Transformations using matrices

Transcription

1 Transformations using matrices 6 sllabusref eferenceence Core topic: Matrices and applications In this cha 6A 6B 6C 6D 6E 6F 6G chapter Geometric transformations and matri algebra Linear transformations Linear transformations and group theor Rotations Reflections Dilations Shears

2 Maths Quest Maths C Year for Queensland Geometric transformations and matri algebra In our junior mathematics studies ou encountered the idea of translation, reflection, rotation and dilation and how these transformations changed the position, size and orientation of the original figure. However, at that stage our investigations were limited to identifing the tpe of transformation that had taken place, the position of the mirror line or centre of rotation, and perhaps the size of the image figure. However, now ou have skills with matrices that will allow much greater detailed eplanation of the position of images or conversel, the transformation necessar to map point (, ) onto point (z, w). The matri algebra used is ver straightforward and because we are limiting our discussion at this stage to -dimensional space, most of our matrices will be of order. Throughout this section ou will be reminded of the properties of groups and how transformations involved in matri algebra can be considered to be a group. Transformations A transformation t is an operation which maps each point of the Cartesian plane onto some other point on the plane. Consider point P(, ). Under a transformation t this point is mapped onto P (, ). The point P(, ) is referred to as the original or pre-image point and P (, ) is the image. This transformation can be written in its most general form as (, ) t(, ). P(, ) t P'(', ') WORKED Eample Find the coordinates of the image points of A(, ) and B(, ) under the transformation defined b the equations: Think of and as functions of and. Substitute and into equations for and. For A(, ) () ()( ) + ( ) ()( ) + () ( ) 6 + +

3 Chapter 6 Transformations using matrices Write the coordinates of the transformed image. The smbol is used to denote maps onto. A(, ) A (9, ) Substitute and for B. For B(, ) () ()() + () ()() + () () Write the coordinates of the transformed image. B(, ) B (, 9) Sketch each original point and its image. Notice that the transformation of A seems quite unconnected with the transformation of B. B'(, 9) A'(9,) t t A(, ) B(, ) Translations The equations used in the previous eample define a general transformation or mapping. A translation is a specific transformation that involves a shift of each point in the same direction. + a + b Each -coordinate is moved a units parallel to the -ais and each -coordinate is moved b units parallel to the -ais. P(, ) t a P'(', ') b The image of P is written P and this translation can be epressed in matri equation form as a + where b. is the vector holding the image coordinates (, ) of point P

4 Maths Quest Maths C Year for Queensland. represents the original coordinates (, ) of point P a. is the translation vector and represents information about the horizontal b and vertical displacement. Note t (lower case) denotes the translation itself and T (upper case) denotes the matri of the translation. Therefore (, ) t(, ) can be written in matri form as + T + a b + a + b WORKED Eample Find the image of triangle PQR with vertices P(, -), Q(, ) and R(-, -) under the translation vector. Sketch the original and image figures. State the general translation matri equation. + a b First sketch the original points, with coordinates, as shown below. Substitute - and -values for each point in turn. For P(, )

5 Chapter 6 Transformations using matrices P(, ) maps to P (7, ). P(, ) P (7, ) For Q(, ) Q(, ) maps to Q (, ). Q(, ) Q (, ) For R(, ) R(, ) maps to R (, ). R(, ) R (, ) Sketch the image points with the original. Q(, ) Q'(, ) R(, ) P(, ) R'(, ) P'(7, ) Note that the image has been moved units to the right and unit down but remains unchanged in shape, area, size and orientation. Such a transformation is said to be congruent. Successive translations The translation above could have been achieved b a succession of translations that have the final effect of across and down. An number of successive translations could achieve this: and or the reverse order, and, and so on.

6 Maths Quest Maths C Year for Queensland Show that the translation T followed b T maps the point P(, -) from the previous eample to the same point P (7, -) as found in worked eample, and that the order of the translation has no effect on the result. Set up the general matri equation. + T Use T followed b T. For P(, ) WORKED Eample + P(, ) P (, ) is the image, P of image P. + T + Use T followed b T. + 7 P (, ) P (7, ) Therefore P(, ) P (, ) P (7, ) P(, ) P (, 6) P (, 6) P (7, ) Therefore P(, ) P (, 6) P (7, )

7 Chapter 6 Transformations using matrices Sketch the translated image in stages. P(, ) P''(7, ) t t P'(, 6) This eample shows, but does not prove that a set of translations is commutative, that is the order of operation does not affect the final result. Translation of a curve WORKED Eample Find the equation of the curve under the translation of T. Set up the general matri equation. + To find the image of the curve we must epress and as found in the original function in terms of and. Substitute for and in the original function to obtain the function in terms of the image coordinates. + + ( ) Rearrange and epand this function. + Factorise to obtain the zeros for the function. -ais intercepts occur when ( )( + ) -ais intercepts occur when or + Find the -ais intercepts. -ais intercepts occur when () Continued over page

