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Matrices and Transformations II

REFLECTION


Under reflection, the object and its image have the same size and shape. The object and its image are equidistant from the mirror line. In addition, a straight line that joins a point on the object and its image is perpendicular to the mirror line. This is shown in the following diagram.

REFLECTION


 

Under reflection, the object and its image have the same size and shape. The object and its image are equidistant from the mirror line. In addition, a straight line that joins a point on the object and its image is perpendicular to the mirror line. This is shown in the following diagram.

M is the mirror line
QQ' is perpendicular to line M
QN = NQ'
Triangle PQR and P'Q'R' have the same shape and size.

ROTATION

Under rotation the object and the image have the same size and shape. The process of rotation involves identification of the centre, angle and direction of rotation.In the above diagram triangle P'Q'R' is the image of triangle PQR under rotation of 95o about O in a clockwise direction. PQR is said to have been rotated through-95o about O.
If triangle P'Q'R' is rotated in an anticlockwise direction to get triangle PQR, then it is said to have been rotated through 95
o about O.
Note that the size and shape of triangles PQR and P'Q'R' are the same.

ENLARGEMENT


Under enlargement the size of an object, increases or decreases in a given ratio. When an object is enlarged:


 
  • Sides of the object are parallel to their corresponding sides on the image
     
  • Lengths of the sides of an object and lengths of their images are in the same ratio.
     
  • Angles do not change

In the illustration below, line AB is parallel to A'B'.



 




 

Angle ADC = Angle A'D'C'
Angle ABC = Angle A'B'C'
Angle BAD = Angle B'A'D'
Angle BCD = Angle B'C'D'

 

LINEAR SCALE FACTOR OF ENLARGEMENT


 

When the linear scale factor is greater than positive one (+ve 1), the image is larger and on the same side of the centre of enlargement as the object. When the linear scale factor is less than +ve 1, but greater than 0 the image is smaller and on the same side of the centre of enlargement as the object.
When the scale factor of enlargement is negative, the centre of enlargement lies between the object and the image. In this case, the image is inverted as shown below

 

Triangle ABC is enlarged using linear scale factor of -2 centre of enlargement O.

 

 

MATRICES AND TRANSFORMATIONS II
Introduction
Life is full of changes both the living and the non living change. Transformation involves changing something in some way. In Mathematics the knowledge of transformation helps us to understand how points and objects change in terms of positions, shape, size and direction.

Transformations

Transformation on a Cartesian plane
The following graphs show various types of transformations. Describe fully the transformation that maps:
Solutions

1.Reflection in the line y = 0
2.Enlargement scale factor positive 2 centre (0,0)
3.Rotation positive 90
O about the origin (0,0)
4.Translation using translation vector (-7)

 

 

Determining the matrix of transformation.
We can obtain a matrix of transformation using the coordinates of the object and its image.
Example 1
A square with vertices O(0,0), A (1,0), B(1,1) and C(0,1) is mapped onto the image O'(0,0), A' (-1,0), B'(-1,1) and C'(0,1). Determine the transformation matrix that maps OABC onto O'A'B'C'.


 

Solution


 


Example 2

Plot the coordinates of the image on the graph below by clicking on the vertices of the image.
 

The transformation is a reflection in the line x = 0

Example 3
a) Use the graph below to determine the matrix that maps:
i) PQR onto P'Q'R'
ii) P'Q'R'onto P''Q''R''
iii) P''Q''R'' onto P'''Q'''R'''
b) Determine a single matrix of transformation that maps PQR onto P''Q''R''


 


 


 


 



 

b) A single matrix of transformation that maps PQR and P''Q''R''can be obtained in two ways, namely:
i) multiplying the matrix that maps PQR onto P'Q'R' by the matrix that maps P'Q'R' onto P''Q''R''.
Let the matrix that maps PQR onto P'Q'R' be A and the one that maps P'Q'R' onto P''Q''R'' be B.

The single matrix of transformation is

 

Matrix and equate to the final image matrix and work out unknowns a, b, c and d. That is


So that a = 2, b = 0, c = 0 and d = -2

 

Relationship between area scale factor and determinant of a matrix
Under the transformation of enlargement, the ratio of the sides of the image and the corresponding sides of the object is constant. This is referred to as the linear scale factor. The square of the linear scale factor gives the area scale factor. The determinant of the matrix of transformation is equal to the area scale factor.

NB, the area scale factor can also be obtained by dividing the area of the image by the area of the object.

Example
Use the following graph to determine:

i)The area of triangle KLM
ii)The area of triangle K'L'M'
iii)Area scale factor.
iv)The matrix P that maps triangle KLM onto K'L'M'.

V)The determinant of matrix P
Vi) State the relationship between area and scale factor and the determinant of P.


 

STRETCH
Stretch is a transformation in which all the points of an object move at right angles to an invariant (i.e. fixed) line. The distance moved by a given point from the fixed line is proportional to its original distance from the line
Example
Use the following graph to;

 

I.Identify the fixed (invariant) line
II.Determine the matrix p, that maps ABCD onto A'B'C'D'
III.Describe the transformation.

Solution
i. The invariant line is y-axis (x=0)

Note: the matrix of the vertices that moves is pre-multiplied by the transformation matrix and the product equated to the matrix of the corresponding image.

The points on the invariant line do not move and therefore should not be used when working out the transformation matrix.

 

SHEAR
A shear is a transformation in which all points of an object move parallel to a fixed or invariant line. The distance moved by a given point is proportional to the distance from the invariant line.

Example
Use the following graph to

i.Identify the fixed (invariant) line
ii.Determine the matrix Q that maps ABCD onto A'B'C'D'
iii.Describe the transformation that maps ABCD onto A'B'C'D'

SOLUTION
I.The invariant line is x axis (y=0)

Note: the matrix of the vertices that moves is pre-multiplied by the transformation matrix and the product equated to the matrix of the corresponding image.

 

.

 

 



 Matrices and Transformations II 

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