Rigid And Non Rigid Transformations

Transformations Math Definition

A transformation is a process that manipulates a polygon or other ii-dimensional object on a plane or coordinate arrangement. Mathematical transformations describe how two-dimensional figures motility around a plane or coordinate system.

A preimage or inverse epitome is the two-dimensional shape before whatever transformation. The image is the figure after transformation.

Transformations Math Definition
Types of Transformations
- Dilation
- Reflection
- Rotation
- Shear
- Translation
Rigid and Not-Rigid Transformations
Rigid and Non-Rigid Transformations
Transformations Examples
Transformations in the Coordinate Plane

Types of Transformations

At that place are five different transformations in math:

Dilation -- The image is a larger or smaller version of the preimage; "shrinking" or "enlarging."
Reflection -- The prototype is a mirrored preimage; "a flip."
Rotation -- The image is the preimage rotated around a fixed point; "a plow."
Shear -- All the points along one side of a preimage remain stock-still while all other points of the preimage move parallel to that side in proportion to the altitude from the given side; "a skew.,"
Translation -- The image is offset by a constant value from the preimage; "a slide."

Dilation

Dilate a preimage of any polygon is done past duplicating its interior angles while increasing every side proportionally. You can think of dilating as resizing. Which triangle prototype, yellow or blue, is a dilation of the orange preimage?

$Dilation Transformation Definition and Example$

The xanthous triangle, a dilation, has been enlarged from the preimage by a cistron of 3.

Reflection

Imagine cutting out a preimage, lifting it, and putting it back face downwards. That is a reflection or a flip. A reflection paradigm is a mirror image of the preimage. Which trapezoid paradigm, crimson or regal, is a reflection of the green preimage?

$Reflection Transformation Definition and Example$

The imperial trapezoid epitome has been reflected along the ten-centrality, but yous practise not need to use a coordinate airplane's axis for a reflection.

Rotation

Using the origin, $(0, 0)$ , as the point around which a ii-dimensional shape rotates, you can easily run across rotation in all these figures:

$Rotation Transformation Definition and Example$

A figure does not accept to depend on the origin for rotation.

Shear

Hither is a square preimage. To shear information technology, you "skew it," producing an image of a rhombus:

$Shear Transformation Definition and Example$

When a figure is sheared, its area is unchanged. A shear does non stretch dimensions; information technology does modify interior angles.

Translation

A translation moves the figure from its original position on the coordinate plane without changing its orientation. Which octagon image beneath, pink or blue, is a translation of the yellowish preimage?

$Translation Transformation Definition and Example$

The blueish octagon is a translation, while the pink octagon has rotated.

Rigid Transformations

A rigid transformation does not change the size or shape of the preimage when producing the paradigm. Three transformations are rigid.

The rigid transformations are reflection, rotation, and translation. The image from these transformations will non alter its size or shape.

Non-Rigid Transformations

A non-rigid transformation can change the size or shape, or both size and shape, of the preimage.

Ii transformations, dilation and shear, are non-rigid. The image resulting from the transformation will change its size, its shape, or both.

$Rigid and Non-rigid Transformations$

Transformation Examples

In that location are v different types of transformations, and the transformation of shapes can exist combined. A polygon can exist reflected and translated, so the prototype appears apart and mirrored from its preimage. A rectangle can be enlarged and sheared, so information technology looks similar a larger parallelogram.

Here are a preimage and an paradigm. What two transformations were carried out on it?

$Combined Transformations Math$

The preimage has been rotated and dilated (shrunk) to make the image.

Transformations in the Coordinate Plane

On a coordinate filigree, you tin utilise the x-axis and y-axis to measure every move. The lines also help with drawing the polygons and flat figures. Focus on the coordinates of the figure's vertices and so connect them to form the image.

Here is a alpine, blue rectangle drawn in Quadrant $Iii$ .

$Translation Transformation Example$

We are asked to translate it to new coordinates. Mathematically, the graphing instructions await similar this:

$(x, y) \to (10 + ix, y + five)$

This tells us to add $9$ to every $10$ value (moving information technology to the correct) and add $9$ to every $Y$ value (moving it upwards):

$(- seven, - 1) \to (- 7 + 9, - 1 + 5) \to (two, 4)$

Do the same mathematics for each vertex and then connect the new points in Quadrants $I$ and $Iv$ .

Rotation using the coordinate filigree is similarly easy using the 10-axis and y-centrality:

To rotate $90 °$ : $(x, y) \to (- y, ten)$ (multiply the y-value times $- ane$ and switch the x- and y-values)
To rotate $180 °$ : $(x, y) \to (- x, - y)$ (multiply both the y-value and x-value times $- ane$ )
To rotate $270 °$ : $(x, y) \to (y, - 10)$ (multiply the x-value times $- ane$ and switch the 10- and y-values)

Reflecting a polygon beyond a line of reflection means counting the distance of each vertex to the line, and so counting that same distance away from the line in the other direction. If the effigy has a vertex at $(- 5, 4)$ and yous are using the y-centrality as the line of reflection, and so the reflected vertex will exist at $(v, 4)$ .

Shearing a figure means fixing one line of the polygon and moving all the other points and lines in a particular direction, in proportion to their distance from the given, fixed-line. Italic letters on a computer are examples of shear.

Mathematically, a shear looks like this, where one thousand is the shear gene y'all wish to apply:

$(x, y) \to (x + one thousand y, y)$ to shear $h o r i z o n t a l l y$
$(x, y) \to (x, y + m x)$ to shear $v e r t i c a fifty l y$

Dilating a polygon ways repeating the original angles of a polygon and multiplying or dividing every side past a calibration factor. If you have an isosceles triangle preimage with legs of $ix f e e t$ , and you employ a scale factor of $\frac{2}{3}$ , the image will have legs of $6 f eastward east t$ .

In summary, a geometric transformation is how a shape moves on a plane or grid. Transformations, and in that location are rules that transformations follow in coordinate geometry.

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Rotational Symmetry