1. Vector Tools for Computer Graphics

2. Vector Tools for Computer Graphics
Two branches of Mathematics
Vector Analysis
Transformation
Why Vectors?

3. Basic Definitions Points specify location in space (or in the plane).
Vectors have magnitude and direction (like velocity).
Points  Vectors

4. Basics of Vectors v Vector as displacement:
v is a vector from point P to point Q. P v Q
The difference between two points is a vector: v = Q - P
Another way:
The sum of a point and a vector is a point : P + v = Q v Q P

5. Multiplication be scalars
Operations on Vectors
Two operations
Addition
a + b
a = (3,5,8), b = (-1,2,-4)
a + b = (2,7,4)
Multiplication be scalars sa
a = (3,-5,8), s = 5
5a = (15,-25,40)
operations are done componentwise

6. Multiplication by scalar
Operations on vectors
Addition a b
a+ b a b
a+ b
Multiplication by scalar a 2a -a

7. Operations on vectors
Subtraction a c -c
a-c

8. Linear Combination of Vectors
Definition
Linear combination of m vectors v1, v2, …, vm:
w = a1v1 + a2 v2 + …+ amvm
where a1,a2,…am are scalars.
Types
Affine Combination
a1 + a2 + …+ am = 1
Examples:
3a + 2b - 4c
(1- t)a + (t)b
Convex Combination
a1 + a2 + …+ am = 1
and ai ≥ 0, for i = 1,…m.
Example: .3a + .2b + .5c

9. Convex Combination of Vectors
a(v2-v1) v3 v v1 v2 v1

10. The standard unit vectors: i = (1,0,0), j = (0,1,0) and k = (0,0,1)
Properties of vectors
Length or size
w = (w1,w2,…,wn)
Unit vector
The process is called normalizing
Used to refer direction
The standard unit vectors: i = (1,0,0), j = (0,1,0) and k = (0,0,1)

11. Dot Product
The dot product d of two vectors v = (v1,v2,…,vn) and w = (w1,w2,…wn):
Properties
Symmetry: a·b = b·a
Linearity: (a+c) ·b = a·b + c·b
Homogeneity: (sa) ·b = s(a·b)
|b|2 = b·b

12. Application of Dot Product
Angle between two vectors: y c b θ Φc Φb x
Two vectors b and c are perpendicular (orthogonal/normal) if
b·c = 0

13. is the counterclockwise perpendicular to a.
2D “perp” Vector
Which vector is perpendicular to the 2D vector a = (ax,ay)?
Let a = (ax,ay).
Then
a┴ = (-ay,ax)
is the counterclockwise perpendicular to a. a┴
-a┴ a a
What about 3D case?

14. Applications: Orthogonal Projection
Mv┴ c c L v┴ A Kv v v A

15. Applications: Reflectionsm -m e a n r θ1 θ2
r = a ( a . n) n

16. Cross Product Also called vector product. Defined for 3D vectors only.
Properties
Antisymmetry: a Χ b = - b Χ a
Linearity: (a +c) Χ b = a Χ b + c Χ b
Homogeneity: (sa) Χ b = s(a Χ b)

17. Geometric Interpretation of Cross Product
axb a b x y z P2 P1 P3
aXb is perpendicular to both a and b
| aXb | = area of the parallelogram defined by a and b

18. Representation of Vectors and Points

19. Coordinate Frame b P v O a c

20. Homogeneous Representation
Basic objects: a,b,c,O
Represent points and vectors using these objects

21. Homogeneous Representation
To go from ordinary to homogeneous
If point append 1 as the last coordinate.
If vector append 0 as the last coordinate.
To go from homogeneous to ordinary
The last coordinate must be made 1 and then delete it
The last coordinate must be made 0 and then delete it

22. Linear Combinations of vectors
The difference of two points is a vector
The sum of a point and vector is a point
The sum of two vectors is a vector
The scaling of a vector is a vector
Any linear combination of vectors is a vector

23. any affine combination of points is a legitimate point
Affine Combinations of Points
Two points P=(P1,P2,P3,1) and R=(R1,R2,R3,1):
E = fP + gR=(fP1+gR1, fP2+gR2, fP3+gR3, f+g)
Since affine combination so f+g=1
So, E is a valid point in homogeneous representation
NOT a valid point, unless f + g = 1
Example: 0.3P+0.7R is a valid point, but P + R is not.
any affine combination of points is a legitimate point

24. Linear Combinations of Points
Linear combination of two points P and R:
E = fP + gR
If the system is shifter by U
P’ = P + U
R’ = R + U
E’ should be equal to E+U i.e E’=E+U
E should also be shifted by U
But,
E’ = fP’ + gR’ = fP + gR + (f+g)U
NOT, unless f + g = 1

25. Affine Combinations of Points [contd]
P1+P2 P2 P1
P1+P2
(P1+P2)/2

26. Point + Vector = Affine Comb of Points
Let,
P = A + t v and v = B – A
Therefore,
P = A + t (B – A)
=> P = t B + (1 – t) A.
“Tweening”
See Hill 4.5.4
Linear Interpolation

27. Representing Lines 3 types of representations: Ray Line Line segment
Two point form
Parametric representation
Point normal form

28. Parametric Representation of a Liney
t > 1 B b
t = 1 C
t = 0
t < 0 x

29. Point Normal Form of a LineB R C

30. All in One : Representations of Line
B = C - n┴
Point
Normal
Two Point
{C, n}
{C, B}
n = (B – C) ┴
b = - n┴
B = C + b
n = b┴
b = B – C
Parametric
{C, b}

31. Planes in 3D 3 fundamental forms Three-point form
Parametric representation
Point normal form

32. Parametric Representation of Planeb

33. Point Normal Form of a PlaneB

34. All in One : Representations of Plane
A=(0,0,n.C/ nz) C=(C-0) B=(0,n.C/ ny,0)
Point
Normal
Three Point
{C, n}
n = (A-C)x(B – C)
{A,C,B}
a= (0,0,1)xn
A = C + a
B = C + b
b = nxa
n = axb
Parametric
a= A – C
b = B – C
{C, a, b}

35. Readings
Hill 4.1 – 4.6