8 6 Maths Quest Maths C Year for Queensland Sketch the original and image functions. Note that the turning point (, ) maps to (, ) which was the translation vector. 6 (, ) ' ' ' (, ) remember remember. A translation T can be written as + T in matri equation form. The matri is the vector representing the coordinates of the point (, ) and represents the coordinates of the point (, ) the image of (, ) after translation.. A translation results in an image congruent to the original object.. A set of translations is commutative the order of operation does not affect the final result. 6A Geometric transformations and matri algebra WORKED Eample WORKED Eample Find the image of each of the following points under the transformation defined b + + a (, ) b (, ) c (, ) d (, ) Sketch the original point and its image. Find the image of each of the following points under the translation T. a (, ) b (, ) c (, ) d (, ) The vertices of a triangle are given b A(, ), B(, ) and C(7, ). Find the image of the vertices under each of the following translations: a b c d

9 Chapter 6 Transformations using matrices 7 WORKED Eample WORKED Eample The line + undergoes a succession of translations defined b T and T the result.. Show that the order in which these translations take place has no effect on The line + undergoes a translation defined b T. Find the equation of the image curve and sketch the original curve and its image. 6 Find the equation of the image of each of the following curves under the following translations. Graph the original curve and its image using a graphics calculator. a b c 6 d + e Use similar methods to those in question 6 to find the translation vector that maps each of the pairs of points: a (, ) (, ) b (, ) (, ) c (6, ) (, ) Linear transformations Have ou ever wondered how programmers who develop computer games move and manoeuvre characters on a screen to get them to spin or shrink as the appear to move further awa from the observer? The stud of linear transformation forms the foundation for these changes of form and size the warping of the plane on which the characters are mapped. There are man different was in which the original, or pre-image, can be changed or moved so that it looks different, or is in a different place. A linear transformation l is a mapping of the pre-image P(, ) onto the image P (, ) where: a + b c + d for all real values of a, b, c, and d. In matri form this sstem is written as: a c b d L where L a b and is called the transformation matri. c d

10 8 Maths Quest Maths C Year for Queensland WORKED Eample Find the images of the vertices of a unit square ABCD under the transformation given b L. Set up the initial matri equation where the image of P is given as P. Investigate the transformation of each For point D(, ) point in turn. (Recall that the smbol is used to denote maps onto.) That is, D(, ) D (, ) For point A(, ) A(, ) D(, ) B(, ) C (, ) That is, A(, ) A (, ) For point B(, ) That is, B(, ) B (, ) For point C(, ) That is, C(, ) C (, ) Plot the image on the same aes as the original. A(, ) B(, ) A'(, ) D(, ) C(, ) B'(, ) C'(, ) This tpe of transformation leaves the origin unchanged and therefore differs from a translation. The transformation matri can also be etracted from information about the original and image points. An eample of this is shown in the following worked eample.

11 Chapter 6 Transformations using matrices 9 WORKED Eample 6 Find the matri of the linear transformation that maps A(, ) onto A (, -) and B(, -) onto B (, -). Set up the initial matri equation. a c b d State matri equations for A, A, B and B. For point A: a b c d and for point B: a c b d Multipl the matrices to arrive at simultaneous equations for unknowns, a, b, c and d. Rearrange these equations and enter them into the graphics calculator as an augmented matri and perform Gaussian elimination on it (using rref as described in chapter ). From the equation for point A: a + b () c + d () From the equation for point B: a b () c d () a + b c + d a b c d Enter this as: Displa shows [I X]. a, b, c --, d -- 6 Use these values to build L, the linear transformation matri. L As hinted at in the introduction to this section, there are two was to conceptualise a transformation. The more obvious wa is to imagine that the points move to new positions on the Cartesian plane. The other less obvious notion is that it is actuall the Cartesian plane on which the original points are plotted that undergoes distortions to ield the transformed image. Perhaps the former is more straightforward, but the end product will be the same.

12 Maths Quest Maths C Year for Queensland remember remember. A linear transformation can be represented b a b or c d L where L a b is the transformation matri that maps c d point (, ) onto the image (, ).. The transformed image can be found using a b. c d. The transformed image is not congruent to the object. 6B Linear transformations WORKED Eample WORKED Eample 6 ii Which of the following transformations are linear? ii Write the transformation matrices for each of these. a + b + + c d e a Find the images of the points A (, ), B (, ) and C(, ) under the following transformations: i ii iii iv b Sketch the original triangle from a and its different images. Find the image of the points (given below) under the transformation defined b: + a A(, ) b B(, ) c C(, ) Plot the original point and its image in each case. Find the image of the pre-image points A(, ), B(, ) and C (, ) under the transformation defined b: + Plot the original and image points. Find the matri of the linear transformation which maps: a (, ) (, ) and (, ) (, ) b (, ) (, ) and (, ) (, ) c (, ) (, ) and (, ) (, 6) d (, ) (, ) and (, ) (, )

13 Chapter 6 Transformations using matrices Linear transformations and group theor Earlier in our Mathematics C course of stud ou were introduced to group theor (chapter ). You found that a sstem formed a group if the properties of closure and associativit applied and an identit element and inverse eisted. These properties appl to man areas of mathematics including linear transformations. In chapter we investigated whether matrices, in general, formed a group; now we will stud groups that perform linear transformations. Closure If l is a linear transformation that maps (, ) (, ) then (, ) l (, ) If l is a linear transformation that maps (, ) (, ) then (, ) l (, ) Therefore it follows that (, ) l [l (, )] where l is followed b l and maps (, ) (, ). This double transformation can be represented as a single, where l l l. This is known as composition of transformations, where the order is significant. From the Mathematics B course ou would be familiar with the idea of composition of functions, where g() h(f()) indicates that f() is the inner function within the structure and general shape of h(). In matri form L L L L L L where L is a matri and L L L We can verif this result b considering the image of the point P(, ) after a linear transformation L followed b a linear transformation L. Show that following this double transformation produces the point P (, ). If we mapped P(, ) directl to P (, ) in a single transformation, find the transformation matri L. Is this transformation matri L equivalent to L L or L L? P l l l P" P' 6

