TRANSFORMATIONS OF 2D SHAPES

There are three kinds of isometric transformations of 2D shapes.

i) Translations

ii) Rotations

iii)  Reflections

Translation of 2D Shapes

A translation is transformation in which every point of figure moves a fixed distance in a given direction.

  • Number of units of horizontal movements can be considered as "x".
  • Number of units of vertical movements can be considered as "y".
  • Translation vector will be in the form (x, y)
  • If x is positive, then we have to move x units to the right
  • If x is negative, then we have to move x units to the left .
  • If y is positive, then we have to move y units up ↑ .
  • If y is negative, then we have to move y units down ↓.

Related Pages

Rotation of 2D Shapes

Rotating the shape means moving them around a fixed point. There are two directions

i) Clockwise 

ii) Counter clockwise (or) Anti clockwise

The shape itself stays exactly the same, but its position in the space will change.

90° clockwise 

90° counter clockwise

180° 

270° clockwise

270° counter clockwise

(x, y) ==> (y, -x)

(x, y) ==> (-y, x)

(x, y) ==> (-x, -y)

(x, y) ==> (-y, x)

(x, y) ==> (y, -x)

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Reflection of 2D Shapes

The reflected image will be congruent to the original.

What is line of reflection ?

A reflection maps every point of a figure to an image across a fixed line. The fixed line is called the line of reflection.

Reflections can be performed easily in the coordinate plane using the general rules below.

Types of reflections

Reflection over x axis

Reflection over y axis

Reflection over y = x

Reflection over y = -x

Reflection about origin

Rules

(x, y) ==> (x, -y)

(x, y) ==> (-x, y)

(x, y) ==> (y, x)

(x, y) ==> (-y, -x)

(x, y) ==> (-x, -y).

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The object A has been translated to give an image B in each diagram. Give the translation in each case.

Problem 1 :

translationofgeofigq1

Solution :

Observing the vertex of A to vertex of rectangle B, only horizontal movement is done.

translationofgeofigq1s.png

Moved 4 units right, so the required translation vector is (4, 0).

Problem 2 :

Translate the shape 2 units right and 1 unit up.

translationof2dshapeq1.png

Solution :

Writing the vertices,

U (-3, 0), Z (-2, 3), F (1, 3) and H (1, 1)

We translate the figure 2 units right and 1 unit up. So, 

(x, y) ===> (x + 2, y + 1)

U (-3, 0) ==> U' (-3+2, 0+1) ==> U' (-1, 1)

Z (-2, 3) ==> Z' (-2+2, 3+1) ==> Z' (0, 4)

F (1, 3) ==> Z' (1+2, 3+1) ==> F' (3, 4)

H (1, 1) ==> H' (1+2, 1+1) ==> Z' (3, 2)

translationof2dshapeq1s.png

Problem 3 :

Rotation 90° clockwise about the origin.

rotationof2dshapeq1.png

Solution :

Marking the coordinate,

Z (0, -4), J (0, -5), T (4, -3) and S (3, -5)

Rotation ==> 90° clockwise

Rule :

(x, y) ==> (y, -x)

Z (0, -4) ==> Z' (-4, 0)

J (0, -5) ==> J' (-5, 0)

T (4, -3) ==> T' (-3, -4)

S (3, -5) ==> S'(-5, -3)

rotationof2dshapeq1s.png

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