1. Linear Algebra (Mathematics for CG) Reading: HB Appendix A
Computer Graphics
Last Updated: 13-Jan-12
Linear Algebra
(Mathematics for CG)
Reading: HB Appendix A

2. Notation: Scalars, Vectors, Matrices
(lower case, italic)
Vector
(lower case, bold)
Matrix
(upper case, bold)

3. Vectors
Arrow: Length and Direction
Oriented segment in 2D or 3D space

4. Column vs. Row Vectors Row vectors Column vectors
Switch back and forth with transpose

5. Vector-Vector Addition
Add: vector + vector = vector
Parallelogram rule
tail to head, complete the triangle
geometric
algebraic
examples:

6. Vector-Vector Subtraction
Subtract: vector - vector = vector

7. Vector-Vector Subtraction
Subtract: vector - vector = vector
argument reversal

8. Scalar-Vector Multiplication
Multiply: scalar * vector = vector
vector is scaled

9. Vector-Vector Multiplication
Multiply: vector * vector = scalar
Dot product or Inner product

10. Vector-Vector Multiplication
Multiply: vector * vector = scalar
dot product, aka inner product

11. Vector-Vector Multiplication
Multiply: vector * vector = scalar
dot product or inner product
geometric interpretation
lengths, angles
can find angle between two vectors

12. Dot Product Geometry Can find length of projection of u onto v
as lines become perpendicular,

13. Dot Product Example

14. Vector-Vector Multiplication
Multiply: vector * vector = vector
cross product

15. Vector-Vector Multiplication
multiply: vector * vector = vector
cross product
algebraic
geometric
parallelogram area
perpendicular to parallelogram

16. RHS vs. LHS Coordinate Systems
Right-handed coordinate system
Left-handed coordinate system
right hand rule:
index finger x, second finger y;
right thumb points up
left hand rule:
index finger x, second finger y;
left thumb points down

17. Basis Vectors
Take any two vectors that are linearly independent (nonzero and nonparallel)
can use linear combination of these to define any other vector:

18. Orthonormal Basis Vectors
If basis vectors are orthonormal (orthogonal (mutually perpendicular) and unit length)
we have Cartesian coordinate system
familiar Pythagorean definition of distance
Orthonormal algebraic properties

19. Basis Vectors and Origins
Coordinate system: just basis vectors
can only specify offset: vectors
Coordinate frame (or Frame of Reference): basis vectors and origin
can specify location as well as offset: points
It helps in analyzing the vectors and solving the kinematics.

20. Working with Frames F1 F1
Coefficient of i: Start from O, move along i and count until the line passing through P parallel to j.
Coefficient of j: Start from O, move along j and count until the line passing through P parallel to i.
Coefficient of i or j will be –ve if the movement is opposite to the direction of i or j, resp.

21. Working with Frames F1 F1
p = (3,-1)

22. Working with Frames F1 F1
p = (3,-1) F2 F2

23. Working with Frames F1 F1
p = (3,-1) F2 F2
p = (-1.5,2)

24. Working with Frames F1 F1
p = (3,-1) F2 F3 F2
p = (-1.5,2) F3

25. Working with Frames F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2) F1 F2

26. Named Coordinate Frames
Origin and basis vectors
Pick canonical frame of reference
then don’t have to store origin, basis vectors
just
convention: Cartesian orthonormal one on previous slide
Handy to specify others as needed
airplane nose, looking over your shoulder, ...
really common ones given names in CG
object, world, camera, screen, ...

27. Lines Slope-intercept form Implicit form y = mx + b y – mx – b = 0
Ax + By + C = 0
f(x,y) = 0

28. Implicit Functions Find where function is 0
An implicit function f(x,y) = 0 can be thought of as a height field where f is the height (top). A path where the height is zero is the implicit curve (bottom).
Find where function is 0
plug in (x,y), check if
= 0 on line
< 0 inside
> 0 outside

29. Implicit Circles circle is points (x,y) where f(x,y) = 0
points p on circle have property that vector from c to p dotted with itself has value r2
points p on the circle have property that squared distance from c to p is r2
points p on circle are those a distance r from center point c

30. Parametric Curves Parameter: index that changes continuously
(x,y): point on curve
t: parameter
Vector form

31. 2D Parametric Lines
start at point p0, go towards p1, according to parameter t
p(0) = p0, p(1) = p1

32. Linear Interpolation Parametric line is example of general concept
p goes through a at t = 0
p goes through b at t = 1
Linear
weights t, (1-t) are linear polynomials in t

33. Matrix-Matrix Addition
Add: matrix + matrix = matrix
Example

34. Scalar-Matrix Multiplication
Multiply: scalar * matrix = matrix
Example

35. Matrix-Matrix Multiplication
Can only multiply (n, k) by (k, m): number of left cols = number of right rows
legal
undefined

36. Matrix-Matrix Multiplication
row by column

37. Matrix-Matrix Multiplication
row by column

38. Matrix-Matrix Multiplication
row by column

39. Matrix-Matrix Multiplication
row by column

40. Matrix-Matrix Multiplication
row by column
Non commutative: AB != BA

41. Matrix-Vector Multiplication
points as column vectors: post-multiply

42. Matrices Transpose Identity Inverse
Note: Not all matrices are invertible

43. Matrices and Linear Systems
Linear system of n equations, n unknowns
Matrix form Ax=b

44. Exercise 1
Q 1: If the vectors, u and v in figure are coplanar with the sheet of paper, does the cross product v x u extend towards the reader or away assuming right-handed coordinate system?
For u = [1, 0, 0] & v = [2, 2, 1]. Find
Q 2: The length of v
Q 3: length of projection of u onto v
Q 4: v x u
Q 5: Cos t, if t is the angle between u & v

45. Exercise 2 F1 F3 F2 F1
p = (?, ?) F2
p = (?, ?) F3
p = (?, ?)

46. Exercise 3
Given P1(3,6,9) and P2(5,5,5). Find a point on the line with end points P1 & P2 at t = 2/3 using parametric equation of line, where t is the parameter.

47. Things to do
Reading for OpenGL practice: HB 2.9
Ready for the Quiz

48. References