Geometry Transformation

In simple terms, transformation is: A displacement or change. In mathematics, transformation is used to move a point or shape on a plane. Meanwhile, geometric transformation is a part of geometry that discusses changes (location, shape, presentation) based on images and matrices.

The transformation in the plane consists of 4 types :

• Shift (Translation)
• Reflection (Reflection)
• Rotation (Rotation)
• Multiplication (Dilation)

Shift Geometry Transformation (Translation)

     Translation is a transformation that moves every point on a plane according to a certain distance and direction. The distance and direction of a translation can be denoted by directed lines for example or vectors (Herynugroho, et al, 2009: 184)

     So that the explanation above is easier to understand, please pay attention to the following simulation:

     Also pay attention to the following simulation:

Translation Point

If Translation maps point A (x, y) to point A '(x', y ') then the relationship applies:

x '= x + a

y '= y + b

or A '(x + a, y + b)

This relationship can be written as:

Pay attention to the following simulation:

 Problems example :

Find the image P (2, 3) by Translation

Answer:

x '= x + a = 2 + 4 = 6

y '= y + b = 3 + 3 = 6

Thus, the image of P (2, 3) by translation is P '(6, 6)

Rotation (Rotation) Geometry Transformation

     Rotation is the process of rotating a geometric shape to a certain point which is called the center of rotation and is determined by the direction of rotation and the angle of rotation.

     The center of rotation is a fixed point or center point that is used as a reference to determine the direction and angle of rotation.

     The direction of rotation is agreed upon by the following rules:

If the rotation is counterclockwise, then this rotation is positive (+).

If the rotation is clockwise, then this rotation is negative (-).

     The amount of rotation angle of rotation determines the distance of rotation. The rotation distance is expressed in terms of the fractional plane with respect to a full rotation (360o) or the angle in terms of degrees or radians (Sartono, 2006: 185-186).

     To make it easy to understand, please pay attention to the following simulation:

     Also consider the following simulation:

     If P (a, b) is rotated by a with the center of rotation at A (x, y), the image that occurs is as follows.

 

Problems example:

• Find the image P (3, -5) if it is rotated 90o with the center of rotation at A (1,2) completed with the picture!

Answer:

P (3, -5) = P (a, b)

A (1, 2) = A (x, y)

a '= (a - x) cos a - (b - y) sin a + x

b '= (a - x) sin a + (b - y) cos a + y

a '= (3 - 1) cos 90o - (-5 - 2) sin 90o + 1 = 0 + 7 + 1 = 8

b '= (3 - 1) sin 90o - (-5 - 2) cos 90o + 2 = 2 + 0 + 2 = 4

So, the image P (3, 5) is P '(8, 4)

Multiplication Geometry Transformation (Dilation)

     Dilation is a transformation that changes the size or scale of a geometric shape (enlargement / reduction), but does not change the shape of the shape.

     Dilation on a flat plane is determined by the following.

Dilation center

Dilation factor

     There are two centers of dilation, namely at point O (0,0) and at point A (x, y). Meanwhile, the dilation factor can be positive (magnification is unidirectional) and can also have negative consequences (magnification in opposite directions). The dilation factor is also called the scale factor (Herynugroho, 2009: 190)

     To make it easy to understand, please pay attention to the following simulation:

If P (a, b) is dilated by a scale factor k, the center of the dilation is at A (x, y), then the image is as follows.

Problems example:

Determine the image of the dilated triangle ABC with the scale factor and center of dilation R (1,2). It is known that the coordinates of points A, B, and C are (4,9), (8,8), and (7,4)!

Answer:

So, the shadow is A'B'C 'with,, and

Shift Geometry Transformation (Translation)

      Translation is a transformation that moves every point on a plane according to a certain distance and direction. The distance and direction of a translation can be denoted by directed lines for example or vectors (Herynugroho, et al, 2009: 184)

      So that the explanation above is easier to understand, please pay attention to the following simulation:

      Also pay attention to the following simulation:

Translation Point

If Translation maps point A (x, y) to point A '(x', y ') then the relationship applies:

x '= x + a

y '= y + b

or A '(x + a, y + b)

This relationship can be written as:

Pay attention to the following simulation:

  Problems example :

Find the image P (2, 3) by Translation

Answer:

x '= x + a = 2 + 4 = 6

y '= y + b = 3 + 3 = 6

Thus, the image of P (2, 3) by translation is P '(6, 6)