14 Maths Quest Maths C Year for Queensland WORKED Eample 7 If l and l are linear transformations such that L and L : a find P, the image of P (, ) under l b find P, the image of P under transformation l c find the single transformation of P such that l l l d verif that P (as found in part b) is equal to LP a Use matri operation to find P, the image of P (, ) under l.. b Find P, the image of P under transformation l. c Find the single transformation of P such that l l l. L d Verif that P (as found in part b) is equal to LP. LP P Therefore P(, ) P (, ) Associativit As seen with matri operations, matri multiplication is associative; that is, (L L )L L (L L ). Therefore linear transformations are associative; that is, (l l )l l (l l ). Identit Remember the identit element (IE) is one which leaves the original number unchanged. When dealing with linear transformation this means that matri multiplication has been performed which leaves the original point unchanged. This is the identit transformation and is denoted b l i and the matri is I. For a matri, I. Inverse transformations An inverse transformation is one that maps the image back to the original point where (, ) (, ) (, ). This transformation is denoted b l. As with other inverses ll l l As with other inverses ll l i l P(, ) l P'(', ')

15 Chapter 6 Transformations using matrices a If L is the linear transformation matri and A is the transformation matri which returns the point to the original, then A L. As with general matri terminolog, the transformation l is non-singular; that is, l has an inverse if it has a matri l that will map the image back to the original. Therefore, onl linear transformations that have an inverse l can be considered to form a group. If l is singular then the set of linear transformations does not form a group. Abelian groups If the composition of linear transformations is commutative, then the set will form an Abelian group. But in general, multiplication of linear transformations is not commutative, that is l l l l. WORKED Eample Find the image of the point P(, ) under l followed b l with L L..6.. b Verif that l l in was. 8 a Set matrices in LP form. a P is the point (, ). Now find P using P L P Since P (, ) P(, ), L has mapped P (, ) back onto the original, therefore L is the inverse linear transformation of L. b Verif this b showing L L I. b L L..6.. Verif b finding the inverse of L. L ad bc d b c a L L..6.. L L State the conclusion. Therefore L L

16 Maths Quest Maths C Year for Queensland WORKED Eample 9 Determine whether the following linear transformation, l is singular or non-singular. + + State l in matri form. Test to determine whether the determinant. State our conclusion. L for L a b c d L ad bc 6 8 Since det L, L is non-singular, that is, it has an inverse. Images of curves non-singular transformations So far we have mainl considered onl the images of individual points under linear transformation where L. Now consider the image of a curve essentiall a set of points. WORKED Eample Find the image of the line under the linear transformation L. We need to epress the original function in terms of the image points so we need to find and substitute image points for the original points and. Evaluate the inverse. L L L L L

17 Chapter 6 Transformations using matrices 6 Epress and in terms of the original points. Simplif and rearrange the image equation. Some tets drop the primes on and at this stage, but if the are left in it reminds us that the graph of this function is the image of the original. Graph the original and image functions. Therefore Substitute for and in terms of the image points, into the original function: -- ( ) ' ' Images of curves singular transformations If a linear transformation L is singular, then L does not have an inverse and the method shown in worked eample cannot be used. We need to use a different approach as shown in the net worked eample. WORKED Eample Find the image of the circle + under the linear transformation L. State the initial transformation in general matri form. L Continued over page

18 6 Maths Quest Maths C Year for Queensland Find values for and. + + Notice that the equation for equals twice the equation for. Therefore this should be stated as the function of the image. Sketch the original and image curves. (, ) ' ' + (, ) (, ) (, ) remember remember. (a) Linear transformations are closed. (b) If (, ) l (, ) where l is a linear transformation that maps (, ) (, ) and l is a linear transformation that maps (, ) (, ) then (, ) l (, ) l [l (, )] where l is followed b l. Linear transformations are associative; that is, (l l )l l (l l ).. The identit transformation is denoted b l i and is represented b the identit matri I.. An inverse transformation is one that maps the image back to the original point where (, ) (, ) (, ) and is denoted b l. As with other inverses ll l l l i Onl linear transformations that have an inverse l can be considered to form a group.. If linear transformations are commutative, then the will form an Abelian group. But in general, multiplication of transformations is not commutative, that is l l l l.

