1. 2IV60 Computer Graphics Basic Math for CG
Jack van Wijk
TU/e

2. Overview
Coordinates, points, vectors
Matrices
H&B A-1

3. nD Space nD Space: n (typically) n : number of dimensions Examples:
1D space: time, along a line or curve
2D space: plane, sphere
3D space: the world we live in
4D space: 3D + time
H&B A-2

4. Coordinates 2D Cartesian coordinates: Standard Screen (output, input)x y
(x,y)
(x,y) y x
Standard Screen (output, input)
H&B A-1

5. Polar coordinates y
(x,y) r  x
H&B A-1

6. 3D coordinates 1 3D Cartesian coordinates: Right-handed Left-handed
(x,y,z) z
(x,y,z) z x y x y
Right-handed Left-handed
H&B A-1

7. 3D coordinates 2 3D Cartesian coordinates: Right-handed Left-handed
(x,y,z)
z (middle)
z (middle)
(x,y,z)
y (index)
x (thumb)
x (thumb)
y (index)
Right-handed Left-handed
H&B A-1

8. 3D coordinates 3
Cylinder coordinates:
(x,y,z) z y x 
H&B A-1

9. 3D coordinates 4
Spherical coordinates:
(x,y,z) z f r y x 
H&B A-1

10. Points Point: position in nD space
Notation: P (H&B), also P, p, p en p
(x,y,z) (H&B), also (x1, x2, x3), (Px, Py, Pz),
(r, , z), ( r, , ), …
H&B A-2

11. Vectors 1 Vector: Notation: V (H&B), also V, v, v en v
“arrow”
multiple interpretations (displacement, velocity, force, …)
has a magnitude and direction
has no position
Notation: V (H&B), also V, v, v en v
(Vx, Vy, Vz) (H&B), also (x, y, z), (x1, x2, x3)
H&B A-2

12. Vectors 2 y P2 y2 P1 y1
V: directed line segment, or difference between two points x1 x2 x
H&B A-2

13. Vectors 3
Length of a vector:
H&B A-2

14. Vectors 4 Direction of a vector: Direction angles. Unit vector V :
Magnitude info is removed, direction is kept. z g b y x a
H&B A-2

15. Vector addition Add components, put vector head to tail y y W V+W W Vx x
H&B A-2

16. Scalar multiplication of vector
Multiplication components with scalar s y y 2V V x x
H&B A-2

17. Combining vector operations
Infinite line through P with direction V:
L(t) = P + Vt, t  V y P t x
t : parameter along line
t [a, b]: line segment
H&B A-2

18. Vector multiplication 1
Scalar product or dot product:
Product of parallel components, gives 1 real value W V q
|W| cos q
|V|
H&B A-2

19. Vector multiplication 2
Scalar product:
H&B A-2

20. Vector multiplication 3
Vector product or cross product: gives a vector (in 3D)
n perpendicular to V and W
VW W q n V
H&B A-2

21. Vector multiplication 5
Scalar product:
H&B A-2

22. Vector multiplication 6
Scalar product:
scalar
Test if vectors are perpendicular
cos
project,…
Vector product:
vector
Get a vector perpendicular to two given vectors
sin
surface area,…
H&B A-2

23. Exercise 1 Given a point P.
Requested: Reflect a point Q with respect to P. Q
W = P – Q
Q’ = Q + 2W
= 2P – Q
or: = P + (P – Q ) W P x y Q’
We don’t need coordinates!

24. Exercise 1 Given a point P.
Requested: Reflect a point Q with respect to P. Q
Alternative
P is halfway Q and Q’:
P = (Q + Q’)/2
2P = Q + Q’
Q’ = 2P – Q P Q’

25. Exercise 2 Given a line L: L(t) = P + Vt .
Requested: Reflect a point Q with respect to L. Q Q’
We know:
Q’ = Q + 2 W
W = L(t) – Q
W. V = 0 W V
L(t) y P t x

26. Exercise 2 We know: L(t) = P + Vt Q’ = Q + 2 W W = L(t) – Q W. V = 0
Substitute to get t:
(L(t) – Q).V = 0
(P + Vt – Q).V = 0
V .V t + (P – Q).V = 0
t = ((Q – P).V) / (V .V)
Then:
Q’ = Q + 2 (P + Vt – Q)
= 2P – Q + V((Q – P).V / V .V)

27. Steps to be made Write down what you know
Eliminate, substitute, etc. to get he result
Check the result
Does it make sense?
Is there a simpler derivation?

