1. Welcome to CSc 830 Advanced Computer Graphics
By Ilmi Yoon
Based on Lecture note from David Luebke & Pradondet Nilagupta

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

3. Mathematical Foundations
FvD appendix gives good review
I’ll give a brief, informal review of some of the mathematical tools we’ll employ
Geometry (2D, 3D)
Trigonometry
Vector and affine spaces
Points, vectors, and coordinates
Dot and cross products
Linear transforms and matrices

4. 2D Geometry Know your high-school geometry:
Total angle around a circle is 360° or 2π radians
When two lines cross:
Opposite angles are equivalent
Angles along line sum to 180°
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:
sin () = x
cos () = y
tan () = x/y
Etc…
(x, y)

6. 3D Geometry To model, animate, and render 3D scenes, we must specify:
Location
Displacement from arbitrary locations
Orientation
We’ll look at two types of spaces:
Vector spaces
Affine spaces
We will often be sloppy about the distinction

7. Vector Spaces Two types of elements: Supports two operations:
Scalars (real numbers): a, b, g, d, …
Vectors (n-tuples): u, v, w, …
Supports two operations:
Addition operation u + v, with:
Identity 0 v + 0 = v
Inverse - v + (-v) = 0
Scalar multiplication:
Distributive rule: a(u + v) = a(u) + a(v)
(a + b)u = au + bu

8. Vector Spaces A linear combination of vectors results in a new vector:
v = a1v1 + a2v2 + … + anvn
If the only set of scalars such that
a1v1 + a2v2 + … + anvn = 0
is a1 = a2 = … = a3 = 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:
Vectors are “arrows” rooted at the origin
Scalar multiplication “streches” the arrow, changing its length (magnitude) but not its direction
Addition uses the “trapezoid rule”:
u+v y x u v

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

11. Vectors And Point We commonly use vectors to represent:
Points in space (i.e., location)
Displacements from point to point
Direction (i.e., orientation)
But we want points and directions to behave differently
Ex: To translate something means to move it without changing its orientation
Translation of a point = different point
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
Point-point subtraction: Q - P = v
Result is a vector pointing from P to Q
Vector-point addition: P + v = Q
Result is a new point
Note that the addition of two points is not defined Q v P

13. Affine Spaces Points, like vectors, can be expressed in coordinates
The definition uses an affine combination
Net effect is same: expressing a point in terms of a basis
Thus the common practice of representing points as vectors with coordinates (see FvD)
Analogous to equating points and integers in C
Be careful to avoid nonsensical operations

14. Affine Lines: An Aside
Parametric representation of a line with a direction vector d and a point P1 on the line:
P(a) = Porigin + ad
Restricting 0  a produces a ray
Setting d to P - Q and restricting 0  a  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:
v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)
Useful for many purposes
Computing the length of a vector: length(v) = sqrt(v • v)
Normalizing a vector, making it unit-length
Computing the angle between two vectors:
u • v = |u| |v| cos(θ)
Checking two vectors for orthogonality
Projecting one vector onto another v θ u

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:
Maps one vector to another
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 Mrc 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
Note: the matrices to the right affect vector first
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
Clip
Project
Rasterize
Model & Camera Parameters
Rendering Pipeline
Framebuffer
Display

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

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

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

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

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

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

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

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

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

32. Rendering: Transformations
Projection transform
Apply perspective foreshortening
Distant = small: the pinhole camera model
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
Denoted [x, y, z, w]
Note that typically w = 1 in model coordinates
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 and projection

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

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

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

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

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

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

41. The Rendering Pipeline: 3-D
Transform
Illuminate
Clip
Project
Rasterize
Model & Camera Parameters
Rendering Pipeline
Framebuffer
Display

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

43. 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
Head
Body
Mouth
Eye
Leg
Trunk
Arm

44. Modeling: The Scene Graph
Traverse the scene graph in depth-first order, concatenating transformations
Maintain a matrix stack of transformations
Robot
Visited
Head
Body
Unvisited
Mouth
Eye
Leg
Trunk
Arm
Matrix Stack
Active
Foot

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

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

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

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

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

50.  2-D Rotation (x’, y’) (x, y) x’ = x cos() - y sin()
y’ = x sin() + y cos()
(Draw it)

51. 2-D Rotations Rotations in 2-D are easy: 3-D is more complicated
Need to specify an axis of rotation
Common pedagogy: express rotation about this axis as the composition of canonical rotations
Canonical rotations: rotation about X-axis, Y-axis, Z-axis

52. 3-D Rotations Basic idea: Objections:
Using rotations about X, Y, Z axes, rotate model until desired axis of rotation coincides with Z-axis
Perform rotation in the X-Y plane (i.e., about Z-axis)
Reverse the initial rotations to get back into the initial frame of reference
Objections:
Difficult & error prone
Ambiguous: several combinations about the canonical axis give the same result

53. 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

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

55. Scaling
Scaling operation:
Or, in matrix form:
scaling matrix

56. 2-D Rotation This is easy to capture in matrix form:
3-D is more complicated
Need to specify an axis of rotation
Simple cases: rotation about X, Y, Z axes

57. 3-D Rotation
What does the 3-D rotation matrix look like for a rotation about the Z-axis?
Build it coordinate-by-coordinate

58. 3-D Rotation
What does the 3-D rotation matrix look like for a rotation about the Y-axis?
Build it coordinate-by-coordinate

59. 3-D Rotation
What does the 3-D rotation matrix look like for a rotation about the X-axis?
Build it coordinate-by-coordinate

60. 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
But possible, see McMillan’s lectures
Standard approach: express general rotation as composition of canonical rotations
Rotations about X, Y, Z

61. Composing Canonical Rotations
Goal: rotate about arbitrary vector A by 
Idea: we know how to rotate about X,Y,Z
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 -)

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

63. Composing Canonical Rotations
Why are we slogging through all this tedium?
A: Because you’ll have to do it on the test

64. 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!

65. Compositing Matrices So we have the following matrices:
p: The point to be rotated about A by 
Ry : Rotate about Y by 
Rx  : Rotate about X by 
Rz : Rotate about Z by 
Rx  -1: Undo rotation about X by 
Ry-1 : Undo rotation about Y by 
In what order should we multiply them?

66. Compositing Matrices
Short answer: the transformations, in order, are written from right to left
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’ = Ry-1 Rx  -1 Rz Rx  Ry p

67. Rotation Matrices Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
How can we calculate Rx  -1?

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

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

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

71. 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

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

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

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

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

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

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