1. MAE152 Computer Graphicsfor Scientists and Engineers Revision of Matrices Relevance to Graphics

2. Where We’re Going §Today’s lectures: l Mathematical Foundations l The graphics pipeline: the big picture l Rigid-body transforms l Homogeneous coordinates

3. Mathematical Foundations §A brief, informal review of some of the mathematical tools employed in graphics l Geometry (2D, 3D) l Trigonometry l Vector and affine spaces Points, vectors, and coordinates l Dot and cross products l Linear transforms and matrices

4. 2D Geometry §Know your high-school geometry: l Total angle around a circle is 360° or 2π radians l When two lines cross: Opposite angles are equivalent Angles along line sum to 180° l Similar triangles: All corresponding angles are equivalent Corresponding pairs of sides have the same length ratio and are separated by equivalent angles Any corresponding pairs of sides have same length ratio

5. Trigonometry §Sine: “opposite over hypotenuse” §Cosine: “adjacent over hypotenuse” §Tangent: “opposite over adjacent” §Unit circle definitions: l sin (  ) = x l cos (  ) = y l tan (  ) = x/y l Etc… (x, y)

6. 3D Geometry §To model, animate, and render 3D scenes, we must specify: l Location l Displacement from arbitrary locations l Orientation §We’ll look at two types of spaces: l Vector spaces l Affine spaces §The distinction is often sloppy or loose

7. Vector Spaces §Two types of elements: Scalars (real numbers):  … l Vectors (n-tuples): u, v, w, … §Supports two operations: l Addition operation u + v, with: Identity 0 v + 0 = v Inverse - v + (- v ) = 0 l Scalar multiplication: Distributive rule:  ( u + v ) =  ( u ) +  ( v ) (  +  ) u =  u +  u

8. Vector Spaces §A linear combination of vectors results in a new vector: v =  1 v 1 +  2 v 2 + … +  n v n §If the only set of scalars such that  1 v 1 +  2 v 2 + … +  n v n = 0 is  1 =  2 = … =  3 = 0 then we say the vectors are linearly independent §The dimension of a space is the greatest number of linearly independent vectors possible in a vector set §For a vector space of dimension n, any set of n linearly independent vectors form a basis

9. Vector Spaces: A Familiar Example §Our common notion of vectors in a 2D plane is (you guessed it) a vector space: l Vectors are “arrows” rooted at the origin l Scalar multiplication “streches” the arrow, changing its length (magnitude) but not its direction l Addition uses the “trapezoid rule”: u+v y x u v

10. Vector Spaces: Basis Vectors §Given a basis for a vector space: l Each vector in the space is a unique linear combination of the basis vectors l The coordinates of a vector are the scalars from this linear combination l Best-known example: Cartesian coordinates Draw example on the board l Note that a given vector v will have different coordinates for different bases

11. Vectors And Point §We commonly use vectors to represent: l Points in space (i.e., location) l Displacements from point to point l Direction (i.e., orientation) §But we want points and directions to behave differently l Ex: To translate something means to move it without changing its orientation l Translation of a point = different point l Translation of a direction = same direction

12. Affine Spaces §To be more rigorous, we need an explicit notion of position §Affine spaces add a third element to vector spaces: points (P, Q, R, …) §Points support these operations l Point-point subtraction: Q - P = v Result is a vector pointing from P to Q l Vector-point addition: P + v = Q Result is a new point l Note that the addition of two points is not defined P Q v

13. Affine Spaces §Points, like vectors, can be expressed in coordinates l The definition uses an affine combination l Net effect is same: expressing a point in terms of a basis §Thus the common practice of representing points as vectors with coordinates

14. Affine Lines: An Aside §Parametric representation of a line with a direction vector d and a point P 1 on the line: P(  ) = P origin +  d  Restricting 0   produces a ray  Setting d to P - Q and restricting 0    1 produces a line segment between P and Q

