Linear transformations Affine transformations Transformations in 3D. Graphics 2009/2010, period 1. Lecture 5: linear and affine transformations

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1 Graphics 2009/2010, period 1 Lecture 5 Linear and affine transformations

2 Vector transformation: basic idea Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiplication of an n n matrix with a vector (i.e. a n 1 matrix): ( a11 a In 2D: 12 a 21 a 22 ) ( ) x = y ( ) a11 x + a 12 y a 21 x + a 22 y a 11 a 12 a 13 x a 11 x + a 12 y + a 13 z In 3D: a 21 a 22 a 23 y = a 21 x + a 22 y + a 23 z a 31 a 32 a 33 z a 31 x + a 32 y + a 33 z The result is a (transformed) vector.

3 Example: scaling Definition Examples Finding matrices Compositions of transformations Transposing normal vectors To scale with a factor two with respect to the origin, we apply the matrix ( ) to a vector: ( ) ( ) 2 0 x 0 2 y = ( ) 2x + 0y = 0x + 2y ( ) 2x 2y

4 Definition Examples Finding matrices Compositions of transformations Transposing normal vectors A function T : R n R m is called a linear transformation if it satisfies 1 T (u + v) = T (u) + T (v) for all u, v R n. 2 T (cv) = ct (v) for all v R n and all scalars c. Or (alternatively), if it satisfies T (c 1 u + c 2 v) = c 1 T (u) + c 2 T (v) for all u, v R n and all scalars c 1, c 2. v T (v) u + v u T (u + v) T (u) can be represented by matrices

5 Example: scaling Definition Examples Finding matrices Compositions of transformations Transposing normal vectors We already saw scaling by a factor of 2. In general, a matrix ( a a 22 ) scales the i-th coordinate of a vector by the factor a ii 0, i.e. ( a a 22 ) ( ) x = y ( ) a11 x a 22 y

6 Example: scaling Scaling does not have to be uniform. Here, we scale with a factor one half in x-direction, and three in y-direction: ( 1 ) ( ) 2 0 x 0 3 y = ( 1 2 x ) 3y Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: what is the inverse of this matrix?

7 Example: projection We can also use matrices to do orthographic projections, for instance, onto the Y -axis: ( ) ( ) 0 0 x 0 1 y = ( ) 0 y Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: what is the inverse of this matrix?

8 Example: reflection Reflection in the line y = x boils down to swapping x- and y-coordinates: ( ) ( ) 0 1 x 1 0 y = ( ) y x Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: what is the inverse of this matrix?

9 Example: more reflections Definition Examples Finding matrices Compositions of transformations Transposing normal vectors

10 Example: shearing Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Shearing in x-direction pushes things sideways : ( ) ( ) 1 1 x 0 1 y = ( ) x + y y Q: What is the matrix for pushing things upwards, i.e. for shearing in y-direction?

11 Example: shearing Definition Examples Finding matrices Compositions of transformations Transposing normal vectors General case for shearing in x-direction: ( ) ( ) 1 s x 0 1 y ( ) x + sy = y Q: What is the inverse of this matrix?

12 Example: rotation Definition Examples Finding matrices Compositions of transformations Transposing normal vectors To rotate 45 about the origin, we apply the matrix ( 1 ) Note: = cos 45 = sin 45, so this is the same as ( cos 45 sin 45 ) sin 45 cos 45

13 Example: rotation Definition Examples Finding matrices Compositions of transformations Transposing normal vectors General case for a counterclockwise rotation about an angle φ around the origin: ( ) cos φ sin φ sin φ cos φ Q: what is the matrix for clockwise rotation?

14 Finding matrices Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Applying matrices is pretty straightforward, but how do we find the matrix for a given linear transformation? ( ) a11 a Let A = 12 a 21 a 22 Q: what is the significance of the column vectors of A?

15 Finding matrices Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Scaling (factors a, b 0): Shearing (x-dir., factor 1): Reflection (in line y = x): ( ) ( ) a 0 x 0 b y ( ) ( ) 1 1 x 0 1 y ( ) ( ) 0 1 x 1 0 y = = = ( ) ax by ( ) x + y y ( ) y x ( ) ( ) Rotation (countercl., 45 ): x y = 1 2 ( ) x y 2 x + y

16 Finding matrices Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Aha! The column vectors of a transformation matrix are the images of the base vectors! That gives us an easy method of finding the matrix for a given linear transformation. Why? Because matrix multiplication is a linear transformation.

