Graphics CSE 581 – Interactive Computer Graphics Mathematics for Computer Graphics CSE 581 – Roger Crawfis (slides developed from Korea University slides)

Graphics

CSE 581 – Interactive Computer Graphics

Mathematics for Computer Graphics

CSE 581 – Roger Crawfis(slides developed from Korea University slides)


Spaces

Scalars (Linear) Vector Space

Scalars and vectors

Affine Space Scalars, vectors, and points

Euclidean Space Scalars, vectors, points Concept of distance

Projections


Scalars (1/2)

Scalar FieldScalar Field Ex) Ordinary real numbers and operations on them

Two Fundamental Operations Addition and multiplication

SSS , ,,

Associative

Commutative

Distributive


Scalars (2/2)

Two Special Scalars Additive identity: 0, multiplicative identity: 1

Additive inverse and multiplicative inverse of

11

00

1

1

01


Vector Spaces (1/4)

Two Entities: ScalarsScalars and VectorsVectors Vectors

Directed line segments n-tuples of numbers Two operations: vector-vector addition, scalar-

vector multiplication

Special Vector: Zero VectorZero Vector

Let u denote a vector

Directed line segments

0

0

uu

uu


Vector Spaces (2/4)

Scalar-Vector Multiplication

u and v: vectors, α and β: scalars

Vector-Vector Addition Head-to-tail axiom: to visualize easily

uuu

vuvu

Head-to-tail axiom

Scalar-vector multi.


Vector Spaces (3/4)

Vectors = n-tuples Vector-vector addition

Scalar-vector multiplication

Vector space:

Linear IndependenceLinear Independence Linear combination:

nvvvv ,,, 21

nn

nn

vuvuvu

vvvuuuvu

,,,

,,,,,,

2211

2121

nvvvv ,,, 21 nR

nnuuuu

2211

Vectors are linear independent if the only set of scalars is

02211 nnuuu

021 n


Vector Spaces (4/4)

DimensionDimension The greatest number of linearly independent vectors

BasisBasis n linearly independent vectors (n: dimension)

Representation Unique expression in terms of the basis vectors

Change of Basis: Matrix M Other basis

nnvvvv

2211

i

nvvv

,,, 21

nnvvvv

2211

nn

2

1

2

1

M


Affine Spaces (1/2)

No Geometric Concept in Vector Space!! Ex) location and distance Vectors: magnitude and direction,

no position

Coordinate SystemCoordinate System Origin: a particular reference point Identical vectors

Arbitraryplacement

of basisvectors

Basisvectorslocatedat theorigin


Affine Spaces (2/2)

Third Type of Entity: PointsPoints New Operation: Point-Point SubtractionPoint-Point Subtraction

P and Q : any two points

vector-point addition

FrameFrame: a Point and a Set of Vectors Representations of the vector and point: n scalars

QPv

QvP

Head-to-tail axiom for points RPRQQP

0P nvvv

,,, 21

nn

nn

vvvPP

vvvv

22110

2211Vector

Point


Euclidean Spaces (1/2)

No Concept of How Far Apart Two Points in Affine Spaces!!

New Operation: Inner (dot) ProductInner (dot) Product Combine two vectors to form a real α, β, γ, …: scalars, u, v, w, …:vectors

Orthogonal:

0

if 0

00

0vvv

wvwuwvu

uvvu

0vu


Euclidean Spaces (2/2)

Magnitude (length) of a vector

Distance between two points

Measure of the angle between two vectors

cosθ = 0 orthogonal cosθ = 1 parallel

QPQPQP

cosvuvu

vvv


Projections

Problem of Finding the Shortest Distance from a Point to a Line of Plane

Given Two Vectors, Divide one into two parts: one

parallel and one orthogonal to the otherProjection of one

vector onto anotheruvw

vvvuvvvw

vv

vw

vvv

vwwvwu


Matrices

Definitions Matrix Operations Row and Column Matrices Rank Change of Representation Cross Product


6.837 Linear Algebra Review

What is a Matrix?