15. Dot Product §The dot product or, more generally, inner product of two vectors is a scalar: v 1 v 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 (in 3D) §Useful for many purposes l Computing the length of a vector: length(v) = sqrt(v v) l Normalizing a vector, making it unit-length l Computing the angle between two vectors: u v = |u| |v| cos(θ) l Checking two vectors for orthogonality l Projecting one vector onto another θ u v

16. Cross Product §The cross product or vector product of two vectors is a vector: §The cross product of two vectors is orthogonal to both §Right-hand rule dictates direction of cross product

17. Linear Transformations §A linear transformation: l Maps one vector to another l Preserves linear combinations §Thus behavior of linear transformation is completely determined by what it does to a basis §Turns out any linear transform can be represented by a matrix

18. Matrices §By convention, matrix element M rc is located at row r and column c: §By (OpenGL) convention, vectors are columns:

19. Matrices §Matrix-vector multiplication applies a linear transformation to a vector: §Recall how to do matrix multiplication

20. Matrix Transformations §A sequence or composition of linear transformations corresponds to the product of the corresponding matrices l Note: the matrices to the right affect vector first l Note: order of matrices matters! §The identity matrix I has no effect in multiplication §Some (not all) matrices have an inverse:

21. 3-D Graphics: A Whirlwind Tour Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

22. The Display You Know Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

23. The Framebuffer You Know Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

24. The Rendering Pipeline Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay §Why do we call it a pipeline?

25. 2-D Rendering You Know Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

26. The Rendering Pipeline: 3-D Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

27. The Rendering Pipeline: 3-D Modeling Transforms Scene graph Object geometry Lighting Calculations Viewing Transform Clipping Projection Transform Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices

28. The Rendering Pipeline: 3-D Scene graph Object geometry Lighting Calculations Clipping Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices Modeling Transforms Viewing Transform Projection Transform

29. Rendering: Transformations §So far, discussion has been in screen space §But model is stored in model space (a.k.a. object space or world space) §Three sets of geometric transformations: l Modeling transforms l Viewing transforms l Projection transforms

30. Rendering: Transformations §Modeling transforms l Size, place, scale, and rotate objects parts of the model w.r.t. each other l Object coordinates  world coordinates Z X Y X Z Y

31. Rendering: Transformations §Viewing transform l Rotate & translate the world to lie directly in front of the camera Typically place camera at origin Typically looking down -Z axis l World coordinates  view coordinates

32. Rendering: Transformations §Projection transform l Apply perspective foreshortening Distant = small: the pinhole camera model l View coordinates  screen coordinates

33. Rendering: Transformations §All these transformations involve shifting coordinate systems (i.e., basis sets) §That’s what matrices do… §Represent coordinates as vectors, transforms as matrices §Multiply matrices = concatenate transforms!

34. Rendering: Transformations §Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector l Denoted [x, y, z, w] Note that typically w = 1 in model coordinates l To get 3-D coordinates, divide by w: [x’, y’, z’] = [x/w, y/w, z/w] §Transformations are 4x4 matrices §Why? To handle translation, rotation and projection uniformly

35. Translations §For convenience we usually describe objects in relation to their own coordinate system l Solar system example §We can translate or move points to a new position by adding offsets to their coordinates: l Note that this translates all points uniformly

36. 2-D Rotation  (x, y) (x’, y’) x’ = x cos(  ) - y sin(  ) y’ = x sin(  ) + y cos(  ) (Draw it)

37. 2-D Rotations §Rotations in 2-D are easy: §3-D is more complicated l Need to specify an axis of rotation l Common pedagogy: express rotation about this axis as the composition of canonical rotations Canonical rotations: rotation about X-axis, Y-axis, Z-axis

38. 3-D Rotations §Basic idea: l Using rotations about X, Y, Z axes, rotate model until desired axis of rotation coincides with Z-axis l Perform rotation in the X-Y plane (i.e., about Z-axis) l Reverse the initial rotations to get back into the initial frame of reference §Objections: l Difficult & error prone l Ambiguous: several combinations about the canonical axis give the same result

39. Scaling §Scaling a coordinate means multiplying each of its components by a scalar §Uniform scaling means this scalar is the same for all components:  2 2

