Tutorial 1

Math for CS Tutorial 1 1

Tutorial 1

Course Outline

Homogeneous Coordinates


Matrix Multiplication Rule

......

....

....

32

11

333231

232221

131211

333231

232221

131211

iii

iii

ba

ba

bbb

bbb

bbb

aaa

aaa

aaa

BA

Matrices make linear transformations of vectors

..323222121

313212111

3

2

1

333231

232221

131211

xaxaxa

xaxaxa

x

x

x

aaa

aaa

aaa

xA


Translation, Scaling, Rotation of Vectors

22

11

2

1

2

1

dx

dx

d

d

x

xdx

2

1

2

1

22

11

0

0

x

x

S

S

xS

xSxS

2

1

12

21

cossin

sincos

sincos

sincos

x

x

xx

xxxR


100

10

01

2

1

d

d

T dxdx

dx

x

x

d

d

xT

11100

10

01

22

11

2

1

2

1

Translation

Now, we can write the translation as the multiplication by specially designed matrix:

Translation in Homogeneous coordinates

1

2

1

x dxxT

1

3

2

1

2

1

100

110

101


gdgd

gd

g

g

d

d

TT

100

10

01

100

10

01

100

10

01

22

11

2

1

2

1

21 Two translations

We can check that the matrix representing two sequential translations can be written as the multiplication of their matrices.

Sequential Translations in Homogeneous Coordinates


Scaling in Homogeneous coordinates

100

00

00

2

1

S

S

S

11100

00

00

22

11

2

1

2

1

xS

xS

x

x

S

S

xS

Scaling

Scaling matrix looks similar to what it was for ordinary coordinates:

1

5.0

2

1

1

1

100

05.00

002

xS

What is the Matrix for Scaling 0.1x1 and 10x2 ?


11100

00

00

1100

00

00

100

00

00

222

1

2

1

2

1 11

12

11

2

1

2

1

xSS

xSS

x

x

SS

SS

x

x

S

S

S

S

xSS AB

AB

AB

AB

A

A

B

B

AB

Two scalings

The matrix of two successful scalings is the multiplication of two scaling matrices:

Several Scalings in Homogeneous coordinates

What is the Matrix for Scaling 0.1x1 and 10x2 and then 20x1 and 0.1x2 ?


1

cossin

sincos

1100

0cossin

0sincos

21

21

2

1

xx

xx

x

x

xR

Rotation

)(coscossinsinsincoscossin

cossinsincossinsincoscos

100

0cossin

0sincos

100

0cossin

0sincos

)()( 2112121212

1212121211

11

22

22

12

RRR

Two Rotations

Easy to check, that clock-wise rotation on angle θ is given by:

Two successful rotations can be represented by multiplication of their matrices:

)cos( 21 )sin( 21

Rotation in Homogeneous coordinates


12

1

x

x

p

1

2

1

x

x

pHomogeneous coordinates scaled by a constant, represent the same point.

x2

x1

W2W3 W1

Scaling of Homogeneous coordinates


Scaling of Homogeneous coordinates

1

2

1

p

2

4

2

2 pp

1.0

2.0

1.0

2

4

2

1

2

1

p


100

10

01

100

0cossin

0sincos

100

10

01

)()( 2

1

2

1

p

p

p

p

xpTRpT

Rotation

How to write the rotation around a point ?

2

1

p

pp

Bring p back

Bring p to the origin

100

10

01

100

00

00

100

10

01

)()( 2

1

2

1

2

1

p

p

S

S

p

p

xpTSpT

ScalingBring p

backBring p to the origin

Bring the point p to the origin; make a rotation, bring it back:

… the same procedure for scaling:

Rotation around arbitrary point


100

0cossin

0sincos

)(

R

1). Write down the matrix for: Rotation θ=90° around p=(2,5), translation (-2,2), scaling (x2) around p=(-1,1).