A matrix is a set of elements, organized into rows and columns

dc

barows

columns


Definitions

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

Transpose: interchanging the rows and columns of a matrix

Column Matrices and Row Matrices Column matrix (n x 1 matrix): Row matrix (1 x n matrix):

mjniaij ,,1 ,,,1 , ijaA

jiT aA

n

i

b

b

b

b2

1

bTb



Basic Operations

Addition, Subtraction, Multiplication

hdgc

fbea

hg

fe

dc

ba

hdgc

fbea

hg

fe

dc

ba

dhcfdgce

bhafbgae

hg

fe

dc

ba

Just add elements

Just subtract elements

Multiply each row by each

column


Matrix Operations (1/2)

Scalar-Matrix MultiplicationScalar-Matrix Multiplication

Matrix-Matrix AdditionMatrix-Matrix Addition

Matrix-Matrix MultiplicationMatrix-Matrix Multiplication A: n x l matrix, B: l x m C: n x m matrix

ija A

ijij ba BAC

l

kkjikij

ij

bac

c

1

ABC


Matrix Operations (2/2)

Properties of Scalar-Matrix Multiplication

Properties of Matrix-Matrix Addition Commutative: Associative:

Properties of Matrix-Matrix Multiplication

Identity MatrixIdentity Matrix I (Square Matrix)

AA

AA

ABBA CBACBA

BAAB

CABBCA

otherwise 0

if 1 ,

jiaa ijijI BIB

AAI



Multiplication

Is AB = BA? Maybe, but maybe not!

Heads up: multiplication is NOT commutative!

......

...bgae

hg

fe

dc

ba

......

...fcea

dc

ba

hg

fe


Row and Column Matrices

Column Matrix

pT: row matrix

ConcatenationsConcatenations Associative

By Row Matrix

z

y

x

p

ABCpp

App

TTTTT

TTT

ABCpp

ABAB



Vector Operations

Vector: 1 x N matrix Interpretation: a line

in N dimensional space

Dot Product, Cross Product, and Magnitude defined on vectors only

c

b

a

v

x

y

v



Vector Interpretation

Think of a vector as a line in 2D or 3D Think of a matrix as a transformation on a line

or set of lines

'

'

y

x

dc

ba

y

xV

V’



Vectors: Dot Product

Interpretation: the dot product measures to what degree two vectors are aligned

A

B

A

BC

A+B = C(use the head-to-tail method to combine

vectors)



Vectors: Dot Product

cfbead

f

e

d

cbaabba T

ccbbaaaaa T 2

)cos(baba

Think of the dot product as a matrix multiplication

The magnitude is the dot product of a vector with itself

The dot product is also related to the angle between the two vectors – but it doesn’t tell us the angle



Vectors: Cross Product

The cross product of vectors A and B is a vector C which is perpendicular to A and B

The magnitude of C is proportional to the cosine of the angle between A and B

The direction of C follows the right hand rule – this why we call it a “right-handed coordinate system”

)sin(baba



Inverse of a Matrix

Identity matrix: AI = A

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

Inversion is tricky:(ABC)-1 = C-1B-1A-1

Derived from non-commutativity property

100

010

001

I



Determinant of a Matrix

Used for inversion If det(A) = 0, then A

has no inverse Can be found using

factorials, pivots, and cofactors!

Lots of interpretations – for more info, take 18.06

dc

baA

bcadA )det(

ac

bd

bcadA

11



Determinant of a Matrix

cegbdiafhcdhbfgaei

ihg

fed

cba

ihg

fed

cba

ihg

fed

cba

ihg

fed

cbaSum from left to rightSubtract from right to leftNote: N! terms


Change of Representation (1/2)

Matrix Representation of the Change between the Two Bases Ex) two bases and

Representations of v

Expression of in the basis

nuuu ,,, 21 nvvv ,,, 21 nnuuuv 2211

nnvvvv 2211

or

Tn 21 a Tn 21 bor

nuuu ,,, 21 nvvv ,,, 21

nivvvu niniii ,,1 ,2211


Change of Representation (2/2)

: n x n matrix

and

By direct substitution

nn v

v

v

u

u

u

2

1

2

1

A

ijA

ia ib

n

T

u

u

u

v2

1

a

n

T

v

v

v

v2

1

bor

n

T

n

T

v

v

v

u

u

u

2

1

2

1

ba

n

T

n

T

v

v

v

v

v

v

2

1

2

1

bAa Aab TT


Cross Product

In 3D Space, a unit Vector, w, is Orthogonal to Given Two Nonparallel Vectors, u and v

Definition

Consistent Orientation Ex) x-axis x y-axis = z-axis

0 vwuw

1221

3113

2332

vuw

321321 ,, ,,, where vu

Documents

Graphics CSE 581 – Interactive Computer Graphics Mathematics for Computer Graphics CSE 581 – Roger Crawfis (slides developed from Korea University slides)