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Transcript
Further Pure 1 Lesson 2 – Transformations Transformations Wiltshire 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 Wiltshire 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) Transformations Wiltshire 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 coordinate becomes (-2,1) You can probably notice that there is a general rule for all the co-ordinates. (-2,1) (2,1) 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`). Reflection in y-axis We can see that x Or x` = -x -x & y Wiltshire y. 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) 1 0 0 1 (2,1) Reflection in x-axis We can see that x Or x` = x x&y Wiltshire -y. 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. 1 0 0 - 1 (2,1) (2,-1) Reflection in y = x We can see that x Or x` = y y&y Wiltshire x. 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 (1,2) equations co-efficient’s by using matrix notation. 0 1 1 0 (2,1) Reflection in y = -x We can see that x Or x` = -y -x & y Wiltshire 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. 0 - 1 -1 0 (-1,-2) (2,1) Enlargement SF 2, centre (0,0) Wiltshire We can see that x Or x` = 2x 2x & y 2y. 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 Finally we can summarise the equations co-efficient’s by using matrix notation. 2 0 0 2 (2,4) (1,2) Two way stretch We can see that x Or x` = 2x 2x & y Wiltshire 3y. 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 Finally we can summarise the equations co-efficient’s by using matrix notation. 2 0 0 3 This is a stretch factor 2 for x and factor 3 for y. (4,3) (2,1) Enlargements Enlargement SF k k 0 0 k Wiltshire Two way stretch Factor a for x Factor b for y a 0 0 b Rotation 90o anti-clockwise We can see that x Or x` = -y -y & y x. y` = x So we can now write these equations as a pair of simultaneous equations as multiples of x and y. x` = 0x – 1y (-2,4) y` = 1x + 0y Finally we can summarise the equations co-efficient’s by using matrix notation. 0 - 1 1 0 Wiltshire (4,2) Rotation 90o clockwise We can see that x Or x` = y y&y -x. 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. 0 1 - 1 0 Wiltshire (4,2) (2,-4) Rotation 180o We can see that x Or x` = -x Wiltshire -x & y -y. 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. -1 0 0 - 1 (-4,-2) (4,2) Rotation through θ anti-clockwise. Wiltshire 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 b J(0,1) I`(a,b) cos θ = a/1 a = cos θ J`(-b,a) sin θ = b/1 b = sin θ a 1 1 Therefore I` is (cos θ, sin θ) and b J` is (-sin θ, cos θ) The transformation matrix is a I(1,0) cosθ - sinθ sinθ cosθ Rotation through θ clockwise. Wiltshire What would be the matrix for a 90o rotation clockwise. cosθ sinθ - sinθ cosθ Transformations - Shears Wiltshire 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 Wiltshire We can see that x Or x` = x + 2y x + 2y & y y. 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. 1 2 0 1 (2,1) (4,1) Shear parallel to y-axis factor 2 Wiltshire We can see that x Or x` = x x&y y + 2x . 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 Finally we can summarise the equations co-efficient’s by using matrix notation. 1 0 2 1 (2,5) (2,1) Two way shear factor 2 We can see that x Or x` = x + 2y x + 2y & y Wiltshire y + 2x. 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 Finally we can summarise the equations co-efficient’s by using matrix notation. 1 2 2 1 (4,5) (2,1) Using multiplication with transformations Wiltshire Lets go back to the first transformation that we looked at. We know that the matrix for reflecting in the y-axis is 1 0 1 2 2 1 1 2 2 1 1 3 3 0 1 1 1 3 3 1 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. Rotation 180o Wiltshire What happens if you rotate 90o cw, twice. 0 1 0 1 1 0 1 0 1 0 0 1 What happens if you reflect in x then in y. 1 0 1 0 1 0 0 1 0 1 0 1 You actually get the same transformation as rotating through 180o. 1 0 0 1 This leads us nicely in to multiple transformations. Composition of transformations Wiltshire 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)