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# Horizontal Flip|Definition & Meaning

**Definition**

A** horizontal flip** is a **transformation** that **reflects** an **image horizontally** or from **left to right**. In other words, it creates a mirror image of the original image along a reference vertical line or axis. This means that the **left and right sides** of the **image** are **reversed** in the **flipped version**. For example, if a picture contains a **car facing to the right**, the **flipped version** of the image would show the **car facing to the left**.

**Conceptual Illustration of Horizontal Flip**

Figure 1 – Illustration of the horizontal flip of a point with x-coordinate 5 and y-coordinate 6, i.e., (5, 6).

**Consider** the following** example**. **Take** the **coordinates** of a **point, P**, and imagine it has these coordinates** (5, 6)**. Our** goal** is to find** its reflection** along the** y-axis** if we want **to flip it horizontally**. This is **accomplished** simply **by changing** the **sign** of the **x-coordinat**e (from **5 to -5**) and the y-coordinate of the point (6 in this case). For a **horizontally flipped point**, we get** (-5, 6)** as its **new coordinates**.

In general, the **coordinates** of a **point P** with respect to the **x and y axes** are represented by** (x, y)**, where **x** is the **distance of the point** from the **y-axis**, and** y** is the** distance of the point from** the **x-axis.** **When** we **horizontally flip** a **point**, we take its y-coordinate and **change** the **sign** of its **x-coordinate**. This** gives us** the **new coordinates** **(-x, y)** for the flipped point.

**Horizontal Flip of Person in Mirror**

Figure 2 – Horizontal reflection of a person raising their left hand in a mirror

Imagine **standing** in** front** of a **mirror** and **watching** your **reflection** in order to **visualize** a **horizontal flip reflection**. In the **event** that **you raised** your** left hand**, your **reflection** would **raise** its **right hand**. The **mirror** **reflects** you **horizontally**, **flipping left and right** without flipping up and down.

**Horizontal Flip in Arts**

**Reflections** on the** horizontal plane** are **commonly used** in art to **create symmetry** and** balance.** An **artist** could, for instance, **paint** a **painting** with **a tree on one side** and **its reflection** on the** other side**. **Flipping** the tree’s **original image** **horizontally** would** result** in the **reflection**. By doing so, the** painting** would **appear** **balanced** and **harmonious.**

**Properties of Horizontal Flip **

A horizontal flip has the following properties:

- It
**preserves distances between points**because it is an**isometry**. Accordingly, any**distance between**any**two points**of the original shape**will match**any space between the**corresponding points**of the**flipped shape**. - Basically, it is its
**own inverse**or**involution**. The**original shape**will be**returned**if you**flip a shap**e**horizontally****and**then**flip**it**again**. - On the shape, it
**reverses the order of**the**points**. When a line has**A, B, and C**at the**points**, then**when**it is**flipped**, it will have**C, B, and A**at the**points.** - By doing this,
**all points**on the shape are**given**a**new sign****for**their**x-coordinates**. Flipping occurs as a result of this. - A
**point**on the shape**will**still**maintain**its**y-coordinates**. By**flipping**the**shape**,**points**are**not affected**in their**vertical position.**

**Horizontal Flip of a Point**

To find the horizontal flip of a point, you can use the following steps:

- You need to
**identify the x-coordinate**of the point. The**horizontal axis**shows this value. **Take**the**x-coordinate**and**multiply**it by**-1**. By doing this, the**x-coordinate**is**flipped horizontally**, effectively**changing its sign.**- As a result, the
**original point**will be**horizontally flipped**. For instance, if the**original point**is**(1, 2)**, the**horizontal flip**will be**(-1, 2)**.

Throughout these steps, the **origin (0, 0)** is **assumed** to be the **center of the flip**. **Adding or subtracting** the **distance** from the origin to the flip center **will adjust** the **x-coordinate** **if** the flip center is **offset** from the origin (i.e., the flip is not about the origin as initially assumed).

**Horizontal Flip using Transformation Matrix**

- Use a
**2D vector representation**of the x and y coordinates of the point in the original image. For example\[ x=\begin{bmatrix} x \\ y \end{bmatrix} \] - In order
**to flip the image**horizontally,**define**the**transformation matrix**. This matrix can be defined as\[ T=\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \] **Using**the**matrix multiplication**operation,**multiply**the**coordinates**of the**point****by**the**transformation matrix**. The**point in**the**flipped image**will now have its new coordinates. For example**x’ = T * x.**\[ x’=\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]**Multiplying x by**a**transformation matrix**will**give**the**point**in the**flipped image**its**new coordinates, x**‘. A point in a flipped image can be located using these coordinates.\[ x’=\begin{bmatrix} -x \\ y \end{bmatrix} \]

**Horizontal Flip of a Geometric Shape**

Figure 3 – Horizontally flipped rectangle

**To find** the **horizontal flip** of any **geometric shape** like a **rectangle, square, parallelogram, or polygon** we can use the following steps:

**Identify the coordinates**of the**corners**of the**geometric shape**. You will need the x-coordinates and y-coordinates of each corner.**Multiply**the**x-coordinates**of all corners**by -1**. This**will change**the**sign of the x-coordinates**, effectively flipping the geometric shape horizontally.- The
**resulting coordinates**will be the**corners**of the**horizontal flip**of the original geometric shape. For example, if the o**riginal rectangle**has corners at**(x1, y1), (x2, y2), (x3, y3), and (x4, y4),**the**horizontal flip**will have corners at**(-x1, y1), (-x2, y2), (-x3, y3), and (-x4, y4).**Illustration is shown above.

**Solved Example**

You are **given** a **triangle** having **three corners** (3.5, 5), (2, 2), and (5, 2). **Compute** the **Horizontal Flip** of the triangle.

**Solution**

Figure 4 – Example of horizontally flipping a triangle

The process of finding the horizontal flip of a triangle involves first **identifying** the** coordinates** of the **triangle’s vertices**. This information is typically given in the form of a list of x-coordinates and a separate list of y-coordinates. In this example, the **triangle has vertices** at **(3.5, 5), (2, 2), and (5, 2),** the coordinates would be **(3.5, 2, 5) for the x-coordinates** and** (5, 2, 2) for the y-coordinates.**

Next, the **x-coordinates** of **each vertex** are **multiplied by -1**. This **changes** the** sign** of the **x-coordinates**, effectively **flipping the triangle** **horizontally**. The resulting coordinates are the vertices of the horizontal flip of the original triangle. In the example above, the **horizontal flip** would **have vertices** at **(-3.5, -2, -5)** for the** x-coordinates** and **(5, 2, 2) for** the **y-coordinates**.

*All mathematical drawings and images were created with GeoGebra.*