40. Scaling §Non-uniform scaling: different scalars per component: §How can we represent this in matrix form? X  2, Y  0.5

41. Scaling §Scaling operation: §Or, in matrix form: scaling matrix

42. 2-D Rotation §This is easy to capture in matrix form: §3-D is more complicated l Need to specify an axis of rotation l Simple cases: rotation about X, Y, Z axes

43. 3-D Rotation §What does the 3-D rotation matrix look like for a rotation about the Z-axis? l Build it coordinate-by-coordinate

44. 3-D Rotation §What does the 3-D rotation matrix look like for a rotation about the Y-axis? l Build it coordinate-by-coordinate

45. 3-D Rotation §What does the 3-D rotation matrix look like for a rotation about the X-axis? l Build it coordinate-by-coordinate

46. Translation Matrices? §We can composite scale matrices just as we did rotation matrices §But how to represent translation as a matrix? §Answer: with homogeneous coordinates

47. Homogeneous Coordinates §Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector (Note that typically w = 1 in object coordinates)

48. Homogeneous Coordinates §Homogeneous coordinates seem unintuitive, but they make graphics operations much easier §Our transformation matrices are now 4x4:

49. Homogeneous Coordinates §Homogeneous coordinates seem unintuitive, but they make graphics operations much easier §Our transformation matrices are now 4x4:

50. Homogeneous Coordinates §Homogeneous coordinates seem unintuitive, but they make graphics operations much easier §Our transformation matrices are now 4x4:

51. Homogeneous Coordinates §Homogeneous coordinates seem unintuitive, but they make graphics operations much easier §Our transformation matrices are now 4x4:

52. Homogeneous Coordinates §How can we represent translation as a 4x4 matrix? §A: Using the rightmost column:

53. Rendering: Transformations §OpenGL caveats: l All modeling transforms and the viewing transform are concatenated into the modelview matrix l A stack of modelview matrices is kept l The projection transform is stored separately in the projection matrix §See Chapter 3 of the OpenGL book

54. The Rendering Pipeline: 3-D Modeling Transforms Scene graph Object geometry Lighting Calculations Viewing Transform Clipping Projection Transform Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices

55. Rendering: Lighting §Illuminating a scene: coloring pixels according to some approximation of lighting l Global illumination: solves for lighting of the whole scene at once l Local illumination: local approximation, typically lighting each polygon separately §Interactive graphics (e.g., hardware) does only local illumination at run time

56. The Rendering Pipeline: 3-D Modeling Transforms Scene graph Object geometry Lighting Calculations Viewing Transform Clipping Projection Transform Result: All vertices of scene in shared 3-D “world” coordinate system All vertices of scene in shared 3-D “world” coordinate system Vertices shaded according to lighting model Vertices shaded according to lighting model Scene vertices in 3-D “view” or “camera” coordinate system Scene vertices in 3-D “view” or “camera” coordinate system Exactly those vertices & portions of polygons in view frustum Exactly those vertices & portions of polygons in view frustum 2-D screen coordinates of clipped vertices 2-D screen coordinates of clipped vertices

57. Rendering: Clipping §Clipping a 3-D primitive returns its intersection with the view frustum: §See Foley & van Dam section 19.1

58. Rendering: Clipping §Clipping is tricky! In: 3 vertices Out: 6 vertices Clip In: 1 polygon Out: 2 polygons

59. The Rendering Pipeline: 3-D Transform Illuminate Transform Clip Project Rasterize Model & Camera Parameters Rendering Pipeline FramebufferDisplay

60. Modeling: The Basics §Common interactive 3-D primitives: points, lines, polygons (i.e., triangles) §Organized into objects l Collection of primitives, other objects l Associated matrix for transformations §Instancing: using same geometry for multiple objects l 4 wheels on a car, 2 arms on a robot

61. Modeling: The Scene Graph §The scene graph captures transformations and object-object relationships in a DAG §Objects in black; blue arrows indicate instancing and each have a matrix Robot BodyHead ArmTrunkLegEyeMouth

