Lecture 4: Transforms. Computer Graphics CMU /15-662, Fall 2016

Transcription

1 Lecture 4: Transforms Computer Graphics CMU /15-662, Fall 2016

2 Brief recap from last class How to draw a triangle - Why focus on triangles, and not quads, pentagons, etc? - What was specific to triangles in what we discussed last class?

3 What in the world is this?

4 Cube (-1, 1, -1) (-1, 1, 1) (1, 1, 1) (1, 1, -1) (-1, -1, 1) (1, -1, 1) (-1, -1, -1) (1, -1, -1)

5 Cube man Stretched out cube moved up The original cube Squishy cube moved to the right Slanty cube moved down and to the left

6 f transforms x to f(x) f(x) x

7 And what is our favorite type of transformation?

8 Linear transforms e 2 e 1 But what does it mean?

9 Linear transforms f(u + v) = f(u) + f(v) f(au) = af(u)

10 Linear transforms If a map can be expressed as f(u) =σ m i=1 u i a i with fixed vectors a i, then it is linear

11 Linear transforms e 2 e 1 Do you know - what u 1 and u 2 are? - what a 1 and a 2 are?

12 Linear transforms e 2 e 1 - u is a linear combination of e 1 and e 2 - f(u) is that same linear combination of a and a a and a are f(e ) and f(e ) by knowing what e and e map to, you know how 1 2 to map the entire space!

13 An example: Coordinate transformations j: Ƹ 2i + j u j i: Ƹ i + j My friend me i My friend says, look at 3 o clock (in their coordinate frame that means one forward and one to the right )! Where should I look? Direction in my friend s coordinate frame f u = f(u 1 Ƹ i + u 2 Ƹ j) = u 1 f i Ƹ + u f 2 1 j Ƹ = u u Same direction in my coordinate frame

14 Linear maps In graphics we often talk about changing coordinate frames (go from local to world to camera to screen coordinates) Equally useful to think about maps transforming a space (and everything in it!)

15 Let s look at some transforms that are important in graphics How do you formally tell a computer that this cube should be squished and slanty?

16 Scale Uniform scale: Non-uniform scale S x = x 1 ae 1 + x 2 be 2 Non axis-aligned scaling??

17 Is uniform scale a linear transform? Yes!

18 Rotation = rotate counter-clockwise by

19 Rotation = rotate counter-clockwise by As angle changes, points move along circular trajectories.

20 Rotation = rotate counter-clockwise by As angle changes, points move along circular trajectories. Shape (distance between any two points) does not change! (Rigid or isometric transformation)

21 Rotation What does look like? r α From x, compute α and r Write down R (x) as a function of α, θ and r θ (i.e. vector (r,0) rotated by α + θ) Apply sum of angle formulae Fine, but remember, we only need to know how e 1 and e 2 are transformed!

22 Rotation So, what happens to vectors (1, 0) and (0, 1) after rotation by θ? Answer: R θ e 1 = cos θ, sin θ = a 1 -sin θ R θ e 2 = sin θ, cos θ = a 2 cos θ sin θ So: cos θ R θ x = x 1 a 1 + x 2 a 2

23 Is rotation linear? Yes!

24 Rotation Note: all points are rotated about the origin - By the way, what are we actually transforming here? What if we want to rotate about another point?

25 Reflection x 3 x 2 Re y x 2 Re y x 3 x 0 x 1 Re y x 1 Re y x 0 Re y x : reflection about y-axis Reflections change handedness Do you know what Re y x looks like? Is reflection a linear transform? Do you know how to reflect about an arbitrary axis?

26 Shear (in x direction) x 3 x 2 H x 3 H x 2 x 0 x 1 H x 0 H x 1 What does H x look like? H a x = x x 2 a 1 Is shearing a linear transformation?