17 Finding matrices Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Let s look at carthesian coordinates, where each vector v can be represented as a linear combination of the base vectors: v = ( ) x = x y ( ) 1 + y 0 ( ) 0 1 If we apply a linear transformation T to this vector, we get: ( x T ( y ) ) = T (x ( ) 1 + y 0 ( ) 0 ) = xt ( 1 ( ) ( ) 1 0 ) + yt ( ) 0 1

18 Finding matrices: example Rotation (counterclockwise, angle φ): Definition Examples Finding matrices Compositions of transformations Transposing normal vectors ( cos φ ) sin φ sin φ cos φ

19 Example: reflection and scaling Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiple transformations can be combined into one. Here, we first do a reflection in the line y = x, and then we scale with a factor 5 in x-direction, and a factor 2 in y-direction: ( ) ( ) = ( ) Remember: matrix multiplication is associative, i.e. A(Bx) = (AB)x.

20 Example: reflection and scaling Definition Examples Finding matrices Compositions of transformations Transposing normal vectors But: Matrix multiplication is not commutative, i.e. in general AB BA, for example: ( ) ( ) ( ) x = y ( ) ( ) ( ) x = y Mind the order! ( ( ) ( ) x = y ) ( ) x = y ( ) 5y 2x ( ) 2y 5x

21 Transposing normal vectors Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Unfortunately, normal vectors are not always transformed properly. E.g. look at shearing, where tangent vectors are correctly transformed but normal vectors not. To transform a normal vector n correctly under a given linear transformation A, we have to apply the matrix (A 1 ) T.

22 Transposing normal vectors Definition Examples Finding matrices Compositions of transformations Transposing normal vectors To transform a normal vector n correctly under a given linear transformation A, we have to apply the matrix (A 1 ) T. We know that tangent vectors are tranformed correctly: At = t A. But this is not necessarily true for normal vectors: An n A. Goal: find a matrix N A that transforms n correctly, i.e. find N A with N A n = n N where n N is the correct normal vector of the transformed surface.

23 Transposing normal vectors Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: obviously, for shearing, normal vectors behave funny. But what about rotations? And scalings (uniform and non-uniform)? exercise in the tutorials

24 Problem and solution: homogeneous coordinates More complex transformations So now we know how to determine matrices for a given transformation. Let s try another one: Q: what is the matrix for a rotation of 90 about the point (2, 1)?

25 Problem and solution: homogeneous coordinates More complex transformations We can form our transformation by composing three simpler transformations: Translate everything such that the center of rotation maps to the origin. Rotate about the origin. Revert the translation from the first step. Q: but what is the matrix for a translation?

26 Problem and solution: homogeneous coordinates More complex transformations Translation is not a linear transformation. With linear translations we get: ( ) a11 x + a Ax = 12 y a 21 x + a 22 y But we need something like: ( ) x + xt x + y t We can do this with a combination of linear transformations and translations called affine transformations.

27 Problem and solution: homogeneous coordinates Homogeneous coordinates Observation: Shearing in 2D smells a lot like translation in 1D...

28 Problem and solution: homogeneous coordinates Homogeneous coordinates... and shearing in 3D smells like translation in 2D (... and shearing in 4D... )

29 Problem and solution: homogeneous coordinates Homogeneous coordinates Idea: Move one dimension higher by adding so called homogeneous coodinates.

30 Problem and solution: homogeneous coordinates Homogeneous coordinates Shearing in 3D based on the z-coordinate is a simple generalization of 2D shearing: 1 0 x t x x + x t z 0 1 y t y = y + y t z z z Or if we only look at the plane z = 1: 1 0 x t x x + x t 0 1 y t y = y + y t

31 Problem and solution: homogeneous coordinates Homogeneous coordinates Translations in 2D can be represented as shearing in 3D by looking at the plane z = 1. ( ) xt The matrix for a translation over the vector t = is 1 0 x t 0 1 y t How should we represent points then? And vectors? y t

32 Problem and solution: homogeneous coordinates Homogeneous coordinates (i.e. linear transformations and translations) can then be done with simple matrix multiplication if we add homogeneous coordinates, i.e. (in 2D): a third coordinate z = 1 x to each location: y 1 a third coordinate z = 0 x to each vector: y 0 a third row ( ) to each matrix:

33 Problem and solution: homogeneous coordinates Homogeneous coordinates Note: this extension is consistent with what we learned so far, e.g. Scaling and moving a location (or an object ): 1 0 x t s x s 1 0 x t x s 1x + x t 0 1 y t 0 s 2 0 y = 0 s 2 y t y = s 2y + y t Scaling (but not moving!) a vector: s 1 0 x t x s 1x 0 s 2 y t y = s 2y

34 Problem and solution: homogeneous coordinates : example Q: What is the matrix for reflection in the line y = x + 5? Hint: move everything to the origin, reflect, and the move everything back.