19 Chapter 6 Transformations using matrices 7 6C Linear transformations and group theor WORKED Eample 7, 8, 9 WORKED Eample WORKED Eample A linear transformation l is defined as + + a What will the image of P(, ) be? b Is this linear transformation singular? c Show that l (l P ) P. d Use this linear transformation to state the image of the following curves: i ii + iii + Find the image of the circle + 9 under each of the following transformations. a b c Find the image of the circle + 9 under each of the following transformations. a b c 6 8 Show that under an linear transformation the image of a straight line is itself a straight line. a Sketch the following curves on separate aes. i ii + b Find the image of each curve under the linear transformation c Sketch each image as well as the original curve. 6 Find the image of each of the following curves under the linear transformation. a b + Rotations A rotation is a transformation in which the plane rotates about a fied point called the centre of rotation. This point is usuall taken as the origin, the rotation in an anticlockwise direction is considered a positive rotation and, in a clockwise direction as a negative rotation. Eamine the diagram at right to note that the centre of rotation is the onl point that doesn t move. C B A B' θ A' C'

20 8 Maths Quest Maths C Year for Queensland In a rotation:. each original point rotates through the same angle of rotation.. the image is congruent to the original the length, angle and area remain unchanged in the image. This is referred to as a congruent transformation.. r q denotes rotation in a positive direction through an angle of θ and R θ is the matri of rotation. With all the transformations that will be discussed we will generate matrices based on where the points (, ) and (, ) are mapped to on the plane, as a result of the transformation. These points are represented b columns and of the identit matri: Special rotations In this section we will discuss transformations involving rotations of 9, 8, 7 and 6, as well as general rotations. Rotation of 9 Consider the figure below. (, ) (, ) 9º (, ) (, ) 9º As the plane rotates through θ 9 about the origin, point (, ) will map to point (, ) and point (, ) will map to point (, ). Hence, the identit matri, I, is altered to to achieve a rotation of 9 about the origin. It is most important that ou recognise the pattern that is displaed b the columns in the matri and the coordinates of the image points. This concept forms the basis of the net section of work and totall eliminates remembering formulas so that ou will be able to understand what is happening to the points. Hence R 9 and is the matri of rotation. In general terms (, ) (, ) P'(', ') P(, )

21 Chapter 6 Transformations using matrices 9 As mentioned earlier, these rotation matrices should not be learned. The are quite similar and can be too readil confused. Sketch the original (, ) and (, ) points and then use their images to build the rotation matrices. Rotation of 8 In the diagrams below, notice that point (, ) is mapped onto point (, ) and point (,) is mapped onto (, ). (, ) P(, ) (, ) 8º (, ) 8º 8º (, ) P'(', ') Therefore R 8 where (, ) (, ). Rotation of 7 In the diagrams below, notice that point (, ) is mapped onto point (, ) and point (, ) is mapped onto point (, ). (, ) P(, ) 7º (, ) (, ) 7º 7º (, ) P'(', ') Therefore R 7 where (, ) (, ). Rotation of 6 R 6 itself). General rotation of θ because R 6 essentiall leaves the original unchanged (or mapped onto Consider the points (, ) and (, ) that are rotated through angle θ about the origin. B' cos θ sin θ θ B(,) Q θ cos θ P A' sin θ A(, )

22 Maths Quest Maths C Year for Queensland Careful eamination of the diagram shows that point (, ) is mapped onto point (cos θ, sin θ) and point (, ) is mapped onto point ( sin θ, cos θ) where sinθ cos θ (horizontal) and sin θ (vertical) θ cos θ R θ cos θ sin θ sin θ cos θ R θ, where θ is taken in a clockwise, negative rotation about the origin, and is shown in the diagram to the right. θ P(, ) R θ cos ( θ) sin ( θ) sin ( θ) cos ( θ) P'(', ') R θ cos θ sin θ sin θ cos θ since cos ( θ) cos θ and sin ( θ) sin θ Both R θ and R θ can be used to confirm the specific cases of R 9, R 8 and R 7. R 9 cos 9 sin 9 sin 9 cos 9 Remember that when ou need to evaluate a trigonometric ratio:. sketch the angle concerned in the correct quadrant. write the coordinates or length of the sides on the right-angled triangle. in the unit circle, the cosine ratio involves onl the -coordinate and the sine ratio involves onl the -coordinate. Verification of the other angle measures is left as a future eercise. WORKED Eample Find the image of the point (, -) under a rotation of π -- c about the origin. Write the general rotation matri and sketch the original point (shown below). R θ cos θ sin θ sin θ cos θ π Substitute -- c cos -- π for θ. R θ sin -- π (Note: The small c is the smbol for sin -- π cos -- π circular or radian measure.)

23 Chapter 6 Transformations using matrices Alwas use a sketch to develop the matri. R θ π π Set up the general matri form for transformations Rationalise the denominator and simplif Sketch the original and the image points. π (, ) (, )

24 Maths Quest Maths C Year for Queensland WORKED Eample Find the image of the line + under the rotation of π -- c 6 about the origin. Write the general R θ matri. R θ cos θ sin θ sin θ cos θ π Substitute -- c for θ and evaluate using 6 the relevant triangle of ratios. R π 6 -- cos π 6 -- sin π 6 -- sin π 6 -- cos π 6 -- π π Set up the general transformation matri model, rearranged so that is the subject. R π 6 -- R π -- 6 Evaluate the inverse of R Multipl out the matrices Substitute for and in the original + becomes function ( ) 8 After appling the Distributive Law and rationalising the denominator, this + + epression can be simplified. ( ) + ( )