100

10

01

2

1

d

d

T

Translation

100

00

00

2

1

S

S

S

RotationScaling

( T(-1,1)·S(2,2)·T(1,-1) ) ·T(-2,2) ·( T(2,5) ·R(90) ·T(-2,-5) )

Example 1. Series of transformations

...))5,2()5,2((

100

110

101

100

020

002

100

110

101

TRT

100

301

710

100

120

102

...


1

z

y

x

p

1000

100

010

001

z

y

x

d

d

d

T dxdz

dy

dx

z

y

x

d

d

d

xTz

y

x

z

y

x

111000

100

010

001Point Translation

1000

000

000

000

z

y

x

S

S

S

T pSzS

yS

xS

z

y

x

S

S

S

pTz

y

x

z

y

x

111000

000

000

000

Scaling

The translation and scaling are very similar in 3D:

Homogeneous coordinates in 3D


The Rotation in 3D can be done around arbitrary axis. Euler angles representation.Any rotation is the composition of three basic rotation, a rotation around the axis x of an angle , a rotation around the axis y of an angle and a rotation around the angle z of an angle are called Euler anglesIn right hand coordinated these rotations are defined as follows

1000

0cossin0

0sincos0

0001

R

1000

0cos0sin

0010

0sin0cos

R

1000

0100

00cossin

00sincos

R

• Simple representation

• Order-dependent:

• Not suitable for animation, because the interpolation between the angles of rotation leads to false locations

RRRRRR ****

Rotation in 3D: Axis needed


2). Rotation θ=90° around x followed by rotation θ=90° around y. find the axis of rotation.

R=R1·R2;If c – rotation axis, then:Rc=c;Solve v.r.t c;c=(a,a,-a,1)

3). Prove that rotation is not commutative: Rx(θ1)·Ry(θ2)≠ Ry(θ2) ·Rx(θ1)

1000

0010

0100

0001

)90(R

1000

0001

0010

0100

)90(R

1000

0100

0001

0010

RR

1000

0010

0001

0100

RR

Examples 2,3. Rotations in 3D


The line equation in Euclidian Coordinates is:

The Euclidian point (x,y) in Homogeneous Coordinates can be written as p=(x,y,1) or

p=(αx, αy, α)

Therefore, denoting Euclidian line (1) by the homogeneous triple u=(a,b,c) we obtain,

that the point p lies on the line u iff:

For example, point p=(1,2,1) lies on the line y=x+1 which can be written by u=(1,-1,1)

)2(0),( 332211 pupupupu

)1(0 cybxa

Lines in Homogeneous Coordinates

)3(0112111),( pu


Now, let us look for the interception point p=(x, y, w) of two lines u1=(a1,b1,c1) and

u2=(a2,b2,c2):

Form the first equation:

Substituting this into the second:

Taking convenient choice of scaling, in which

)1(0

0

222

111

wcybxa

wcybxa

Intersection of two lines 1/2

1

11

a

ybwcx

.

,)(

,0

1221

2112

21121221

221

112

baba

wcawcay

wcawcababay

wcyba

ybwca

1221 babaw


We obtain

On the other hand, writing formally

We have showed, that the point p of the intersection of two lines u1 and u2 is described

by

Intersection of two lines 2/2

21211

2112112211

1

11 )()(bccb

a

cacabbabac

a

ybwcx

,1221 babaw 2112 cacay

],,[][][ 122121212121

222

11121 wyxbabacaacbccb

cba

cba

kji

uu

][],,[ 21 uuwyxp


Now, let us look for the line u=(a, b, c) passing through two points p1=(x1,y1,w1) and

p2=(x2,y2,w2):

Similarly to the solution of the lines intersection, we obtain

The third point p3=(x3,y3,w3) lies on the line u if (p3,u)=0, which via definitions of

determinant can be written as

Due to duality of representation of lines and points in Homogeneous Coordinates, if the

three lines u1, u2 and u3 intersect in a single point, they satisfy

)1(0

0

222

111

wcybxa

wcybxa

)2(21 ppu

)3(0],,det[ 321 ppp

)4(0],,det[ 321 uuu

Three points on the line

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