Math Review

OpenGL Demos

double.c

alpha.c

unproject.c

Demo shapes example from Nate Robins’ tutorial

Math Review

A vector describes a length and a direction.

Length and direction gives offset

a

Vector Operations

A vector’s length is denoted |a| and is computed as

A unit vector is any vector whose length is one

The zero vector is the vector of zero length

222|| aaa zyxa

Vector Operations

Two vectors are added according to the parallelogram rule.

a

b

a+b

a

b a + b = b + a

Vector Addition is commutative

Vector Subtraction

A unary minus for a vector: -a is a vector with the same

length but opposite direction

This allows us to define subtraction

b – a = – a + b

a

b

b – a

Dot Product

Simplest way to multiply two vectors is the dot product

The dot product of two vectors a and b is denoted by a·b

and is also known as the scalar product because it returns

a scalar

a· b = |a| |b| cos φ

b

a

φ

Dot Product

The dot product is associative

a· b = b· a

The dot product is distributive

a· (b+c) = a· b + a· c

The dot product of two 3D vectors a and b

expressed in Cartesian coordinates is given by

a· b = xa xb + ya yb + za zb

Cross Product

The cross product of two vectors returns a 3D

vector perpendicular to both

Right-hand rule dictates direction of cross product

Cross Product

The cross product has a nice property that

a × ( b + c ) = a × b + a × c

However, as a consequence of the right-hand rule

a × b = – ( b × a )

The cross product of two 3D vectors a and b

expressed in Cartesian coordinates is given by

a × b = (ya zb - za yb, za xb - xa zb, xa yb - ya xb)

Orthogonal and Normal

Orthogonal = perpendicular: u· v = 0

Normal = unit-length : u· u = 1

Orthonormal: set of vectors both orthogonal and

normal

Orthogonal matrix: rows (& columns)

orthonormal For orthogonal matrices:

A-1 = AT

ATA = AAT = I

Examples?

Orthonormal Bases

Any set of three 3D vectors u, v and w form an

orthonormal basis provided they are

orthogonal and are each of unit length.

|u| = |v| = |w| = 1 and

u· v = v· w = w· u = 0

This orthonormal basis is right-handed if

w = u × v

otherwise it is left-handed

Coordinate systems

The Cartesian orthonormal basis has an implicit

origin

The vectors x, y and z are never explicitly stored and

neither is the canonical origin location O. It is also

called the global coordinate system.

To use another coordinate system with origin p and

orthonormal basis vectors u, v and w, we do store the

vectors explicitly.

Coordinate systems

The coordinate system associated with the object, such as the plane,

is called a local coordinate system

Local coordinate system

At a low level, the local frame is stored in canonical

coordinates, for eg

u = (xu , yu , zu) = xux + yuy + zuz

The origin is stored as an offset from the canonical origin

p = (xp , yp , zp) = o + xpx + ypy + zpz

Linear Interpolation

Perhaps the most common mathematical

operation in graphics is linear interpolation

To form a line segment between two points a and

b a parameter t is used for interpolation

p = ( 1 –t ) a + t b

This is interpolation because p goes through a and b

for t=0 and t=1 respectively

It is linear because the weighting terms t and 1-t

are linear polynomials of t.

Questions and Break!

Recommendation – Create a vector manipulation library

SIGGRAPH 2010 Papers video:

http://www.youtube.com/watch?v=-PMf3XrwPKo

Basis Vectors

A 2D vector can be written as a combination of any two

non-zero vectors which are not parallel

This property of the two vectors is called linear

independence

Two such vectors which are linearly independent form a

2D basis and the vectors are thus referred to as basis

vectors

Basis Vectors

For example, a vector c may be expressed as a

combination of two basis vectors a and b

c = aca + bcb

Vectors ac and bc are unique. More useful if they are

orthonormal.

Basis Vectors

In the Cartesian coordinate system, we have such ‘special’ vectors x, y and z (in case of 3D), which can be used to represent all other vectors.

Each vector is represented as three real numbers

a = xax + yay + zaz

where xa, ya and za are the real Cartesian coordinates of the 3D vector

Vector space

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

Note that a given vector will have different coordinates for

different bases

Example with Cartesian coordinate system

Slide courtesy: David Luebke

Linear Transformations

A linear transformation:

Maps one vector to another (useful when using local coordinate systems)

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

Slide courtesy: David Luebke

Matrices

A matrix is an array of numeric elements that follow

certain arithmetic rules

Matrices are frequently used in computer graphics for a

variety of purposes

Demo from Nate Robins’ tutorial

http://www.youtube.com/watch?v=deSpf9EGcvw

Matrices

mnm2m1

2n2221

1n1211

MMM

MMM

MMM

M

3

2

v

v

v

v

1

• By convention, matrix element Mrc is located

at row r and column c:

• By (OpenGL) convention,

vectors are columns:

