Dilation

Dilation is the fundamental step to many areas of work and research. The term is quite simple to define. When a specific entity’s size is changed (decreased or increased) for some purpose, then this step is called dilation. Specific to mathematics, dilation is defined as the expansion or reduction in the size of a geometrical shape. This is a simple yet significant concept in mathematics and applies to many subfields as well. 


The Basic Definition of Dilation with an Example

Using a few scalar factors, the size of a given object can be extended or contracted. Specific to geometry, in the stance of changing the shape of an image, there are 4 major forms of Transformations.

  1. Translation 

  2. Rotation 

  3. Reflection 

Resizing (also called b)

From designing logos and posters to editing drawings and photographs, many domains make use of dilation regularly. Henceforth, we will discuss more into the 4th size transformation, i.e., dilation. 

To state a quick example, consider a semi-circle of radius 20 cms. The radius of the structure can be cut-down to 14, 10, 6, or even 3 cms. Also, the semi-circle’s radius can increase as 23, 27, 32, and so on. Both the former and the latter is a form of dilation. 


Dilation and its Scope in Mathematics 

As we saw the example of resizing a semi-circle, any geometrical figure can be subjected to change in its size using dilation. The Scalar Factor is the term used to denote either the contraction or extension in the mathematical model. 

  • The result of dilation is ‘Reduction’ if the outcome is smaller than the original. 

  • Enlargement’ is used to denote the formation of a dilated image larger to the original one.

Note that the dilated image (result) is the same as the original figure (before dilation). Only the size will vary.

(Image to be added soon)


What is a Scalar Factor?

In terms of calculating the reduction or enlargement of an object, the scalar factor is used. This is represented to be the ratio of an old image to that of the newly dilated image. The ‘centre of dilation’ is commonly present in the mid-point for a given object or plane. But there is also a limitation, since when 2 figures are combined or present 1 upon the other, then the centre of dilation can exist even outside either figure as well. 

Now, dilation from this point;

Scalar factor > 1: Image is stretched. Hence dilation enlargement occurs. 

  • Scalar factor =1: Both the original and the new image is equal or congruent. 

Scalar factor = 0 to 1: Image is contracted. Hence dilation reduction occurs. 

One can also calculate the slope of a line from a graph chart using the centre of dilation. By taking both the X and Y coordinates of a geometrical figure, you can evaluate the slope of the positive or negative line using the Scale Factor, Plotted Points, and the Centre of Dilation. 


A Few Important Properties Of Dilation

We noted that only the size of an image varies, and the resulting figure is a replica of the original. However, there are a few mathematical properties that do not change before and after dilation. Apart from the image, 3 other unchanged variables of dilation are given below:

  1. Perpendicular and Parallel lines of both the image remain equal and same. 

  2. The dilated image’s side mid-points will be equal to that of the original figure’s.

  3. Both images have equal and constant angles. 

The dilation will change the length of the image subjective to the required size. This can be of 2 forms. 

  • Horizontal dilation: Y = f(Cx)

  • Vertical Dilation: Y = C * f(x)

Where C denotes the Scalar factor and f(X) represents the function of dilation transformation to that image. 


Conclusion

Dilation is the process of increasing or decreasing the size of an image. The decrease is regarded as Reduction, and the increase is said to be Enlargement. Scalar Factor or scale factor is present at the midpoint of a figure and helps in transforming the image to a larger or smaller figure. Both dilation and scale factors together have good applications in areas such as graphic designing, photography, engineering, architecture, interior designing, mathematical constructions, and so on. 

FAQ (Frequently Asked Questions)

1. How Can I Calculate The Dimensions Of A Square Using Dilation?

You can calculate the dimensions of a square using dilation by multiplying the original dimensions before dilating the square using a scale factor. This will give you the present dimensions of the square.

2. Can A Dilated Circle Have Both The Size And Similar To That Of The Original Figure?

Yes. A dilated circle can have both its size and shape similar to that of the original, only if the procedure’s scalar factor is exactly measuring 1.0. Otherwise, the resulting circle will either be small or big to that of the original. 

3. What Will The Resulting Image Of A Polygon With A Scale Factor Of 0.5 Look Like?

The resulting image of a polygon with a scale factor of 0.5 will be exactly halved to that of the original polygon. This is observed for any geometrical figure regardless of its original size (C=0.5, the image will be half the size of the original figure).

4. What Is The Difference Between A Preimage And Image Respect To Dilation?

‘Preimage’ is a mathematical entity used to denote the image before scaling. It can be reduced, enlarged, or even inverted in shape. The ‘image’ is none other than the result of a figure’s dilation. For easy remember, the new/resulting figure is called ‘image,’ and the old/original figure is the ‘preimage.’ 

5. What Happens To A Triangle With Dilation Scale Factor Value In A Negative Integer?

A triangle with its dilation scale factor value with a negative integer will be upside down to that of the original figure. When a preimage is assigned a negative number to its dilation, then the final image will always be an inverted shape.