Linear Algebra Review for Computer Graphics

Matt BoggusCSE 3451/5541

Outline

• Coordinate Spaces• Primitives– Scalars– Vectors– Points

• Operations• Matrices

Coordinate system

Boardwork examples: 1D, 2D, 3D coordinate systems

Coordinate system in Unity

• 3D, left-handed coordinate system

Scalars• Scalar value – a number (integer or real)• Ex: the scale, weight, or magnitude of something

• Scalar field – 2D array of scalar values

C# example:

float [,] temps;temps = new float[50,50];

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

Scalar operations and terms

• Two fundamental operations– Addition and multiplication

SSS , ,,

Associative

Commutative

Distributive

Scalar operations and terms• Additive identity 0

• Multiplicative identity: 1

• Additive inverse

• Multiplicative inverse

11

1

0

11

00

Vectors• Vector – a quantity possessing both magnitude

and direction• Visualized as directed line segments

• Mathematically described as n-tuples of scalars– Examples: (0,0,0) (x,y) (r,g,b,a)

Vector operations – mathematically• Vectors = n-tuples– Vector-vector addition

– Scalar-vector multiplication

– Vector decomposition

110 ,,, nvvvv

111100

110110

,,,

,,,,,,

nn

nn

vuvuvu

vvvuuuvu

110 ,,, nvvvv

111100 nn uuuu

Numeric examples of vector operations

Vector operations – intuitively• Vector-Vector Addition– Visualize using head-to-tail axiom

• Scalar-vector multiplication– Resize (scale) the directed line segment α > 1 length increases 0 > α > 1 length decreases α < 0 reverse direction

Head-to-tail axiom

Scalar-vector multi.

Points• Point – a location or position• Visualized as dots

• Mathematically described as n-tuples of scalars– Examples: (0,0,0) (x,y) (r,g,b,a)– Note: points and vectors have the same

representation!

Coordinate system or frame

• Origin – a point– Indicates the “center” of the coordinate systems– Other points are defined relative to the origin

• Basis vectors– A set of linearly independent vectors– None of them can be written as a linear

combination of the others

Arbitraryplacement

of basisvectors

Basisvectorslocatedat theorigin

Vectors and points in a coordinate system

• Coordinate frame defined by point P0 , set of vectors

• A vector v is

• A point P is

• Note: α and β indicate scalar values

110 ,,, nvvv

1111000 nn vvvPP

111100 nn vvvv

Practice problem

• Given a point P = (2,-5,1) in a right handed coordinate system, what is the point is a left handed coordinate system?

Point-Vector operations• P and Q are points, v is a vector• Point-point subtraction operation

• Vector-point addition operation

QPv

QvP

RPRQQP

Working toward a distance metric

• The previous operations do not include a way to estimate distance between points

• Create a new operation: Inner (dot) Product– Input: two vectors Output: scalar

– Properties we want:

For orthogonal vectors

0

if 0

00

0vvv

wvwuwvu

uvvu

0vu

Dot product logic

• If we can multiply two n-tuples, this implies

– Magnitude (length) of a vector

– Distance between two points

QPQPQP

vvv

Dot product computation

• Dot product of two vectors, u and v

• Ex: (5,4,2) (1,0,1) = 5 * 1 + 4 * 0 + 2 * 1 = 7∙• With some algebra we find that the dot product

is equivalent to where θ is the angle between the two vectors–cosθ = 0 orthogonal–cosθ = 1 parallel

111100

1

0

...

nn

n

iii vuvuvuvuvu

cosvuvu

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,

one parallel to v and one orthogonal to v

Projection of onevector onto another

uvw

vvvuvvvw

vv

vw

vvv

vwwvwu

Cross product

• Another formula that varies as the angle between vectors change

http://en.wikipedia.org/wiki/Cross_product

Matrices

• Definitions• Matrix Operations• Row and Column Matrices• Change of Representation• Relating matrices and vectors

What is a Matrix?

dc

ba

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

rows

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):

1,,0 ,1,,0 , mjniaij

ijaA

jiT aA

1

1

0

n

i

b

b

b

b

b

110 ... nT bbbb

Matrix Operations

• Scalar-Matrix Multiplication– Multiply every element by the scalar

• Matrix-Matrix Addition– Add elements with same index

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

ija A

ijij ba BAC

1

0

l

kkjikij

ij

bac

cABC cij = the sum of multiplying elements in row i of matrix a times elements in column j of matrix b

Matrix Operation Examples

hdgc

fbea

hg

fe

dc

ba

hdgc

fbea

hg

fe

dc

ba

dhcfdgce

bhafbgae

hg

fe

dc

ba

Matrix Operations

• Properties of Scalar-Matrix Multiplication

• Properties of Matrix-Matrix Addition– Commutative:– Associative:

• Properties of Matrix-Matrix Multiplication

• Identity Matrix I (Square Matrix)

AA

AA

ABBA CBACBA

CABBCA

otherwise 0

if 1 ,

jiaa ijijI

BIB

AAI

1000

0100

0010

0001

Matrix Multiplication Order

• Is AB = BA? Try it!

• Matrix multiplication is NOT commutative!• The order of series of matrix multiplications

is important!

hdgbhcga

fdebfcea

dc

ba

hg

fe

dhcfdgce

bhafbgae

hg

fe

dc

ba

Inverse of a Matrix

• Identity matrix: AI = A

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

100

010

001

I

100

110

101

A

100

110

1011A

Inverse of a Matrix

• Do all matrices have a multiplicative inverse? Consider this example, try to solve for A-1:

AA-100 =

100

010

000

A

ihg

fed

cba

A 1

0 * a + 0 * d + 0 * g = 0 ≠ 1

Note: 1 is the element at 00 in the identity matrix

Inverse of Matrix Concatenation

• Inversion of concatenations(ABC)-1 = ?

A * B * C * X = IA * B * C * C-1 = A * B

A * B * B-1 = AA * A-1 = I

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

Row and Column Matrices + points

• Column Matrix

– Row matrix• Concatenations– Associative

• By Row Matrix

z

y

x

p

ABCpp

App

TTTTT

TTT

ABCpp

ABAB

zyxT p

By convention we will use column matrices for points

Summary

• Primitives: scalars, vectors, points

• Operations: addition and multiplication

• Matrix representation and operations