27 Translation T b x 3 T b x 2 x 3 x 2 T b x 0 T b x 1 x 0 x 1 Let s write T b x in the form? T x = x b 1? + x 2?? such that T b x = x + b

28 Is translation linear? T b x +T b y x b T b x x + y b T b x + y b T b y y No. Translation is affine.

29 When at first you don t succeed We ll turn affine transformations into linear ones via Homogeneous coordinates (aka projective coordinates) But first, let s use matrix notation to represent linear transforms

30 Linear transforms as matrix-vector products A x a 11 a 12 a 21 a 22 x 1 x 2 = a 11x 1 + a 12 x 2 a 21 x 1 + a 22 x 2 = x 1 a 11 a 21 + x 2 a 12 a 22 = x 1 a 1 + x 2 a 2 f(x) =σ m i=1 x i a i = Ax

31 Linear transforms as matrix-vector products Change of coordinate systems f x = x x = x j: Ƹ 2i + j u j i: Ƹ i + j My friend me i

32 Linear transforms as matrix-vector products Non-uniform scale S x = x 1 ae 1 + x 2 be 2 = a 0 0 b x

33 Linear transforms as matrix-vector products Rotation R θ e 1 = cos θ, sin θ = a 1 R θ e 2 = sin θ, cos θ = a 2 R θ x = x 1 a 1 + x 2 a 2 = cos θ sin θ sin θ cos θ x

34 Linear transforms as matrix-vector products Shear H x = x x 2 a 1 = 1 a 0 1 x

35 Linear transforms as matrix-vector products Translation Not a linear map* *when we re using Cartesian coordinates

36 2D homogeneous coordinates (2D-H) Key idea: lift 2D points to a 3D space x 1 So the point (x 1, x 2 ) is represented as the 3-vector: x 2 1 And 2D transforms are represented by 3x3 matrices For example: 2D rotation in homogeneous coordinates: cos θ sin θ 0 sin θ cos θ x 1 x 2 1 Q: how do the transforms we ve seen so far affect the last coordinate?

37 Translation in 2D-H coords Translation expressed as 3x3 matrix multiplication: 1 0 b 1 x 1 x 1 + b 1 T x = x + b = 0 1 b x 2 1 = x 2 +b 2 1 In homogeneous coordinates, translation is a linear transformation!

38 Translation in 2D-H coords What is this magic? 1 0 b b x 1 x 2 x 3 = x 1 + b 1 x 2 +b 2 x 3 Translation in 2D homogeneous coordinates is equivalent to shearing along x and y axes a linear operation. But why is x 3 set to 1? Could it not be instead?

39 Homogeneous coordinates w wx 1, wx 2, w, w > 0 - Homogenous coordinates are scale invariant w = 1 (x 1, x 2, 1) - x and wx correspond to the same 2D point (divide by w to convert 2D-H back to 2D) y - 2D-H points with w = 0 correspond to 2D vectors (technically, points at infinity) x (x 1, x 2 ) - In homogenous coordinates, points and vectors are distinguishable from each other!

40 Homogeneous coordinates: points vs. vectors w wx 1, wx 2, w, w > 0 w = 1 (x 1, x 2, 1) 1 0 b b x 1 x 2 1 y vs (x 1, x 2, 0) 1 0 b b x 1 x 2 0 x

41 Visualizing 2D transformations in 2D-H Original shape in 2D can be viewed as many copies, uniformly scaled by w. 2D rotation rotate around w 2D scale scale x and y; preserve w (Question: what happens to 2D shape if you scale x, y, and w uniformly?) 2D translate shear in xy

42 Summary so far We know how to transform (scale, rotate, reflect, shear, translate) 2D points and vectors - All these transforms are linear maps expressed as matrix-vector products when using (slightly) higher-dimensional homogenous coordinates - How about other types of transforms (e.g. rotate about an arbitrary point)? - How about 3D transforms?

43 Onto more complex transforms How would you transform this object such that it gets twice as large? - but remains where it is

44 Composition of basic transforms Scale by 0.5, then translate by (3,1) Translate by (3,1), then scale by 0.5 Note 1: order of composition matters! Note 2: common source of bugs!

45 How do we compose linear transforms? Compose linear transforms via matrix multiplication. Enables simple & efficient implementation: reduce complex chain of transforms to a single matrix.

46 How would you perform these transformations?

47 Common pattern: rotation about point x Step 1: translate by -x Step 2: rotate Step 3: translate by x Q: In homogenous coordinates, what does the corresponding transformation matrix look like?

48 Exercise Reflection about an arbitrary line