35 Problem and solution: homogeneous coordinates : example 1. Translation of y = x + 5 to y = x 1 0 x t x x + x t 0 1 y t y = y + y t Reflection on y = x: ( 0 ) Translate y = x back to y = x + 5

36 Problem and solution: homogeneous coordinates Q: The matrix for reflection in the line y = x + 5 is Q: what is the significance of the columns of the matrix?

37 Problem and solution: homogeneous coordinates Q: what is the significance of the columns of the matrix? Aha! The last column vector is the image of the origin after applying the affine transformation (Remember: The other columns are the image of the base vectors under the linear transformation) (Also remember: The coordinates in the last row are called homogeneous coordinates)

38 Problem and solution: homogeneous coordinates : example Does that give us a faster way to find matrices? E.g.: What is the matrix for rotation about the point (2, 2)? Rotation in general: cos φ sin φ? sin φ sin φ? rotation around (2,2):

39 Problem and solution: homogeneous coordinates : example Q: What is the matrix for rotation about the point (2, 2)?

40 Moving up a dimension Rotations in 3D Reflections in 3D For scaling, we have three scaling factors on the diagonal of the matrix. s x s y s z Shearing can be done in either x-, y-, or z-direction (or a combination thereof). For example, shearing in x-direction: 1 d y d z x x + d y y + d z z y = y z z For translations (and affine transformations in general), we use 4 4 matrices.

41 Moving up a dimension Rotations in 3D Reflections in 3D You see: transformations in 3D are very similar to those in 2D. Scaling and shearing are straightforward generalizations of the 2D cases. Reflection is done with respect to planes Rotation is done about directed lines.

42 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the z-axis? Rotation in 2D (x y-plane): ( ) cos φ sin φ sin φ cos φ

43 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the z-axis? Rotation in 2D (x y-plane): ( cos φ ) sin φ sin φ cos φ Rotation in 3D around z-axis: cos φ sin φ 0 sin φ cos φ

44 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the z-axis? Q: What is the matrix for a rotation of angle φ about the vector (3, 0, 4)?

45 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the z-axis? Q: What is the matrix for a rotation of angle φ about the vector (3, 0, 4)? Q: What is the matrix for a rotation of angle φ about the directed line (0, 1, 2) + t(3, 0, 4) t > 0?

46 Moving up a dimension Rotations in 3D Reflections in 3D : rotations We need a 3D transformation that rotates around an arbitrary vector w. How can we do that? Idea: 1 Rotate vector w to z-axis 2 Do 3D rotation around z-axis 3 Rotate everything back to original position

47 Moving up a dimension Rotations in 3D Reflections in 3D : rotations We already know how to rotate around the z-axis: cos φ sin φ 0 sin φ cos φ Note: Knowing that the inverse of an orthogonal matrix is always it s transposed saves you some calculation here.

48 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Hmm, but we only have the vector w. How do we get a whole coordinates system u, v, w?

49 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: Is such a rotation unique? Does it need to be?

50 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane?

51 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane?

52 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 0?

53 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 0? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 11?

54 Moving up a dimension Rotations in 3D Reflections in 3D

55 Moving up a dimension Rotations in 3D Reflections in 3D Important dates Wednesday, Sept 23: Tutorials and programming labs, but no lecture Monday, Sept 28, 8:30-10:30h: Midterm exam and therefore Programming labs, but no lecture and tutorials Wednesday, Sept 30: Lecture and programming labs, but no tutorials Friday, Oct 2, 12h00: Deadline programming assignment P1

56 Moving up a dimension Rotations in 3D Reflections in 3D Midterm exam Monday, Sept 28, 8:30-10:30h, EDUC-ALFA Crosscheck time and location in OSIRIS Come in time Bring a pen (no pencil, no red pens), and your student ID Coverage: lecture 1-5, tutorials 1-3 If you followed the lecture,... read the textbook,... and actively did the exercises you should be fine

57 Moving up a dimension Rotations in 3D Reflections in 3D Midterm exam No books, notes, electronic tools, etc. allowed Note: that includes cellphones (even if you just want to use them as a watch!) Note on formulas: We expect that you know the important ones We expect that you have enough knowledge to judge what formulas are considered important for graphics A hint: scaler product, cross product, sin/cos are important and graphics programmers should know them out of their head Complex relationships and trigonometrical characteristics are also important but it s not required to know them out of your head (i.e. they will be provided or not be necessary to solve the exam questions)

58 Moving up a dimension Rotations in 3D Reflections in 3D Programming assignment P1 Deadline: Friday, Oct 2, 12h Online submission (see website) System closes later (13h), but don t rely on that Grading: Based on your delivery, but we will do crosschecks and draw random groups for a personal meeting