25 Chapter 6 Transformations using matrices 8 Use our calculator onl at the end to simplif surds for sketching purposes. ' ( )' + ( ) + remember remember. For general rotation θ in an anticlockwise direction about the origin R θ cos θ sin θ. sin θ cos θ. Use the special right-angled triangles to obtain the trigonometric ratios.. Rotation is a congruent transformation. 6D Rotations WORKED Eample WORKED Eample Construct matrices for the following anticlockwise rotations about the origin (the angles are given in radians). π π a -- b π c d π Find the image of the following points under the given anticlockwise rotations about the origin. a (, ) π c θ -- b (, ) θ π c c (6, ) π c θ -- d (, ) θ 6 e (, ) θ 9 f (, ) π c θ -- 6 a Find the equation of the image of the line + as a result of the following rotations: π c π i θ ii θ -- iii θ -- c b Sketch each original line and its image. Find the equation of the image of the circle + π c after a rotation of --. What do ou notice? Can ou eplain wh this is so? WorkSHEET 6.

26 Maths Quest Maths C Year for Queensland Reflections A reflection is a linear transformation in which ever point of the original is reflected through a straight line called a mediator. This line can be thought of as a mirror. The diagram at right shows LABC reflected through the mediator m, at. A' A In a reflection:. corresponding points of the image and original figures are equidistant from and perpendicular to the mediator. length, angle and area of the image and C' B' original are unchanged, hence it is a congruent transformation. an points of the original on the mediator are left unchanged. m B C We usuall let m denote the reflection transformation and M the reflection matri.

27 Chapter 6 Transformations using matrices Reflection in the -ais (where ) Again, sketch the points (, ) and (, ) from the identit matri I. Under a reflection in the -ais, point (, ) will map to (, ) and point (, ) will map onto itself because it is on the mediator. (, ) Therefore M. (, ) m (, ) Reflection in the -ais (where ) If ou sketch the original points (, ) and (, ) ou will notice that if these points are reflected in the -ais then point (, ) will map to (, ) and point (, ), which is on the mediator, will map onto itself. Therefore, M. (, ) m (, ) (, ) Find the image of point (, ) under reflection M. Sketch the original and its image. WORKED Eample Sketch the diagram to construct our reflection matri. (, ) (, ) m (, ) M Write the initial transformation matri statement. M Continued over page

28 6 Maths Quest Maths C Year for Queensland Substitute the necessar values and evaluate. Sketch the original and image points. The image is the point (, ). P(, ) m P'(, ) Find the image of under reflection in the -ais. Sketch the original and its image. WORKED Eample Sketch the diagram to construct our reflection matri. m (, ) (, ) (, ) M Write the initial transformation matri statement and rearrange it to have the original points as the subject. M M Substitute for M and evaluate the inverse

29 Chapter 6 Transformations using matrices 7 Multipl to give epressions for and. Sketch the original and image graphs. Note the origin is left unchanged. Substitute for and into the original equation. m ' ' Reflection in line To find this reflection, sketch the situation as described. Remember to note the main points from the introduction to this section:. corresponding points of the image and original figures are equidistant from and perpendicular to the mediator. length, angle and area of the image and original are unchanged, hence it is a congruent transformation. an points of the original on the mediator are left unchanged. We find that (, ) and (, ) map to each other, therefore M. WORKED Eample 6 Find the equation of the image reflected in the line. (, ) (, ) Sketch the relevant diagram to establish the reflection matri. (, ) (, ) M Continued over page

30 8 Maths Quest Maths C Year for Queensland Set up the initial matri equation and rearrange to have and as the subject. M M Find the inverse of M Multipl matrices to determine and. Substitute for and in the original epression. Sketch the original and image curves. Note that the points (, ) and (, ) are unchanged as the are on the mediator. 6 becomes ± (, ) ' ' ' ' Reflection in the line tan q This line might be more easil recognised as m, where m is the gradient of the line which passes through the origin. Remember that the gradient m and tangent ratio Therefore the tangent and gradient ratios provide the rise same information: run Carefull eamine these diagrams that illustrate reflection of the points (, ) and (, ) in the line tan θ. θ θ B(, ) θ 9 θ A' (cos, sin ) θ θ tanθ A(, ) m tanθ 9 θ B'(cos(9 ), sin(9 ) ) θ θ

31 Chapter 6 Transformations using matrices 9 Note the following from these diagrams. For the point (, ):. point A is reflected to a point equidistant from and perpendicular to the line. the angle from the -ais to A is θ. the -coordinate of the right-angled triangle is cos θ. the -coordinate of this triangle is sin θ.. Hence point (, ) (cos θ, sin θ). For the point (, ):. point B is reflected to a point equidistant from and perpendicular to the line. MOB 9 θ therefore MOB 9 θ. therefore XOB (9 θ) θ 9 θ. the -coordinate cos (9 θ). the -coordinate sin (9 θ) because the angle is in the fourth quadrant. 6. Hence point (, ) [cos (9 θ), sin(9 θ)]. 7. Using trigonometric ratios, this simplifies to ield (sin θ, cos θ). (Remember that sin cos 6, etc.) Using all this information from the reflection of points (, ) and (, ) in the line tan θ ields: M tan θ cos θ sin θ. sin θ cos θ WORKED Eample Find the matri for the reflection in the line. (Note that the sign applies onl to the.) Use a sketch to epress as the tangent ratio of some angle. 7 π 6 π π tan -- State the general reflection matri in the line π tan θ, then substitute -- for θ. M tan θ M cos θ sin θ sin θ cos θ cos π sin π sin π cos π Continued over page

