Transformations

In geometrical terms, when objects move in the coordinate plane, they go through transformations. In other terms, a set of coordinate points change into a different set of coordinates by the process of transformation. As in English, “transformation” means “to change,” in geometry, it refers to changes in the geometric properties of an object. There are different kinds of transformations that can occur; some of them do not change the size of the object while some of the transformations resize the original object. We will look at the different kinds of transformation in this article with examples.


Definition of Transformation

One can perform different operations on any image in a plane to transform it. In the process of transformation, an object can change its size, position, or orientation. It involves taking a pre-image of an object and producing its image with a transformation of some sort. 


Categories of Transformation

A transformation can be broadly categorized into 2 different types:

  • Rigid or isometric Transformation – The transformation in which size of the pre-image is unchanged is called a rigid transformation

  • Non-Rigid or non-isometric Transformation – A transformation that will change the size, but not the shape of the pre-image is called a non-rigid transformation.


Types of Transformations with Examples

Based on how the image is changed, transformations have 5 different types. One of them falls in the non-rigid transformation and the rest are all rigid transformations. 

  1. Translation (also called slide)

In this type of transformation, the object is only moved in space by sliding it, without any rotation, resizing, etc. There is no change in the object’s dimensions or shape. In translation, every point of the shape must move the same distance and in the same direction. You could use angle-distance or x-y coordinates to translate an object. This is a rigid transformation. Below is a pictorial representation of translation:


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  1. Rotation (also called Turn)

In this transformation, the object is turned around a fixed point, called the center of rotation, either clockwise or anti-clockwise by a certain angle. All the lines of the shape go through the same angle of rotation, changing the orientation and position of the object but not its size or shape. This is a rigid transformation. The distance at any point on the object and the center remains the same in rotation. Below is an example of rotation:


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  1. Reflection (also called Flip)

A mirror image of an object, along a line, is produced by reflection transformation. This line, along which reflection happens, is called the “mirror line,” and all points on the original object, as well as the mirror image, are at the same distance from this line. Reflection does not alter the size or shape of the object, just the position; hence it is again a rigid transformation. You can observer this in the image below:


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  1. Glide Reflection

This is a type of reflection in which the final image also goes through a translation. In the figure below, the blue object undergoes a reflection along the center black axis and then a translation of 6 units down, so glide reflection of the blue object is the pink figure here:


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  1. Enlargement (also called dilation)

This is a non-rigid transformation. In this transformation, there is a resizing of the image, but no change in shape. This kind of transformation is also known as compression, expansion, resizing, and contraction. In this, enlargement or reduction in the size of the object happens. The relative sizes of the image, as well as all angles, remain the same. You can see below that the pink figure is the dilated image of the blue figure:


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Conclusion

When a geometric figure is moved around on a coordinate plane, then transformations happen. Transformations do not change the shape of an object but can change its position, size, or orientation. Transformation can be rigid or isometric, where the size of the image does not change, or they can be non-rigid or non-isometric when the size of the image changes due to the transformation. Transformations are mostly done on a coordinate plane, as it makes it easy to count and draw. A common and easiest way of performing transformation is by performing the required operation on the vertices of the preimage, and when you connect the dots, you will be able to get the final transformed image.

FAQ (Frequently Asked Questions)

1. When do we say 2 Shapes are Congruent?

When one shape can be transformed into another by only using turns (rotations), flips (reflections), or slides (translation) then those shapes are called congruent.

2. When do we say 2 Shapes are Similar?

When we need to resize or dilate a shape in order to get another shape, then those shapes are called similar. It might also involve turning, flipping, or sliding of the shapes.

3. What are the Different Categories of Transformations?

Transformations are classified into 2 types: rigid or isometric and non-rigid or non-isometric. In the rigid transformation size of the object does not change but in non-rigid transformation, the size of the object is either enlarged or reduced.

4. What are the Different Types of Rigid Transformations?

The rigid transformations are rotation, translation, reflection, and glide reflection.

5. What are the Different Types of Non-Rigid Transformations?

There is only one non-isometric or non-rigid transformation called dilation. It is also called expansion, contractions, compressions, or enlargement.

6. How do you Perform a Transformation?

The most known way to perform a transformation on a shape is by doing the requested operation on the vertices of the preimage and then joining the dots to get the image.

7. What is a “Mirror Line” in the Context of Transformations?

In the reflection transformation the shape is flipped along a line which is called the “mirror line” and all the points on both the preimage and the image are at equal distance from the mirror line.

8. What is a Glide Reflection and How is it Different from Reflection Transformation?

A glide reflection is the one in which the shape is flipped across the mirror line and then some kind of slide or translation is done on the shape. So glide reflection is different from reflection as it involves “reflection + translation.”