There are five types of transformation that we will be dealing with in this section:

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Vivien Cooper
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1 Further oncepts for dvanced Mathematics - FP Unit Matrices Sectionb Transformations Transformations There are five types of transformation that we will be dealing with in this section: Reflection Rotation Enlargement Two-way stretch Shear You are probably familiar with the first from G..S.E. Reflection (i) In the line x = 0 0 (,) (-,) (,) (-,) (,) (-,) (,0) (-,0) You can see that each time the x coordinate is being multiplied by -. If we let the new x coordinate be called then we can say that = x + 0y The new y coordinate remains unchanged so we can say that it is simply the old y value or = 0 x + y You may wonder about the inclusion of the other value in each formula. This will be explained later (see ). The two equations that allow us to calculate the new coordinates are = x + 0y = 0 x + y 0 These can be written as the matrix where the top line corresponds to the first 0 ( ) equation the bottom line corresponds to the second ( y ) equation.

2 (ii) In the line y = 0 0 (,) (,-) (,) (,-) (,) (,-) (,0) (,0) You can see that each time the x coordinate is being multiplied by the y coordinate by -. So = x + 0y = 0x y 0 (iii) In the line y = x 0 (,) (,) (,) (,) (,) (,) (,0) (0,) is the old y coordinate. The new y coordinate is the old x coordinate. So = 0 x + y = x + 0y 0 0 You are probably beginning to realise that the transformations in this section can all be written using equations matrices like the ones you ve just seen for reflections. Transformations that can be written in this way are called linear transformations. Linear transformations can be expressed as a pair of equations of the form = ax + cy = bx + dy or as a matrix of the coefficients like this a b c d It is worth noting that reflections can only be described in this way if the reflection line passes through 0, rotations must have the centre of rotation at 0 enlargements must have the centre of enlargement at 0. See if you can think why this is. Try some examples for yourself.

3 Rotation (i) lockwise through 90º with centre 0 0 (,) (,-) (,) (,-) (,) (,-) (,0) (0,-) is the old y coordinate. The new y coordinate is the old x coordinate. So = 0 x + y = x + 0y 0 0 (ii) nticlockwise through θ with centre 0 In order to show how this works, two new (to you) trigonometric identities will have to be used. These are: cos( + ) = cos cos sin sin sin( + ) = sin cos + cos sin To find our transformation matrix, we will only consider one point; the general point ( x, y). To make things as clear as possible, we will only work in the first quadrant. (x, y ) The angle that the line joining the original point ( x, y) to 0 makes with the x axis is called α. The length of the line is r. Using the original coordinates some simple trigonometry, 0 r θ α r (x, y) x = r cosα y = r sinα The angle that of rotation is θ so the angle that the line joining ( x, ) to 0 makes with the horizontal is (θ + α). Using the new coordinates the same simple trigonometry = r cos( α + θ ) = r sin( α + θ ) but using our two identities, = r cosα r sinα sinθ = r sinα + r cosα sinθ ut we know from our first equations that x = r cosα y = r sinα so we can substitute these in to find in terms of x y.

4 So = x y sinθ which is rewritten as = y + xsinθ = x y sinθ. = xsinθ + y This gives the matrix for an anticlockwise rotation through θ with centre 0 as sinθ sinθ. You can learn this result simply use it as when necessary. an you find a general matrix for a clockwise rotation through θ with centre 0? Enlargement Scale factor, centre 0 0 (,) (,) (,) (,) (,) (,) (,0) (,0) is the old x coordinate the new y coordinate is the old y coordinate. So = x + 0y = 0 x + y 0 What would happen for a negative scale factor? What about a fractional scale factor? Two way stretch Scale factor horizontally, ½ vertically 0 (,) (,½) (,) (,½) (,) (,) (,0) (,0) is the old x coordinate the new y coordinate is the old y coordinate. So = x + 0y = 0 x + y y

5 In matrix form: Shear 0 Example Each point is moved parallel to the x axis Each point has its distance from the x axis multiplied by Points above the x axis are moved to the left Points below the x axis are moved to the right (0,) (-,) (,) (-,) (,) (-,) (,-) (,-) 0 is the old x coordinate minus the y coordinate the y coordinate is unchanged. So = x y y = 0 x + y

Designing Information Devices and Systems I Discussion 2A

EECS 16A Spring 218 Designing Information Devices and Systems I Discussion 2A 1. Visualizing Matrices as Operations This problem is going to help you visualize matrices as operations. For example, when