32 Maths Quest Maths C Year for Queensland Evaluate these ratios using the following triangle π π WORKED Eample 8 Find the image of the line as reflected in the line. Use the matri from the previous eample as M. M Set up the initial matri transformation and inverse statement. M M Find the inverse and multipl the matrices

33 Chapter 6 Transformations using matrices 6 Substitute for and into the original equation. Make sure ou carr through the minus sign from the function. Simplif and rationalise the denominators to find the equation of the image line. The last line is included for ease of graphing onl. Sketch the original and its image. becomes ( ) '.7'.7 remember remember. Reflection is a congruent transformation.. Reflection occurs through a mediator, m.. Reflection in the line tan θ is represented b M cos θ sin θ. sin θ cos θ 6E Reflections Write the matrices for the following reflections: a m b m c m d m e m f m WORKED Eample Find the images of each of the following points under the reflection given below. Sketch each original and its image. a -ais b -ais c d i (, ) ii (, ) iii (, ) iv (, ) v (, ) vi (, )

34 Maths Quest Maths C Year for Queensland WORKED Eample,6,7,8 Find the image of the following curves under each of the reflections given below. a b c + d Dilations i -ais ii -ais iii iv (part a onl) So far we have investigated kinds of transformations. The translation shifted the figure on the plane, the general linear transformation produced an image that, on occasions, bore little resemblance to its original. The rotation and reflection transformations are P' congruent transformations with the original basicall repositioned on the plane. A dilation P is a transformation in which point P and image P are collinear from a fied point, usuall the origin O, as shown in the figure at right. O The length OP kop where k is referred to as the dilation factor. If k >, a dilation ma be an enlargement (for k > ) or a reduction (for < k < ). A' A A B B' A' B' B O C k > C' O C' C < k < If k < then the image of the original has been mapped through the origin in a reverse direction. In this diagram, k --, therefore the image appears half the distance from the fied point O and on the opposite side of O to the original points. B' C' O C A A' k < B In a dilation:. length and area are not preserved; the shape will appear similar, but not congruent to the original. the dilation d is denoted b the matri D k, with the dilation factor of k given parallel to the -ais and the anchor line being the -ais.

35 Chapter 6 Transformations using matrices Dilation parallel to the -ais The dilation matri D k, of the points (, ) and (, ) under the dilation d k, is given b D k, k (, ) where (, ) (k, ). (, ) is left unchanged since it is on the anchor line. the -coordinate is mapped k units awa from the anchor line. This is shown graphicall in the figure at right. Dilation parallel to the -ais can be thought of as pulling the plane awa from the fied point or anchor line in this case, the -ais. Dilation parallel to the -ais This dilation pulls the plane awa from the -ais so all points on the -ais are anchored. The figure at right shows that (, k) D k, k where. (, ) is left unchanged since it is on the anchor line. the -coordinate is mapped k units awa from the anchor line (the -ais). (, ) (, ) WORKED Eample 9 Find the coordinates of the image of point (, ) under the dilation factor of - parallel to the -ais. Sketch the dilation and construct the matri from the sketch. D, (, ) (, ) (, ) Write the general transformation matri equation. D, Continued over page

36 Maths Quest Maths C Year for Queensland Multipl the matrices to produce the image coordinates. P'( 8, ) P(, ) 8 Point (, ) maps to image point ( 8, ) under a dilation of parallel to the -ais. WORKED Eample Find the equation of the image of + under the dilation d,. Sketch the dilation and construct the matri from the sketch. D, (, ) (, ) (, ) Set up the general transformation matri equation and rearrange to have as the subject. D, D, Find the inverse of D, and substitute it into the equation Multipl the matri equations. -- Substitute and in the original equation ( + ). + ( -- ) + +

37 Chapter 6 Transformations using matrices 6 Sketch the original and its image. Note that (, ) remains unchanged since it is on the anchor line of the -ais. (, ) + ' ' + WORKED Eample Find the image of the circle + 9 with a dilation factor of -- parallel to the -ais. Sketch the situation and use this to construct the dilation matri. D --, -- D --, (, ) (, ) (, ) Rearrange to put as the subject. D --, Calculate the inverse of D and substitute it into the equation. Multipl the matrices and write epressions for and. Substitute and into the original equation and rearrange to fit the general equation of an ellipse becomes ( ) + ( ) Continued over page

38 6 Maths Quest Maths C Year for Queensland Sketch the original and its image. This can be written as a b since is the general equation of an ellipse about the origin. Therefore a So the length of the semi-major ais is. b So the length of the semi-minor ais is. + 9 ' + ' 9 If ou think about the original shape and its image as shown in this eample ou will understand that the dilation factor of --, in effect, shrinks the original shape, parallel to the -ais so the figure falls back towards the anchor line (the -ais) and leaves all points on the -ais unchanged. Dilation about the origin, d k The previous dilations have been mapped parallel to an ais, where that ais has provided the anchor line for the stretching of the plane. However, a dilation about the origin does not anchor to a line, but rather to a point the origin. The diagram at right shows this stretch that results in both and coordinates being mapped a dilation factor of k from the origin. Therefore, if the original point is on the origin it will map onto itself. Onl the dilation factor is given in the dilation matri: (, k) (, ) P'(k, k ) P(, ) (, ) (k, ) D k k k

