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Matrix Math

©Anthony Steed 1999

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Overview

To revise• Vectors • Matrices

New stuff• Homogenous co-ordinates• 3D transformations as matrices

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Vectors and Matrices

Matrix is an array of numbers with dimensions M (rows) by N (columns)• 3 by 6 matrix• element 2,3

is (3)

Vector can be considered a 1 x M matrix•

zyxv

100025114311212003

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Types of Matrix

Identity matrices - I

Diagonal

1001

1000010000100001

Symmetric

• Diagonal matrices are (of course) symmetric

• Identity matrices are (of course) diagonal

4000010000200001

fecedbcba

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Operation on Matrices

Addition• Done elementwise

Transpose• “Flip” (M by N becomes N by M)

s d r cq b p a

s rq p

d cb a

379724651

376825941

T

Anthony Steed:

EQ needs fixing

Anthony Steed:

EQ needs fixing

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Operations on Matrices

Multiplication• Only possible to multiply of dimensions– x1 by y1 and x2 by y2 iff y1 = x2

• resulting matrix is x1 by y2

– e.g. Matrix A is 2 by 3 and Matrix by 3 by 4• resulting matrix is 2 by 4

– Just because A x B is possible doesn’t mean B x A is possible!

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Matrix Multiplication Order

A is n by k , B is k by m

C = A x B defined by

BxA not necessarily equal to AxB

k

l

ljilij bac1

*

*****

*****

*.

*.*.*.*.*.

*****

*...

*.*.*.*.*.

*****.....

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Example Multiplications

____

____

1011

12

101132

__________________

010001100

111013322

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Inverse

If A x B = I and B x A = I then

A = B-1 and B = A-1

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3D Transforms

In 3-space vectors are transformed by 3 by 3 matrices

Example?

ziyfxczfyexbzgydxaihgfedcba

zyx

Anthony Steed:

EQ needs fixing

Anthony Steed:

EQ needs fixing

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Scale

Scale uses a diagonal matrix

Scale by 2 along x and -2 along z

zcybxac

ba

zyx

000000

1046200

010002

543

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Rotation

Rotation about z axis

Note z values remain the same whilst x and y change

1000)θcos()θsin(-0)θsin()θcos(

Y

X

θ yx

yθ cosxθ sin yθ sin-xθ cos

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Rotation X, Y and Scale

About X

About Y

Scale (should look familiar)

)θcos()θsin(0)θsin(-)θcos(0

001

)θcos(0)θsin(010

)θsin(-0)θcos(

cb

a

000000

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Homogenous Points

Add 1D, but constrain that to be equal to 1 (x,y,z,1)

Homogeneity means that any point in 3-space can be represented by an infinite variety of homogenous 4D points• (2 3 4 1) = (4 6 8 2) = (3 4.5 6 1.5)

Why?• 4D allows as to include 3D translation in matrix

form

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Homogenous Vectors

Vectors != Points Remember points can not be added If A and B are points A-B is a vector Vectors have form (x y z 0) Addition makes sense

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Translation in Homogenous Form

Note that the homogenous component is preserved (* * * 1), and aside from the translation the matrix is I

1

1010000100001

1 czbyax

cba

zyx

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Putting it Together

R is rotation and scale components T is translation component

1000

321

987

654

321

TTTRRRRRRRRR

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1432

1432010000100001

1

1432010000100001

1000001001000001

1

YZXYZXZYX

1442

1000011001000001

1442

1000001001000001

1432010000100001

1

YZXZYXZYX

Order Matters

Composition order of transforms matters• Remember that basic vectors change so

“direction” of translations changed

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Exercises

Show that rotation by/2 about X and then /2 about Y is equivalent to /2 about Y then /2 about X

Calculate the following matrix /2 about X then /2 about Y then /2 about Z (remember “then” means multiply on the right). What is a simpler form of this matrix?

Compose the following matrix translate 2 along X, rotate /2 about Y, translate -2 along X. Draw a figure with a few points (you will only need 2D) and then its translation under this transformation.

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Summary

Rotation, Scale, Translation Composition of transforms The homogenous form