62. Modeling: The Scene Graph §Traverse the scene graph in depth-first order, concatenating transformations §Maintain a matrix stack of transformations ArmTrunk Leg EyeMouth HeadBody Robot Foot Matrix Stack Visited Unvisited Active

63. Modeling: The Camera §Finally: need a model of the virtual camera l Can be very sophisticated Field of view, depth of field, distortion, chromatic aberration… l Interactive graphics (OpenGL): Pinhole camera model –Field of view –Aspect ratio –Near & far clipping planes Camera pose: position & orientation

64. Modeling: The Camera §These parameters are encapsulated in a projection matrix l Homogeneous coordinates  4x4 matrix! l See OpenGL Appendix F for the matrix §The projection matrix premultiplies the viewing matrix, which premultiplies the modeling matrices l Actually, OpenGL lumps viewing and modeling transforms into modelview matrix

65. Rigid-Body Transforms §Goal: object coordinates  world coordinates §Idea: use only transformations that preserve the shape of the object l Rigid-body or Euclidean transforms l Includes rotation, translation, and scale §To reiterate: we will represent points as column vectors:

66. Vectors and Matrices §Vector algebra operations can be expressed in this matrix form l Dot product: l Cross product: Note: use right-hand rule!

67. 3-D Rotation §General rotations in 3-D require rotating about an arbitrary axis of rotation §Deriving the rotation matrix for such a rotation directly is difficult l But possible, see McMillan’s lectures §Standard approach: express general rotation as composition of canonical rotations l Rotations about X, Y, Z

68. Composing Canonical Rotations §Goal: rotate about arbitrary vector A by  l Idea: we know how to rotate about X,Y,Z l So, rotate about Y by  until A lies in the YZ plane, Then rotate about X by  until A coincides with +Z, Then rotate about Z by  Then reverse the rotation about X (by -  ) Then reverse the rotation about Y (by -  )

69. Composing Canonical Rotations §First: rotating about Y by  until A lies in YZ l Draw it… §How exactly do we calculate  ? l Project A onto XZ plane l Find angle  to X:  = -(90° -  ) =  - 90 ° §Second: rotating about X by  until A lies on Z §How do we calculate  ?

70. 3-D Rotation Matrices §So an arbitrary rotation about A composites several canonical rotations together §We can express each rotation as a matrix §Compositing transforms == multiplying matrices §Thus we can express the final rotation as the product of canonical rotation matrices §Thus we can express the final rotation with a single matrix!

71. Compositing Matrices §So we have the following matrices: p: The point to be rotated about A by  R y  : Rotate about Y by  R x  : Rotate about X by  R z  : Rotate about Z by  R x  -1 : Undo rotation about X by  R y  -1 : Undo rotation about Y by  §In what order should we multiply them?

72. Compositing Matrices §Short answer: the transformations, in order, are written from right to left l In other words, the first matrix to affect the vector goes next to the vector, the second next to the first, etc. §So in our case: p’ = R y  -1 R x  -1 R z  R x  R y  p

73. Rotation Matrices §Notice these two matrices: R x  : Rotate about X by  R x  -1 : Undo rotation about X by  §How can we calculate R x  -1 ?

74. Rotation Matrices §Notice these two matrices: R x  : Rotate about X by  R x  -1 : Undo rotation about X by  §How can we calculate R x  -1 ? l Obvious answer: calculate R x (-  ) l Clever answer: exploit fact that rotation matrices are orthonormal

75. Rotation Matrices §Notice these two matrices: R x  : Rotate about X by  R x  -1 : Undo rotation about X by  §How can we calculate R x  -1 ? l Obvious answer: calculate R x (-  ) l Clever answer: exploit fact that rotation matrices are orthonormal §What is an orthonormal matrix? §What property are we talking about?

76. Rotation Matrices §Orthonormal matrix: l orthogonal (columns/rows linearly independent) l Columns/rows sum to 1 §The inverse of an orthogonal matrix is just its transpose: