Mathematics for Computer Graphics

Outline – Linear Algebra topicsOutline  Linear Algebra topics

• Scalars• Vector Space– Scalars and vectors

• Affine Space– Scalars, vectors, and points

• Euclidean Space• Euclidean Space– Scalars, vectors, points– Distance metric

• Projections• Matrix representations and operations

Scalars• Scalar – a number

Ex: the scale weight or magnitude of something– Ex: the scale, weight, or magnitude of something

• Two Fundamental Operationsdd d l l– Addition and multiplication

SSS , ,,

Commutative

Associative

Distributive

Scalars• Two Special Scalars

Additi id tit 0 lti li ti id tit 1– Additive identity: 0, multiplicative identity: 1

00 11

– Additive inverse  0

–Multiplicative inverse 1 11

2D scalar field2D scalar field

2D array of scalar values2D array of scalar values

In C# the data structureIn C#, the data structure might look something like this:

float [,] temps;[,] p ;temps = new float[50,50];// (allocate after declaring)// (allocate after declaring)

http://upload.wikimedia.org/wikipedia/commons/a/a8/Scalar_field.png

Vector spaceVector space

• Two Entities: Scalars and VectorsTwo Entities: Scalars and Vectors• Vectors

Di t d li t– Directed line segments– n‐tuples of scalars– Two operations• vector‐vector addition

l t lti li ti• scalar‐vector multiplication

• Zero Vector0

Directed line segments 00

vv

vv

Vector spaceVector space

• In vector space two vectors are equal if they areIn vector space, two vectors are equal if they are represented with the same n‐tuple

Ex: (1 2 3) = (1 2 3)– Ex: (1,2,3) = (1,2,3)(1,2,3) ≠ (3,2,1)

• A vector, by itself, does not have a starting position– Visually you can think of a vector starting at (0,0,0)

Vector space operationsVector space operations

• Vectors = n‐tuples vvvvVectors = n tuples– Vector‐vector addition

110 ,,, nvvvv

111100

110110

,,,,,,,,,

nn

nn

vuvuvuvvvuuuvu

– Scalar‐vector multiplication

– Vector decomposition 110 ,,, nvvvv

111100 nn uuuu

Vector space operations

• Scalar‐Vector Multiplicationu and v: vectors, α and β: scalars

vuvu

Scalar-vector multi.

uuu

• Vector Vector Addition• Vector‐Vector Addition– Visualize using head‐to‐tail axiom

Head-to-tail axiom

Vector space limitations

• Vector space has no geometric concept– Vectors indicate magnitude anddirection, not position

• Coordinate SystemId ti l t

y– Origin: a particular reference point

Identical vectors

Arbitraryplacementof basis

Basisvectorslocated

of basisvectors

at theorigin

Coordinate systemCoordinate system

d k l 2 3 diBoardwork examples: 1D, 2D, 3D coordinate axes

Unity uses left‐handed coordinate system with    +x being right and +y being upg g y g p

Affine SpacesAffine Spaces• Scalars, vectors, and points• A point is represented as an n‐tuple

• Operations [P and Q are points, v is a vector]– Point‐point subtraction operation

• Vector point addition operation

QPv

• Vector‐point addition operation

QvP Head-to-tail axiom for points

RPRQQP

• Frame: a Point and a Set of Vectors– All vectors and points in the space are defined relative to these

0P 110 ,,, nvvv

V t

1111000

111100

nn

nn

vvvPPvvvv

Vector

Point

Euclidean SpacesEuclidean Spaces• Affine space does not include an operation or metric for distance between points

• Create a new operation: Inner (dot) Product• Create a new operation: Inner (dot) Product– Input: two vectors  Output: scalar– α, β : scalars u, v, w : vectors

– Properties we want:

wvwuwvu

uvvu

0 if 0

000vvv

wvwuwvu

For orthogonal vectors0 00

0 vu

Euclidean Spaces – dot productEuclidean Spaces  dot product• If we can multiply two n‐tuples, this implies

– Magnitude (length) of a vector vvv

– Distance between two points QPQPQP

– Measure of the angle between two vectors• cosθ = 0 orthogonal

cosvuvu

• cosθ = 1 parallel

• Computation of dot productp p

111100

1

0...

nn

n

iii vuvuvuvuvu

Projections• We can determine if two points are “close” to each other, what about vectors?,

• How much of w is in the same direction as v?• Given vectors v and w, decompose w into two parts,Given vectors v and w, decompose w into two parts, one parallel to v and one orthogonal to v

uvw

vvvuvvvw

Projection of onevector onto anothervv

vw

vvvvwwvwu

MatricesMatrices

• DefinitionsDefinitions• Matrix Operations

d C l i• Row and Column Matrices• Change of Representation• Relating matrices and vectors

What is a Matrix?What is a Matrix?• A matrix is a set of elements, organized into rows and columns

b

rows

dcba

columns

dc

DefinitionsDefinitions• n x m Array of Scalars (n Rows and m Columns)– n: row dimension of a matrix, m: column dimension– m = n square matrix of dimension n– Element 1010 mjniaElement 

T i h i h d l f i

1,,0 ,1,,0 , mjnia ij ijaA

– Transpose: interchanging the rows and columns of a matrix

jiT aA

b• Column Matrices and Row Matrices– Column matrix (n x 1 matrix):Row matrix (1 x matrix)

1

0

i

bb

b

b– Row matrix (1 x n matrix):

1nb

110 ... nT bbbb

Matrix OperationsMatrix Operations

• Scalar‐Matrix Multiplication a AScalar Matrix Multiplication–Multiply every element by the scalar

• Matrix Matrix Addition

ijaA

ba BAC• Matrix‐Matrix Addition– Add elements with same index

ijij ba BAC

• Matrix‐Matrix Multiplication– A: n x l matrix, B: l x m C: n x m matrix

1l

ijcABC cij = the sum of multiplying elements in row i of matrix a

1

0

l

kkjikij bac times elements in column j of

matrix b

Matrix Operation ExamplesMatrix Operation Examples

fbeafeba

hdgcf

hgf

dc

hdfbea

hfe

dba

hdgchgdc

dhcfdgcebhafbgae

hgfe

dcba

dhcfdgcehgdc

Matrix OperationsMatrix Operations

• Properties of Scalar‐Matrix Multiplication AA

p p

• Properties of Matrix‐Matrix Addition

AA

ABBA– Commutative:– Associative:

• Properties of Matrix‐Matrix Multiplication

ABBA CBACBA

p p

BAAB

CABBCA

• Identity Matrix I (Square Matrix)

BAAB

if1 ji AAI 0001

otherwise 0if 1

,ji

aa ijijIBIBAAI

1000010000100001

Matrix Multiplication OrderMatrix Multiplication Order

• Is AB = BA? Maybe but maybe not!Is AB = BA?  Maybe, but maybe not!

...bgaefeba

...fceabafe

......hgdc

......dchg

• In general, matrix multiplication is NOT commutative!

• The order of matrix multiplications is important!important!

Row and Column Matrices + pointsRow and Column Matrices + points

• Column Matrix

x By convention

we will useColumn Matrix

zyp we will use

column matrices for points

– Row matrix

C t ti zyxT p

• Concatenations– Associative ABCpp

App

• By Row Matrix TTT ABAB TTTTT ABCpp

Vector OperationsVector Operations

• Vector: 1 x N matrix

Vector: 1 x N matrix• Interpretation: a line in N dimensional

ba

in N dimensional spaceD P d d

cbv

• Dot Product and Magnitude 

i

cyoperations y

v

x

Vectors: Dot ProductVectors: Dot Product

d Thi k f th d t cfbead

fed

cbaabba T

Think of the dot product as a matrix multiplication

bbT 2The magnitude is the dot

d t f t ithccbbaaaaa T product of a vector with itself

)cos(baba

If a and b are both length one, the angle between them is the cos-1 of their dot product

Inverse of a MatrixInverse of a Matrix

• Identity matrix: 001Identity matrix: AI = A

• Some matrices have an

010001

I• Some matrices have an inverse, such that:AA‐1 = I

100

AA 1 = I

101 101

110101

A

110101

1A 100

100

Inverse of a MatrixInverse of a Matrix

• Some matrices do not have an inverseSome matrices do not have an inverse

000

cba

100010A

ihfedA 1

100 ihg

AA‐100 =  0 * a + 0 * d + 0 * g = 0 ≠ 1

Inverse of Matrix ConcatenationInverse of Matrix Concatenation

• Inversion of concatenationsInversion of concatenations(ABC)‐1 = C‐1B‐1A‐1

A * B * C * X = IA * B * C * C 1 A * BA * B * C * C‐1 = A * B

A * B * B‐1 = AA * A‐1 = I

Order is important so X = C‐1B‐1A‐1Order is important, so X C B A

SummarySummary

• Primitives: scalars, vectors, points

• Operations: addition and multiplication

• Matrix representation and operationsMatrix representation and operations