28. Exercise 3
Given a triangle with vertices P, Q en R, where the angle PQR is perpendicular.
Requested: Rotate the triangle around the line PQ over an angle a . What is the new position R’ of R? P  R’ Q R

29. Circle in space |A|=1, |B|=1, A.B = 0y
C()
C()  P A B r P A B  x
|A|=1, |B|=1, A.B = 0
C() = (r cos , r sin , 0) = (0,0,0) + r cos  (1,0,0) + r sin  (0,1,0) = P + r cos  A + r sin  B

30. Exercise 3 A = (R – Q) / |R – Q|
B = (R – Q)(P – Q) / | (R – Q)(P – Q) |
R’ = Q + |R – Q| cos  A + |R – Q| sin  B P B  R’ Q A R

31. Use: Scalar product, cross product coordinate independent definitions
vector algebra
Don’t use:
arccos, arcsin
y = f(x)

32. Very short intro to Linear Algebra
System of linear equations:
Such systems occur in many, many applications.
They are studied in Linear Algebra.
H&B A-5

33. Very short intro to Linear Algebra
System of linear equations:
Typical questions:
Given u, v, w, what are x, y, z?
Can we find a unique solution?
H&B A-5

34. Very short intro to Linear Algebra
System of linear equations:
Crucial in computer graphics:
Transforming geometric objects
Change of coordinates
H&B A-5

35. Example transformationy
P: (x,y) x y’ P’ V U y x x’
H&B A-5

36. What is a matrix? Matrix: Mathematical objects with operations
Matrix in computer graphics:
Defines a coordinate frame
Defines a transformation
Handy tool for manipulating transformations
H&B A-5

37. Matrix Matrix: rectangular array of elements
Element: quantity (value, expression, function, …)
Examples:
H&B A-5

38. Matrix r  c matrix: r rows, c columns
mij : element at row i and column j.
H&B A-5

39. Matrix r  c matrix: r rows, c columns
mij : element at row i and column j.
r  c elements
H&B A-5

40. Matrix r  c matrix: r rows, c columns
mij : element at row i and column j.
c columns
H&B A-5

41. Matrix r  c matrix: r rows, c columns
mij : element at row i and column j.
r rows
H&B A-5

42. Matrix Column vector: matrix with c =1 Used for vectors (and points)
Row vector:
matrix with r =1
Scalar:
matrix with r=1and c=1
H&B A-5

43. Matrix Matrix as collection of column vectors:W
Matrix contains an axis-frame:
H&B A-5

44. Operations on matrices
Multiplication with scalar
simple
Addition
Matrix-matrix multiplication
More difficult, but the most important
H&B A-5

45. Scalar Matrix multiplication
Matrix M multiplied with scalar s:
H&B A-5

46. Scalar Matrix addition
H&B A-5

47. Scalar Matrix addition
Just add elements pairwise
A and B must have the same number of rows and columns
Generalization of vector and scalar addition
H&B A-5

48. Matrix Matrix multiplication
i = 2 ; k
j = 1 ; k
cij: dot product of row vector ai and column vector bj
#columns A must be the same as #rows B: n = p
C: m  q matrix
H&B A-5

49. Matrix Matrix multiplication
Example:
i = 3
j = 2
i=3, j=2
H&B A-5

50. Matrix Matrix multiplication
Example:
H&B A-5

51. Matrix Matrix multiplication
Example:
H&B A-5

52. Matrix Matrix multiplication
Example:
AB  BA
H&B A-5

53. Matrix Matrix multiplication
AB  BA
Matrix matrix multiplication is not commutative!
Order matters!
A(B+C) = AB+AC
Matrix matrix multiplication is distributive
H&B A-5

54. Example transformation sequence (coordinate version)
Unclear!
Error-prone!
H&B A-5

55. Example transformation sequence (matrix vector version)
H&B A-5

56. Example transformation sequence (compact matrix vector version)
Matrix vector notation allows for compactness
and genericity!
H&B A-5

57. Matrix Transpose Transpose matrix: Interchange rows and columns.
r  c matrix M  c  r matrix MT
H&B A-5

58. Matrix Inverse Simple algebra puzzle. Let ax = b. What is x?
Linear algebra puzzle. Let MU = V. What is U?
H&B A-5

59. Matrix Inverse
H&B A-5

60. Matrix Inverse examples
H&B A-5

61. Matrix Inverse examples
Does not exist, cannot be inverted.
H&B A-5

62. Matrix Inverse Does not always exist
In general: if determinant = | M | = 0, matrix cannot be inverted
Inverse for n = 1 and n = 2: easy
Inverse for higher n: use library function
Important special case: orthonormal matrix.
H&B A-5

63. Overview Coordinates, points, vectors Matrices
definitions, operations, examples
Matrices