Tranformations

Math Meaning of Transformation

A transformation is that geometric process of mathematics that maneuvers a polygon, quadrant or other 2-D object on a plane or coordinates system. These transformation maths helps to explain how two-dimensional figures move around about a coordinate plane. Vectors are most commonly used to describe translations in mathematics. Thus, in simple terms, a translation of an image takes place when a shape is moved from one place to another. It is just like gathering up the shape from its original place and putting it down somewhere else. That being said, any object or image in a plane or a coordinate system could be manipulated by utilizing different operations of transformations.


Two - Dimensional Shapes

A preimage or an inverse image is the 2D shape that we used to denote before any transformation. Image is the figure that we get after transformation.


Types of Transformations

We have 5 different transformations or operations of transformations in math which are as follows:


  1. Reflection - The image is a mirrored preimage; what we mathematically call a "a flip."

  2. Rotation - The image is the preimage rotated around a specified point; what we mathematically call "a turn."

  3. Dilation - The image is a smaller or a larger version of the preimage; what we mathematically call a "shrinking" or "enlarging."

  4. Shear - All the points across one side of a preimage does not change (remain fixed) while all other points of the preimage move parallel to that side in distribution to the distance from the given side; what we mathematically call "a skew.,"

  5. Translation - The image is equalized by a constant value from the preimage; what we mathematically call "a slide."

Now that you are aware of the different transformations that can occur, let us learn about them in detail as to appropriately use them further.


Reflection

Imagine snipping off a preimage, picking it up, and putting it down again at face. That is what the process of a reflection or a flip is. You already know that a reflection image is an exact duplicate of the preimage. So, now tell us if the below  triangle image in purple is a reflection of the purple preimage or object?


(image will be uploaded soon)


Since the purple triangle image is being reflected through the x-axis, it is a reflection image. Here, you also need NOT to use a coordinate plane's axis for to identify a reflection.


Rotation

A two-dimensional figure does not have to be dependent on the origin for rotation. You can easily see rotation in these figures using the coordinate grid which is as easy as using the x-axis and y-axis. For example,


In order to create rotation in 90° 90°: (x, y) → (−y, x)(x, y) → (-y, x) (multiply the y-value times −1-1 and shift the x- and y-values)


(image will be uploaded soon)


Dilation

Dilate a preimage of any polygon is performed by making a carbon copy of its interior angles while increasing every side in proportion. Imagine dilating as resizing and you will master over the concept of dilation geometry. Below is the image green which is a dilation of the purple preimage. 


(image will be uploaded soon)


Shear

To shear an image, you "skew it," meaning that fixing one line of the figure whereas moving all the other lines & points in a specified proportion, direction to their distance without changing its area, interior angles or stretching dimensions.


Below you will find a square figure transformed into a parallelogram without changing dimensions and angles. 


(image will be uploaded soon)


Translation

A translation moves the object from its native place on the coordinate grid without having to alter its orientation. 


(image will be uploaded soon)


Solved Examples

Translation Transformation Example 1:

If you are asked to translate a figure into a coordinate plane, how will you do it?

For translating a polygon MNO on the coordinate plane, your mathematical graphing instructions look like this:


Taking a Triangle MNO + 9 units, we will translate the triangle on the coordinate plane in the x-direction and -4 in the Y direction.


Using Method by Writing down the original coordinates,

                 X           Y

M (-8, 6) -- 8 + 9   6 - 4

N (-8, 9) -- 8 + 9         9 - 4

O (-4, 6) -- 4 + 9         6 - 4


Thus, we get

M (1, 2)

N (1, 5)

O (5, 2)

Hence, translated polygon should be congruent to the original


Translation Transformation Example 2:

Problem

Determine the components of the translation, and identify where P point' ends up:


(image will be uploaded soon)


Solution

To identify the translated vector, we need to choose a labeled point and then subtract the old coordinates from the new. That is to say:-

Translation

A'(-1, 2) - A (1, 4)

(-2, -2)

P' = P (-1, 3) + (-2, -2)

Your answer is (-3, 1)


(image will be uploaded soon)


Fun Facts

  • You can easily Rotate 90 ° Clockwise or 270 ° Counterclockwise About the Origin using the mathematical formula (x, y) ⟶ (y, -x).

  • Reflecting a polygon along x-axis means measuring the distance of each vertex to the line of reflection.

  • Perfect examples of shear are Italic letters on a computer.

FAQ (Frequently Asked Questions)

1. What is meant by the Rigid Transformations?

A rigid transformation is one such geometric translation that remains the same with respect to both shape and size of the preimage while generating the image. There are chiefly three transformations that are accounted for as rigid. Types of transformations included under the rigid transformations are reflection, rotation, and translation. The image from all these three transformations will not alter its shape or size.

2. What is Meant by the Non-Rigid Transformations?

A mathematical non- rigid transformation is one such that can either manipulate the size or shape, or both the size and shape of the preimage while generating the image. There are chiefly two transformations that are accounted as non-rigid. Types of transformations included under the non-rigid transformations are dilation and shear. The images from all these two transformations are liable to change the shape and size both.

3. What does Transformations in the Coordinate Plane suggest?

Along the coordinate grid or a plane, you can use the x-axis and y-axis in order to keep a track of every move. The lines also provide good assistance while drawing the polygons and flat figures. You need to concentrate on the coordinates of the object’s vertices and then join them to form the image. To take an example, you will be able to reflect and translate a polygon, so the image that will appear is apart and duplicated from its preimage. Similarly, a rectangle can be shrinked or enlarged, so it looks like a larger parallelogram.