换至关重要。系统地完成例题，始终清楚地识别变换类型、其定义参数以及对坐标的影响。使用描图纸进行旋转，仔细计算方格进行反射，并始终通过验证像是否满足给定的变换规则来检查你的答案。通过持续的练习，变换将成为你的第二天性。

---

更多咨询请联系16621398022（同微信）