39 Chapter 6 Transformations using matrices 7 WORKED Eample Find the image of under a dilation factor of about the origin. Sketch the situation to construct the matri. D (, ) (, ) (, ) (, ) Set up the initial transformation matri equation. D D Evaluate the inverse and multipl Substitute for and into the original equation and simplif. Sketch the original and its image. The minus sign results in the image reversing its position with respect to the origin, and the factor of results in the broader parabola becomes -- ( -- ) ' '

40 MQ Maths C Yr - 6 Page 8 Monda, October, 6:8 AM 8 M a t h s Q u e s t M a t h s C Ye a r f o r Q u e e n s l a n d Histor of mathematics M AU R I T S C O R N E L I U S E S C H E R ( ) Portrait of M. C. Escher Cordon Art, Baarn, Holland. All rights reserved. During his life... World War I and World War II take place. Flight technolog develops from the Wright brothers first flight in 9 to the moon landing in 969. Israel is established. Quoted as having said I never got a pass mark in math. And just imagine mathematicians now use m prints to illustrate their books. I guess the are quite unaware that I am ignorant about the whole thing. Maurits Escher was born on 7 June 898 in the Netherlands. His earl work was mainl concerned with the representation of visible realit, such as landscapes and buildings. However, he graduall became more interested in studing the abstract space-filling patterns used b the Moors in mosaics found in Spain. He also studied a paper b Pola on 7-plane crstallographic groups; although instead of using geometrical motifs, Escher used animals, plants or people to fill the space on his intricate prints. Even though he professed ignorance of all things mathematical, Escher incorporated man mathematical ideas in his works infinit, Mobius strips, stellations, deformations, reflections, rotations, Platonic solids, spirals and the hberbolic plane. Original Escher prints are highl prized possessions now, but it was not until 9 that he actuall began to earn a reasonable income from his prints. Widel regarded as a graphic artist, his designs have appeared on postage stamps, bank notes, T-shirts, jigsaw puzzles, record album covers, and, as he remarked, man scientific and mathematical publications. Relativit b M. C. Escher Cordon Art, Baarn, Holland. All rights reserved. His work has been held in high regard b both artists and mathematicians. He died in 97, in the Netherlands. Research. Research Mobius strips, stellations, deformations, reflections, rotations, Platonic solids, spirals and the hberbolic plane.. Look through scientific and mathematical publications to see if an use Escher s prints as covers or illustrative pages. remember remember. Dilation occurs in relation to an anchor line or point.. Dilation is not a congruent transformation.

41 Chapter 6 Transformations using matrices 9 6F Dilations WORKED Eample 9 WORKED Eample WORKED Eample WORKED Eample Sketch the original and its image for all questions. Find the image of each of the following points under the dilations given: a, parallel to the -ais b, parallel to the -ais i (, ) ii (, ) iii (, ) iv (, ) v (, ) vi (--, ) Find the image of the line under the following dilations: a dilation factor parallel to the -ais b dilation factor parallel to the -ais Find the image of the ellipse with a dilation factor of 9 a --, parallel to the -ais b, parallel to the -ais Find the image of with a dilation factor of a parallel to the -ais b parallel to the -ais Find the image of the line under a dilation factor of -- about the origin. 6 Find the image of each of the points in question under the following dilations: a --, about the origin b, about the origin 7 Find the image of the ellipse in question with a dilation factor of a, about the origin b --, about the origin Shears The final transformation discussed in this chapter is that of shears, which can be thought of as a push from one side that results in a change in shape. An eample of this is seen when changing a rectangle into a parallelogram. Where a dilation pulls the plane from a certain anchor point or line, a shear pushes from one side and an points on the anchor line again remain unchanged. A shear parallel to the -ais (see the figure shown at right) moves ever point in the plane parallel to the -ais b a distance proportional to its distance from the -ais. That is, points on the -ais remain anchored while points further awa are pushed further from their original position. Push

42 6 Maths Quest Maths C Year for Queensland Similarl a shear parallel to the -ais (see the figure shown at right) moves ever point in the plane parallel to the -ais b a distance proportional to its distance from the -ais. The shear transformation matri uses similar notation to other linear transformations, where S k, denotes the shear with a shear factor of k parallel to the -ais and S k, denotes the shear with a factor of k parallel to the -ais. That is, S k, k, S k,. k Shears parallel to the -ais As can be seen from the figure at right, point (, ) remains unchanged because it is anchored to the -ais while point (, ) is mapped to (k, ). This means that point (, ) will map to (k, ). (, ) (k, ) (, ) Shears parallel to the -ais The figure at right shows the point (, ) unchanged b the shear because it lies on the -ais while point (, ) is mapped to (, k). (, ) (, k) WORKED Eample The vertices of a triangle are O(, ), A(, ) and B(, ). Find the image of these points O A B under a shear factor of parallel to the -ais. Sketch the original and its image. (, ) Sketch the initial unit diagram and use this to determine the shear matri. Sketch the original (shown below). S, (, ) (, ) (, )

