Vector Tools forComputer Graphics

Computer Graphics

Vector Tools for Computer Graphics

Two branches of Mathematics Vector Analysis Transformation

Why Vectors?

Basic Definitions

Points specify location in space (or in the plane). Vectors have magnitude and direction (like

velocity).

Points Vectors

Basics of Vectors

v

v

Q

PVector as displacement:

v is a vector from point P to point 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

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

Operations on vectors

Addition

a

b

a+ b

ab

a+ b

Multiplication by scalar

a2a

-a

Operations on vectors

Subtraction

a

c

a

-c

a-ca

c

a-c

Linear Combination of Vectors

Linear combination of m vectors v1, v2, …, vm:

w = a1v1 + a2 v2 + …+ amvm

where a1,a2,…am are scalars.

Definition

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

Convex Combination of Vectors

v1

v2

v

a(v2-v1)

v1v2

v3

Properties of vectors

Length or size

222

21 ... nwww |w|

w = (w1,w2,…,wn)

Unit vector

aa

a

• 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)

Dot Product

The dot product d of two vectors v = (v1,v2,…,vn) and w = (w1,w2,…wn):

Properties

1. Symmetry: a·b = b·a

2. Linearity: (a+c) ·b = a·b + c·b

3. Homogeneity: (sa) ·b = s(a·b)

4. |b|2 = b·b

Application of Dot Product

Angle between two vectors:

c

b

y

xΦb

Φc

θ^^

)cos( cb

Two vectors b and c are perpendicular (orthogonal/normal) ifb·c = 0

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?

Applications: Orthogonal Projection

A

C

v

cL

A

C

v

c

v┴

Mv┴

Kv

Applications: Reflections

a

n

r

m -m

e e

a

n

r

θ1 θ2

r = a - 2 ( a . n) n

Cross Product

Also called vector product. Defined for 3D vectors only.

zyx

zyx

bbb

aaa

kji

ba

Properties

1. Antisymmetry: a Χ b = - b Χ a

2. Linearity: (a +c) Χ b = a Χ b + c Χ b

3. Homogeneity: (sa) Χ b = s(a Χ b)

Geometric Interpretation of Cross Product

axb

a

b

x

y

z

P2

P1P3

1. aXb is perpendicular to both a and b 2. | aXb | = area of the parallelogram defined by a and b

Representation of Vectors and Points

Coordinate Frame

P

v

c

a

b

O

Homogeneous Representation

Basic objects: a,b,c,O Represent points and vectors using these objects

Homogeneous Representation

To go from ordinary to homogeneous

1.If point append 1 as the last coordinate.2.If vector append 0 as the last coordinate.

To go from homogeneous to ordinary

1.The last coordinate must be made 1 and then delete it2.The last coordinate must be made 0 and then delete it

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

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

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)UNOT, unless f + g = 1

Affine Combinations of Points [contd]

P2

P1 P1+P2

P1+P2

(P1+P2)/2

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.

Linear Interpolation

“Tweening”

See Hill 4.5.4

Representing Lines

Line Line segment Ray

3 types of representations:1. Two point form

2. Parametric representation

3. Point normal form

Parametric Representation of a Line

y

x

t = 1

B

Cb

t = 0

t < 0

t > 1

Point Normal Form of a Line

n

B

R

C

All in One : Representations of Line

Two PointPoint

Normal

Parametric

B = C - n┴

n = (B – C) ┴

n =

b┴

b =

- n┴B = C + bb

= B – C

{C, B}{C, n}

{C, b}

Planes in 3D

3 fundamental forms Three-point form Parametric representation Point normal form

Parametric Representation of Plane

C

a

b

Point Normal Form of a Plane

B

n

All in One : Representations of Plane

Three PointPoint

Normal

Parametric

A=(0,0,n.C/ nz) C=(C-0) B=(0,n.C/ ny,0)

n = (A-C)x(B – C)

n =

axbb =

nxa

A = C + a

B = C + ba=

A – C

b = B

– C

{A,C,B}

{C, n}

{C, a, b}

a= (0

,0,1

)xn

Readings

Hill 4.1 – 4.6