Further Pure 1

Transformations

Transformations

2 × 2 matrices can be used to describe transformations in a 2-d plane.

Before we look at this we are going to look at particular transformations in the 2D plane.

A transformation is a rule which moves points about on a plane.

Every transformation can be described as a multiple of x plus a multiple of y.

Transformations

Lets look at a point A(-2,3) and map it to the co-ordinate (2x+3y,3x-y)

This gives us the co-ordinate

(2×-2 + 3×3, 3×-2–3)

=(5,-9) Where would the

co-ordinate (2,1) map to?

(-2,3)

(5,-9)

(2,1)

(7,5)

Transformations

Take the transformation reflecting an object in the y-axis. The black rectangle is the object and the orange one is the image. What has happened to the co-ordinates in the reflection? Lets look at one specific co-ordinate, (2,1).

Under the reflection the co-ordinate becomes (-2,1)

You can probably notice that there is a general rule for all the co-ordinates.

For each co-ordinate the x becomes negative and the y stays the same.

Lets use the general co-ordinate (x,y) and let them map to (x`,y`).

(2,1)(-2,1)

Reflection in y-axis We can see that x -x & y y. Or x` = -x

y` = y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = -1x + 0yy` = 0x + 1y

Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1)(-2,1)

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Reflection in x-axis We can see that x x & y -y. Or x` = x

y` = -y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 1x + 0yy` = 0x + -1y

Finally we can summarise the equations co-efficient’s by using matrix notation.

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(2,1)

(2,-1)

Reflection in y = x We can see that x y & y x. Or x` = y

y` = x So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 0x + 1yy` = 1x + 0y

Finally we can summarise the equations co-efficient’s by using matrix notation.

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10(2,1)

(1,2)

Reflection in y = -x We can see that x -x & y y. Or x` = -y

y` = -x So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 0x + -1yy` = -1x + 0y

Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1)

(-1,-2)

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1-0

Enlargement SF 2, centre (0,0)

We can see that x 2x & y 2y. Or x` = 2x

y` = 2y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 2x + 0yy` = 0x + 2y

Finally we can summarise the equations co-efficient’s by using matrix notation.

(1,2)

(2,4)

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Two way stretch We can see that x 2x & y 3y. Or x` = 2x

y` = 3y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 2x + 0yy` = 0x + 3y

Finally we can summarise the equations co-efficient’s by using matrix notation.

This is a stretch factor 2 for x and factor 3 for y.

(2,1)

(4,3)

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Enlargements

Enlargement

SF k

Two way stretch Factor a for x Factor b for y

b0

0a

k0

0k

Rotation 90o anti-clockwise We can see that x -y & y x. Or x` = -y

y` = x So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 0x – 1yy` = 1x + 0y

Finally we can summarise the equations co-efficient’s by using matrix notation.

(4,2)

(-2,4)

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1-0

Rotation 90o clockwise We can see that x y & y -x. Or x` = y

y` = -x So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 0x + 1yy` = -1x + 0y

Finally we can summarise the equations co-efficient’s by using matrix notation.

(4,2)

(2,-4)

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Rotation 180o

We can see that x -x & y -y. Or x` = -x

y` = -y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = -1x + 0yy` = 0x – 1y

Finally we can summarise the equations co-efficient’s by using matrix notation.

(4,2)

(-4,-2)

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We are going to think about this example in a slightly different way. The diagram shows the points I(1,0) and J(0,1) and there images after a rotation through θ

anti-clockwise. You can see OI = OJ = OI` = OJ` From the diagram we can see that

cos θ = a/1 a = cos θ sin θ = b/1 b = sin θ

Therefore I` is (cos θ, sin θ) andJ` is (-sin θ, cos θ)

The transformation matrix is

Rotation through θ anti-clockwise.

cosθsinθ

sinθ-cosθ

I(1,0)

J(0,1) I`(a,b)J`(-b,a)

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a

ba

b

Rotation through θ clockwise.

What would be the matrix for a 90o rotation clockwise.

cosθsinθ-

sinθcosθ

Transformations - Shears

For the next example you need to understand the concept of a shear.

Here is an example of a shear parallel to the x-axis factor 2. Each point moves parallel

to the x-axis. Each point is moved twice

its distance from the x-axis. Points above the x-axis

move right. Points below the x-axis

move left. You can see that the point (2,1)

moves to (2 + 2 × 1,1) = (4,1) A shear parallel to the y-axis

factor 3 would move every point 3 times its distance from y parallel to the y-axis.

Shear parallel to x-axis factor 2

We can see that x x + 2y & y y. Or x` = x + 2y

y` = y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 1x + 2yy` = 0x + 1y

Finally we can summarise the equations co-efficient’s by using matrix notation.

(2,1) (4,1)

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Shear parallel to y-axis factor 2

We can see that x x & y y + 2x . Or x` = x

y` = 2x + y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 1x + 0yy` = 2x + 1y

Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1)

(2,5)

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Two way shear factor 2 We can see that x x + 2y & y y + 2x. Or x` = x + 2y

y` = 2x + y So we can now write these equations as a pair of

simultaneous equations as multiples of x and y.x` = 1x + 2yy` = 2x + 1y

Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1)

(4,5)

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Using multiplication with transformations

Lets go back to the first transformation that we looked at. We know that the matrix for reflecting in the y-axis is

Now lets write down the co-ordinates of the object as a matrix.

What happens if we multiply the two matrices together.

The multiplication performs the transformation and the new matrix is the co-ordinates of the image.

3311

1221

3311

1221

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Rotation 180o

What happens if you rotate 90o cw, twice.

What happens if you reflect in x then in y.

You actually get the same transformation as rotating through 180o.

This leads us nicely in to multiple transformations.

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Composition of transformations

Notation: A single bold italic letter such as T is often used to

represent a transformation. A bold upright T is used to represent a matrix

itself. If you have a point P with position vector p The image of p can be denoted

P` = p` = T(P) If you transform p by a transformation X then by

a transformation Y the result would be:Y(X(p)) = YX(p)