2D TRANSFORMATIONS. Course Website:

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1 Unit-III PPT Slides Tet Books:. Computer Graphics C version, Donald Hearn and M.Puline Baker, Pearson Education 2. Computer Graphics Principles & Practice second edition in C,Fole VanDam, Fiener and Hughes, Pearson Educatio

2 2D TRANSFORMATIONS Course Website:

3 What is transformations? 2D Transformations The geometrical changes of an object from a current state to modified state. Wh the transformations is needed? To manipulate the initiall created object and to displa the modified object without having to redraw it.

4 2 was 2D Transformations Object Transformation Alter the coordinates descriptions an object Translation, rotation, scaling etc. Coordinate sstem unchanged Coordinate transformation Produce a different coordinate sstem

5 Matri Math Wh do we use matri? More convenient organization of data. More efficient processing Enable the combination of various concatenations Matri addition and subtraction a b ± c d = a ± c b ± d

6 Matri Math How about it? a ± c d e f b Matri Multiplication Dot product a c b d e f. g h = a.e b.g a.f b.h c.e d.g c.f d.h

7 Matri Math What about this? = = tak boleh!! Tpe of matri a Row-vector b a b Column-vector

8 Matri Math Is there a difference between possible representations? a c b e ae bf = d f ce df [ ] e f a c b d = [ ] ae cf be df [ ] e f a c b d = [ ] ae bf ce df

9 Matri Math We ll use the column-vector representation for a point. Which implies that we use pre-multiplication of the transformation it appears before the point to be transformed in the equation. A C B A B = D C D What if we needed to switch to the other convention?

10 Translation A translation moves all points in an object along the same straightline path to new positions. The path is represented b a vector, called the translation or shift vector. We can write the components: p' = p t? t =4 p' = p t or in matri form: P' = P T (2, 2) t = 6 = t t

11 A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. First, we ll assume the pivot is at the origin. P Rotation θ P

12 Rotation Review Trigonometr => cos φ = /r, sin φ= /r = r. cos φ, = r.sin φ P (, ) => cos (φ θ) = /r = r. cos (φ θ) θ = r.cosφcosθ -r.sinφsinθ =.cos θ.sin θ r P(,) =>sin (φ θ) = /r = r. sin (φ θ) = r.cosφsinθ r.sinφcosθ =.sin θ.cos θ θ r φ Identit of Trigonometr

13 Rotation We can write the components: p' = p cos θ p sin θ p' = p sin θ p cos θ or in matri form: P' = R P θ can be clockwise (-ve) or counterclockwise (ve as our eample). Rotation matri cosθ R= sinθ sinθ cosθ P (, ) θ r φ θ P(,)

14 Eample Find the transformed point, P, caused b rotating P= (5, ) about the origin through an angle of 9. Rotation = θ θ θ θ θ θ θ θ cos sin sin cos cos sin sin cos = cos9 sin 9 5 sin 9 cos9 5 = 5 5 = 5

15 Scaling Scaling changes the size of an object and involves two scale factors, S and S for the - and - coordinates respectivel. Scales are about the origin. We can write the components: p' = s p p' = s p or in matri form: P' = S P Scale matri as: P P S = s s

16 Scaling If the scale factors are in between and the points will be moved closer to the origin the object will be smaller. Eample : P(2, 5), S =.5, S =.5 Find P? P(2, 5) P

17 Scaling If the scale factors are in between and the points will be moved closer to the origin the object will be smaller. P Eample : P(2, 5), S =.5, S =.5 Find P? P(2, 5) If the scale factors are larger than the points will be moved awa from the origin the object will be larger. P Eample : P(2, 5), S = 2, S = 2 Find P?

18 Scaling If the scale factors are the same, S = S uniform scaling Onl change in size (as previous eample) P If S S differential scaling. Change in size and shape Eample : square rectangle P(, 3), S = 2, S = 5, P? P(, 2) What does scaling b do? What is that matri called? What does scaling b a negative value do?

19 Combining transformations We have a general transformation of a point: P' = M P A When we scale or rotate, we set M, and A is the additive identit. When we translate, we set A, and M is the multiplicative identit. To combine multiple transformations, we must eplicitl compute each transformed point. It d be nicer if we could use the same matri operation all the time. But we d have to combine multiplication and addition into a single operation.