43 Chapter 6 Transformations using matrices 6 Set up the initial general transformation matri equation. S, Substitute S, and solve for each point in turn. O(, ) is unchanged because it is on the anchor ais. For O(, ), the equation is: So O(, ) O (, ) For A(, ), the equation is: So A(, ) A (, ) For B(, ), the equation is: Note that the -coordinate is not actuall multiplied b. 7 So B(, ) B (, 7) Sketch the image points with the original. B'(, 7) A'(, ) B(, ) O(,) A(, )

44 6 Maths Quest Maths C Year for Queensland WORKED Eample Find the image of the parabola under the shear factor of, parallel to the -ais. Sketch the original and its image. Sketch the initial unit diagram and use this to determine the shear matri. Sketch the original (shown below). S, (, ) (, ) (, ) Set up the initial general transformation matri equation. S, Rearrange in terms of and. S, Find the inverse of S, Multipl to find epressions for and. + Substitute for and into the original equation and solve for to find zeros. Sketch the image with the original. Notice that ii(i) the origin is anchored i(ii) the positive -values are pulled up from the -ais (iii) the negative -region seems to have slipped back from the -ais. becomes + + ( + ) is zero when and ' '(' + ) --

45 remember remember Chapter 6 Transformations using matrices 6. A shear can be thought of as a push parallel to an anchor line that transforms all points on the plane b a distance proportional to their distance from the anchor line.. A shear is not a congruent transformation. 6G Shears WORKED Eample WORKED Eample Find the image of each of the following points under a shear factor of i parallel to the -ais ii -- parallel to the -ais a (, ) b (, ) c (, ) d (, ) e (, ) f (, ) Sketch each pair of original and image points. A parallelogram has vertices A(, ), B(, ), C(6, ) and D(, ). Sketch the original shape and its image under the shear factor of a parallel to the -ais b parallel to the -ais c parallel to the -ais d parallel to the -ais Find the image of each of the following curves under a shear factor of i parallel to the -ais ii parallel to the -ais a b c d + Sketch the original and image curves. SkillSHEET SkillSHEET WorkSHEET Transformations Transform the ellipse under the dilation of d. Give the equation of the new ellipse, full supporting our response with matri operations and a full labelled diagram. Design a set of or transformations that map a shape of our own choosing to another shape on the plane. Your response should include all working and full labelled diagrams. With our current knowledge of transformations using matri applications, investigate whether the following transformations are possible. You ma need to consider a series of transformations. a A square into a straight line b A triangle into a square c A circle into a straight line d A square into a circle e A kite into a square

46 6 Maths Quest Maths C Year for Queensland summar Geometric transformations and matri algebra A general transformation maps each point of the Cartesian plane onto some other point of the plane. A translation, t, moves each -coordinate a units parallel to the -ais and each -coordinate b units parallel to the -ais, such that + T, where (, ) is the image of point (, ) and T is the transformation matri. The translation t results in a congruent transformation. Linear transformations A linear transformation, l, warps the plane such that L. Linear transformations are not congruent transformations. Rotations A rotation, r, rotates the plane about a fied point to result in a congruent transformation. The matri representing the rotation is R θ cos θ sin θ. sin θ cos θ Reflections A reflection, m, reflects ever point of the original through a straight line called a mediator and results in a congruent transformation. The reflection matri M tan θ cos θ sin θ. sin θ cos θ Dilations A dilation, d, transforms each point P to P where P and P are collinear with a fied point O. The matri D k, represents a dilation of k units parallel to the -ais anchored from the -ais. The matri D k represents a dilation factor of k units through the origin and D k, represents the same dilation in the reverse direction. Shears A shear, s, is a transformation like a push from one side. The matri S k, moves ever point in the plane parallel to the -ais b a distance proportional to its distance from the -ais. Points on the -ais remain unchanged.

47 Chapter 6 Transformations using matrices 6 CHAPTER review State all congruent transformations. Find the image of the following points under the translation T a (, ) b (, ) c (, ) 6A 6A The line undergoes a succession of translations defined b T and 6A T. Show that the order in which these occur has no effect on the result. Find the equation of the image of under the translation of. 6A Find the image of points A(, ) and C(, ) under the transformation. 6 Find the matri of the linear transformation which maps (, ) to (, ) and (, ) to (, ). 7 Find the image of + under the linear transformation. Sketch the original and image curves. 8 Find the image of the following points under the given anticlockwise rotations about the origin. π a (, ) where θ π b (, ) where θ -- π c (, ) where θ -- d (, ) where θ 6 π 9 Find the image of the line + through a rotation of --. Sketch the original and the image. Find the image of each of the following points under the reflection as given: a (, ) in the -ais b (, ) in the line c (, ) in the line d (, ) in the line 6B 6B 6C 6D 6D 6E

48 66 Maths Quest Maths C Year for Queensland 6E Find the image of the line under reflection in the line. Sketch the original and the image. 6F Find the image of each of the following points under the dilation factors given: 6F 6G 6G test ourself CHAPTER 6 a (, ), units parallel to the -ais b (, ), -- unit about the origin c (, ), units about the origin d (, ), units parallel to the -ais Find the image of under the dilation factor of parallel to the -ais. Find the image of each of the following points under the given shear factor: a (, ), -- unit parallel to the -ais b (, ), units parallel to the -ais Find the image of the curve under a shear factor of parallel to the -ais.