Reflection

Transformation is defined to be the process of changing something in its form, texture, shape, size, model, colour, etc. In the same context, when we take mathematical transformation, this process is done to increase or decrease the size of an object or figure. For example, when the length of a rectangle is reduced from 8 cm to 3 cm, then this is a form of geometrical transformation. There are 4 major forms of transforming shapes namely, translation, rotation, dilation or resizing, and reflection. Let’s take account of the 4th type Reflection and understand the highlighted concepts from the below sections.


Important Points Regarding the ‘Reflection’ Definition 

We understood what a mathematical transformation. Now, let us learn the reflection definition by using the following pointers for a better understanding. 

  • Flipping an image is called a Reflection in geometry. So, the resulting image will be the mirror image to the origins structure. 

  • Only the direction of the resulting image is the opposite. However, the size and shape remain the same. 

  • Since the position is changed in this transformation, there are chances for Translation as well. 

  • Both the figures (before and after reflection) are equidistant from all their points over their surfaces.

  • There is 1 line that helps in reflecting the object and this line is said to be the line of reflection. 

All About Reflection in the Coordinate Plane

Now, let us move onto the consideration of reflection with a graph. The sub-headings followed down are details covered on reflection in the coordinate plane, with the X-Axis and Y-Axis as the references.


What is the Reflection over X-Axis?

Imagine that a point is reflecting over the X-Axis. Hence, the Y-coordinates will transform in their signs oppositely but the X-coordinates stay constant. 

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So, if the point of reflection is labelled as [X, Y] then the same coordinates across the X-Axis would be [X, - Y].

The Reflection upon the Y-Axis

As the contrary case of X-Axis, the Y-Axis here will stay the same while the X-coordinates transform with their opposite symbols when the reflection takes place across the Y-Axis. 

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Therefore, [X, Y] is the reflection of point and is changed as [- X, Y] in the region of Y-Axis.


A Condition of Reflection when Y = X

Take the case where a point is reflecting across a line Y=X. Now, the X and Y coordinates will interchange their positions. However, the signs get negated/cancelled when the point of reflection takes place over a line Y = - X, but the point of coordinates still changes places. 

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The interpretations are as follows:

  • Reflection of point [X, Y] is [Y, X] in the case of line Y = X.

  • Reflection of point [X, Y] is [- Y, X] in the case of line Y = - X.


Gist about Reflection on a Point

The point of reflection is also referred to as the centre of a figure. Any structure is built using this reflection point as its single reference. When you consider points over a figure,  you will note exact points on the opposite direction of the original image, on the other side. Note that the shape and size are left unchanged at the point of reflection. 


What is the Reflection in the origin (0, 0)?

Any random area is used as the point of reflection in a coordinate plane. ‘Origin’ is the frequently used point.

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As you observed in the diagram above, the preimage triangle (original) has coordinates [1, 2, 3] and the reflected image is [- 1, - 2, - 3]. By drawing a line segment at the point of reflection, you should note the line of origin, which is the figure’s mid-point. Note that the point of reflection, in the case of origin reflection, changes the coordinates to a negative value, after the transformation of the shape.

Conclusion

A reflection is a form of geometrical transformation, where the figure is flipped. The points on the surface of both the images are equidistant. The point of reflection is said to be the mid-point where the reflection transformation takes place. Usually, the size and shape of the figure remain the same and only the direction gets opposite with the preimage. When the point of reflection is taken with the origin, then the coordinates turn their signs negative, due to the opposite direction of reflection.

FAQ (Frequently Asked Questions)

1. Can The Mirror Line Or The Preimage Be Reflected In The Vertical Direction?

Yes. The mirror line or the preimage can be reflected in both the horizontal and vertical directions. Moreover, this form of reflection transformation can be performed in any direction even upside or downside. This is simply a flip of the image and will result in whichever diagonal transformed.

2. How To Mark The Coordinates Of A Reflected Image On A Graph Paper?

To mark the coordinates of a reflected image on a graph paper, simply put dashes over the preimage’s coordinates. Say, that your shape before reflection is marked coordinates as [1, 3, 5] then your reflected image’s coordinates are [1’, 3’, 5’] respectively. The symbol dash denotes ‘Prime’ value and marks every point of reflection after the preimage.

3. Why Is The Process Of Reflection Transformation Referred To As An Involution?

The process of reflection transformation is referred to as an involution because when the same process is applied twice, that is, if the reflection occurs 2 times continuously, the final image is similar to the original state, similar to the restoration of the figure.

4. State Any 3 Properties Of The Reflection Transformation.

The reflection transformation has a matrix (the array of integers) as orthogonal, having a determinant value of - 1. The rotation transformation can be represented here using the product of 2 orthogonal matrices. When the reflection takes place with a given odd number, then the result of such a process is called ‘improper rotation’.

5. What Are The Real-life Instances Where The Transformation Process Of Reflection Is Observed?

The most common and beautiful form of reflection is when the moon reflects over the water surfaces. Even the mirror image that your face reveals in the mirror is a flip of your original face and is a direct form of reflection. All shiny surfaces and glossy or colourless elements like that of water and diamond undergo reflection.