Further Pure 1

1. Further Pure 1 Transformations

2. 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.

3. 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? (7,5) (-2,3) (2,1) (5,-9)

4. 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`). Transformations (-2,1) (2,1)

5. 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 + 0y y` = 0x + 1y Finally we can summarise the equations co-efficient’s by using matrix notation. Reflection in y-axis (-2,1) (2,1)

6. 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 + 0y y` = 0x + -1y • Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1) (2,-1)

7. 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 + 1y y` = 1x + 0y • Finally we can summarise the equations co-efficient’s by using matrix notation. (1,2) (2,1)

8. 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 + -1y y` = -1x + 0y • Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1) (-1,-2)

9. 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 + 0y y` = 0x + 2y (2,4) • Finally we can summarise the equations co-efficient’s by using matrix notation. (1,2)

10. 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 + 0y y` = 0x + 3y (4,3) • 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)

11. Enlargement SF k Two way stretch Factor a for x Factor b for y Enlargements

12. 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 – 1y y` = 1x + 0y (-2,4) • Finally we can summarise the equations co-efficient’s by using matrix notation. (4,2)

13. 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 + 1y y` = -1x + 0y • Finally we can summarise the equations co-efficient’s by using matrix notation. (4,2) (2,-4)

14. 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 + 0y y` = 0x – 1y • Finally we can summarise the equations co-efficient’s by using matrix notation. (4,2) (-4,-2)

15. Rotation through θ anti-clockwise. • 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) and J` is (-sin θ, cos θ) • The transformation matrix is b J(0,1) I`(a,b) J`(-b,a) a 1 1 b a I(1,0)

16. Rotation through θ clockwise. • What would be the matrix for a 90o rotation clockwise.

17. 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. Transformations - Shears

18. 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 + 2y y` = 0x + 1y • Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1) (4,1)

19. 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 + 0y y` = 2x + 1y (2,5) • Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1)

20. 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 + 2y y` = 2x + 1y (4,5) • Finally we can summarise the equations co-efficient’s by using matrix notation. (2,1)

21. 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. Using multiplication with transformations

22. 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.

23. Composition of transformations • Notation: • A single bold italic letter such as T is often used to represent a transformation. • A bold upright T isused 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)