20 Homogenous Coordinates w Let s move our problem into 3D. Let point (, ) in 2D be represented b point (,, ) in the new space. Scaling our new point b an value a puts us somewhere along a particular line: (a, a, a). A point in 2D can be represented in man was in the new space. (2, 4) (8, 6, 4) or (6, 2, 3) or (2, 4, ) or etc.

21 Homogenous Coordinates We can alwas map back to the original 2D point b dividing b the last coordinate (5, 6, 3) --- (5, 2). (6, 4, ) -?. Wh do we use for the last coordinate? The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate sstem its name homogeneous coordinates.

22 Matri Representation Point in column-vector: Our point now has three coordinates. So our matri is needs to be 33. Translation = t t

23 Rotation Matri Representation = cos sin sin cos θ θ θ θ Scaling = s s

24 Composite Transformation We can represent an sequence of transformations as a single matri. No special cases when transforming a point matri vector. Composite transformations matri matri. Composite transformations: Rotate about an arbitrar point translate, rotate, translate Scale about an arbitrar point translate, scale, translate Change coordinate sstems translate, rotate, scale Does the order of operations matter?

25 Composition Properties Is matri multiplication associative? (A.B).C = A.(B.C) = l k j i dh cf dg ce bh af bg ae l k j i h g f e d c b a? = dhl cfl dgj cej dhk cfk dgi cei bhl afl bgj aej bhk afk bgi aei = = dhl dgj cfl cej dhk dgi cfk cei bhl bgj afl aej bhk bgi afk aei hl gj hk gi fl ej fk ei d c b a l k j i h g f e d c b a

26 Is matri multiplication commutative?? A. B = B. A Composition Properties a c b d e g f h = ae bg af bh ce dg cf dh e g f h a c b d = ea ga fc hc eb gb fd hd

27 Order of operations So, it does matter. Let s look at an eample:. Translate 2. Rotate. Rotate 2. Translate

28 Composite Transformation Matri Arrange the transformation matrices in order from right to left. General Pivot- Point Rotation Operation :-. Translate (pivot point is moved to origin) 2. Rotate about origin 3. Translate (pivot point is returned to original position) T(pivot) R(θ) T( pivot) t t cosθ -sinθ sinθ cosθ.. - t - t t cosθ -sinθ -t cosθ t t sinθ. sinθ cosθ -t sinθ - t cosθ cosθ -sinθ -t cosθ t sinθ t sinθ cosθ -t sinθ - t cosθ t

29 Composite Transformation Matri Eample Perform 6 rotation of a point P(2, 5) about a pivot point (,2). Find P? cosθ -sinθ -t cosθ t sinθ t sinθ cosθ -t sinθ - t cosθ t. Sin 6 =.866 Kos 6 = / = P = (-, 4)

30 Eample Without using composite homogenus matri Perform 9 rotation of a point P(5, ) about a pivot point (2, 2). Find P?. Translate pivot point ke asalan ( t = -2, t = -2) Titik P(5, ) P (3, -) 2. Rotate P = 9 degree P (3, -) -- > kos 9 -sin 9 3 = - 3 = sin 9 kos Translate back ke pivot point (t = 2, t = 2) titik (, 3 ) titik akhir (3, 5)

31 Composite Transformation Matri General Fied-Point Scaling Operation :-. Translate (fied point is moved to origin) 2. Scale with respect to origin 3. Translate (fied point is returned to original position) T(fied) S(scale scale) T( fied) Find the matri that represents scaling of an object with respect to an fied point? Given P(6, 8), S = 2, S = 3 and fied point (2, 2). Use that matri to find P?

32 t t S S.. t t. S -t S S -t S - t - t = S -t S t S -t S t Answer =6, = 8, S = 2, S = 3, t =2, t = 2 2-2( 2) 2 3-2(3) = 2

33 Composite Transformation Matri General Scaling Direction Operation :-. Rotate (scaling direction align with the coordinate aes) 2. Scale with respect to origin 3. Rotate (scaling direction is returned to original position) R( θ) S(scale) R(θ) Find the composite transformation matri b ourself!!

