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Further Pure 1 Transformations

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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.

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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)

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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)

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

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

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

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

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Enlargement SF 2, centre (0,0) We can see that x 2x & y 2y. Orx` = 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 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. Orx` = 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 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

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

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

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Rotation 180 o We can see that x -x & y -y. Orx` = -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)

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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) and J` is (-sin θ, cos θ) The transformation matrix is Rotation through θ anti-clockwise. I(1,0) J(0,1) I`(a,b) J`(-b,a) 11 a b a b

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

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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.

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Shear parallel to x-axis factor 2 We can see that x x + 2y & y y. Orx` = 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)

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Shear parallel to y-axis factor 2 We can see that x x & y y + 2x. Orx` = 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 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. Orx` = 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 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.

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Rotation 180 o What happens if you rotate 90 o cw, twice. What happens if you reflect in x then in y. You actually get the same transformation as rotating through 180 o. 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)

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Part7: Geometric Transformations

Part7: Geometric Transformations

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