Slide courtesy: David Luebke

Matrix Arithmetic

Matrix times a constant

222120

121110

020100

aaa

aaa

aaa

A

then bA is

222120

121110

020100

bababa

bababa

bababa

babA ij

where b is a constant

Matrix Addition

222120

121110

020100

aaa

aaa

aaa

A

222120

121110

020100

bbb

bbb

bbb

B

• With matrices A, B

• To compute C = A + B

ijijij bac

Matrix Multiplication

• With matrices A, B

• To compute C = A B

222120

121110

020100

aaa

aaa

aaa

A

222120

121110

020100

bbb

bbb

bbb

B

jijijiij bababac 221100

Matrix Multiplication

• With matrices A, B

• To compute C = A B

• For example

222120

121110

020100

aaa

aaa

aaa

A

222120

121110

020100

bbb

bbb

bbb

B

jijijiij bababac 221100

20121011001010 bababac

Transpose of a Matrix

The transpose AT of a matrix A is one whose rows are

switched with its columns

2120

1110

0100

aa

aa

aa

A

121110

020100

aaa

aaaA

T

Determinants

If we have two 2D vectors r and s, we denote the

determinant as |rs|. This value is the signed area of the

parallelogram formed by the vectors

For two 2D vectors with cartesian coordinates (x1, y1)

and (x2, y2) the determinant can be written as

2121

21

21

2

2

1

1

xyyxyy

xx

y

x

y

x

Laplace’s expansion

The standard way of managing the algebra of

computing determinants

The key part is to find cofactors

Each element of a square matrix has a cofactor which

is the determinant of a matrix with one fewer row

and column

The smaller matrix is obtained by eliminating the row

and column that the element is in.

The sign of a cofactor is +ve is the sum of the row

and column indices is even and –ve otherwise

444342

343332

242322

11

aaaaaaaaa

a

444341

343331

242321

12

aaaaaaaaa

a

Cofactor

For example, for a 5 by 5 matrix, the cofactor of a42 is the

determinant of the 4 by 4 matrix with the 4th row and 2nd

column eliminated

For a 4 by 4 matrix, the cofactor for a11 is

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A

c

c

The determinant of a matrix is found by taking the sum of

products of the elements of any row or column with

their cofactors

Matrix is singular if the determinant is 0

For example, for the 4 by 4 matrix in the previous slide

Determinant of a Matrix

4242323222221212 aaaaaaaaA c c c c

Lets solve an example of a 3x3 matrix on the whiteboard

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A

Inverse of a Matrix

Determinants give us a great tool to compute the inverse

of a matrix

For a 4 by 4 matrix A, the inverse A-1 is given by

44342414

43332313

42322212

41312111

cccc

cccc

cccc

cccc

1

aaaa

aaaa

aaaa

aaaa

A

1A

Inverse of a Matrix

44342414

43332313

42322212

41312111

cccc

cccc

cccc

cccc

aaaa

aaaa

aaaa

aaaa

• This is the transpose of Matrix A where its

elements are replaced by their respective

cofactors

• This matrix is called the Adjoint of A

Inverse of a Matrix

If we multiple A with its adjoint,we get

....

....

....

...

...

...

...

...

....

....

....

A

a

a

a

aaaaa

14

13

12

1114131211

c

c

c

c

....

....

...

....

...

...

...

...

....

....

....

0

a

a

a

a

aaaa

14

13

12

11

24232221

c

c

c

c

On dividing the resultant matrix by |A|, we get the identity matrix.

Linear systems

We often encounter linear systems in computer graphics

with “n equations and n unknowns”.

We can write this in matrix form as

1zy2x5

1z3y4x2

4z2y7x3

1

1

4

z

y

x

125

342

273

Linear systems

A common shorthand for such systems is Ax = b

Many ways to solve such systems, but for small

systems, the Cramer’s rule is very useful.

125

342

273

121

341

274

x

125

342

273

125

142

473

z

125

342

273

115

312

243

y

Eigenvalues and Eigenvectors

Eigen-analysis is very important in science and

engineering

For example, eigenvalues are related to frequencies

The stability of a bridge can be predicted by eigen analysis of a

system modeling the bridge

Tacoma Narrows bridge collapse

http://www.youtube.com/watch?v=P0Fi1VcbpAI

Eigen in German means characteristic or special

Eigenvalues and Eigenvectors

Square matrices have eigenvalues and eigenvactors

associated with them

Eigenvectors are those non-zero vectors whose

directions do not change when multiplied by the matrix

Image credits: Wikipedia

Eigenvalues and Eigenvectors

For example, for a matrix A and a vector a,

λ is the eigenvalue associated with the eigenvector a

It means that vector a has been stretched/compressed,

but its direction is still the same

aAa

Eigenvalues and Eigenvectors

Extremely helpful for geometric transformations

Let us assume that a matrix has at least one eigenvector a

Let us find it using a standard manipulation,

0)(

0

aIA

IaAa

IaAa

I is the identity matrix

Matrix multiplication is distributive,

so we can group the matrices

Eigenvalues and Eigenvectors

can only be true if (A-λI) is singular

Singular = determinant is zero

For example, for a 2x2 matrix, the eigenvalues obey

Since this is a quadratic equation, there are exactly two

solutions to λ, which may or may not be unique or real

0)( aIA

0)()( 211222112211

2

2221

1211

aaaaaa

aa

aa

Recommended Exercises

Highly recommend writing a matrix operation library too

Read example on Page 105 (FCG 3rd edition)

Read about Singular Value Decomposition

Very useful technique in computer graphics, machine learning

and data analysis tasks

45

Trivia

When did Computer generated graphics make their

feature film debut and what was the name of the movie?

In the 1973 film West World (starring scary bald robot-

gone-mad Yul Brunner), when a pixilated image was used

to portray a shot from an android's point of view.