34 Dapatkan titik akhir bagi P(5, 8) jika titik tersebut diputarkan sebanak 9 darjah, kemudian ditranslate sebanak (-6, 9) dan akhirna diskala dengan faktor skala (2,.5). latihan

35 S. T. R S S t t.. cosθ -sinθ sinθ cosθ S S. cosθ -sinθ t sinθ cosθ t Scosθ S(-sinθ) S t S sinθ S cosθ S t

36 Reflection: Other transformations -ais -ais

37 Reflection: Other transformations origin line =

38 Shear: Other transformations -direction - direction sh sh

39 Coordinate Sstem Transformations We often need to transform points from one coordinate sstem to another:. We might model an object in non-cartesian space (polar) 2. Objects ma be described in their own local sstem 3. Other reasons: tetures, displa, etc

40 Computer Graphics 3: 2D Transformations Course Website:

41 Contents In toda s lecture we ll cover the following: Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications Combining transformations

42 Wh Transformations? In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it Images taken from Hearn & Baker, Computer Graphics with OpenGL (24)

43 Translation Simpl moves an object from one position to another new = old d new = old d Note: House shifts position relative to origin

44 Translation Eample (2, 3) 2 (, ) (3, )

45 Scaling Scalar multiplies all coordinates WATCH OUT: Objects grow and move! new = S old new = S old Note: House shifts position relative to origin

46 Scaling Eample (2, 3) 2 (, ) (3, )

47 Rotation Rotates all coordinates b a specified angle new = old cosθ old sinθ new = old sinθ old cosθ Points are alwas rotated about the origin π θ =

48 Rotation Eample (4, 3) 2 (3, ) (5, )

49 Homogeneous Coordinates A point (, ) can be re-written in homogeneous coordinates as ( h, h, h) The homogeneous parameter h is a non- zero value such that: = h We can then write an point (, ) as (h, h, h) We can convenientl choose h = so that (, ) becomes (,, ) h = h h

50 Wh Homogeneous Coordinates? Mathematicians commonl use homogeneous coordinates as the allow scaling factors to be removed from equations We will see in a moment that all of the transformations we discussed previousl can be represented as 3*3 matrices Using homogeneous coordinates allows us use matri multiplication to calculate transformations etremel efficient!

51 Homogeneous Translation The translation of a point b (d, d) can be written in matri form as: d d Representing the point as a homogeneous column vector we perform the calculation as: d * * d* d d = = * * d* d * * *

52 Remember Matri Multiplication Recall how matri multiplication takes place: = z f e d z c b a f e d c b a * * * * * * z i h g z i h g * * *

53 Homogenous Coordinates To make operations easier, 2-D points are written as homogenous coordinate column vectors d d d d v T v d d ), ( ' : = = v s s S v s s s s ), ( ' : = = Translation: Scaling:

54 Homogenous Coordinates (cont ) v R v ) ( ' : cos sin sin cos cos sin sin cos θ θ θ θ θ θ θ θ θ = = Rotation:

55 Inverse Transformations Transformations can easil be reversed using inverse transformations = d d T = d T = s s S = cos sin sin cos θ θ θ θ R

56 Combining Transformations A number of transformations can be combined into one matri to make things eas Allowed b the fact that we use homogenous coordinates Imagine rotating a polgon around a point other than the origin Transform to centre point to origin Rotate around origin Transform back to centre point

57 Combining Transformations (cont ) House (H ) T ( d, d) H 2 R (θ ) T ( d, d) H T ( d, d) R( θ ) T ( d, d) H 3 4

58 Combining Transformations (cont ) The three transformation matrices are combined as follows d cosθ sinθ d d sinθ cosθ d v ' = T( d, d) R( θ) T( d, d) v REMEMBER: Matri multiplication is not commutative so order matters

59 In this lecture we have taken a look at: 2D Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications Combining transformations Summar Net time we ll start to look at how we take these abstract shapes etc and get them onscreen

60 Eercises Translate the shape below b (7, 2) (2, 3) 2 (, 2) (3, 2) (2, )

61 Eercises 2 Scale the shape below b 3 in and 2 in (2, 3) 2 (, 2) (3, 2) (2, )

62 Eercises 3 Rotate the shape below b 3 about the origin (7, 3) 2 (6, 2) (8, 2) (7, )

63 Eercise 4 Write out the homogeneous matrices for the previous three transformations Translation Scaling Rotation

64 Eercises 5 Using matri multiplication calculate the rotation of the shape below b 45 about its centre (5, 3) 5 4 (5, 4) 3 (4, 3) (6, 3) 2 (5, 2)

65 Scratch

66 Equations Translation: new = old d new = old d Scaling: new = S old Rotation new = old cosθ old sinθ new = S old new = old sinθ old cosθ

67 Shear Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions 6 7

68 Shear Matri Consider simple shear along ais = cot θ = z = z 6 8 θ cot H(θ) =