kom graf 04 transform as i

8/12/2019 Kom Graf 04 Transform as i

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Gasal 2013/2014KOMPUTER GRAFIK (3SKS)

Cucun Very Angkoso,ST.,MT

:: Pertemuan ke ::


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TRANSFORMASI 2D

Transformasi adalah fungsi yang memetakan suatu set X ke set yanglain ataupun ke set X sendiri. Terdapat banyak jenis operasi

transformasi: translasi, refleksi, rotasi, scaling, shearing.

A transformation maps points to other points and/or vectors to

other vectors

v=T(u)


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A1. Matriks Translasi

1

y

x

.t

t

'y

'x

y

x

100

10

01

1

A2. Matriks Rotasi

1

yx

.

100

0cossin0sincos

1

''

y

x

A3. Matriks Skala

1

y

x

.s

s

'y

'x

y

x

100

00

00

1

atau P = T(tx, ty) . P

atau P = R() . P

atau P = S(sx, sy) . P


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Rotasi

Arah rotasi berlawanan dengan arah putar jarum jam, arahputarnya dikatakan Positif

Sebaliknya, jika arah rotasi searah dengan arah putar

jarum jam, arah putarnya dikatakan negatif


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Transformasi Geometri (Rotasi)

1. Titik Q(-1, 4) diputar searah jarum jam terhadap titik pusat O,tentukanbayangan titik Q oleh rotasi (O, 450)

Jawab:

= - 450

x = x cos - y sin= -1 Cos (- 450)4 sin (- 450)

= - - 4 . (- )

= - + 2

=

y = x sin + y cos= -1 Sin(-450) + 4 Cos(-450)

= -1(- ) + 4 .

= + 2

= 5/2

2

2

3

2

2

22

Jadi Q (3/2 , 5/2 )22

2

2

2

2

2


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Bagaimana urutan transformasi yang diperlukan untuk

mewujudkannya?


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10

Pipeline Implementation

transformation rasterizer

u

v

u

v

T

T(u)

T(v)

T(u)

T(u)

T(v)

T(v)

vertices vertices pixels

frame

buffer

(from application program)


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11

Translation Matrix

Express translation using a

4 x 4 matrix Tin homogeneous coordinates

p=Tpwhere

T = T(dx, dy, dz) =

This form is better for implementation because all affinetransformations can be expressed this way and multipletransformations can be concatenated together

1000

d100

d010

d001

z

y

x


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12

Rotation (2D)

Consider rotation about the origin by degrees radius stays the same, angle increases by

x=x cos y sin

y = x sin + y cos

x = r cos f

y = r sin f

x = r cos (f + )

y = r sin (f + )


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13

Rotation about the zaxis

Rotation about zaxis in three dimensions leavesall points with the same z

Equivalent to rotation in two dimensions in planes of

constant z

or in homogeneous coordinates

p=Rz()p

x=x cos y sin

y = x sin + y cos z =z


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14

Rotation Matrix

1000

0100

00cossin

00sincos

R= Rz() =


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15

Rotation about xand yaxes

Same argument as for rotation about zaxis For rotation about xaxis, xis unchanged

For rotation about yaxis, yis unchanged

R= Rx() =

R= Ry() =

1000

0cossin0

0sin-cos0

0001

1000

0cos0sin-

0010

0sin0cos


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16

Scaling

1000

000000

000

z

y

x

s

s

s

S = S(sx, sy, sz) =

x=sxx

y=syx

z=szx

p=Sp

Expand or contract along each axis (fixed point of origin)


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17

Reflection

corresponds to negative scale factors

originalsx= -1 sy= 1

sx= -1 sy= -1 sx= 1 sy= -1


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18

Inverses

Although we could compute inverse matrices bygeneral formulas, we can use simple geometric

observations

Translation: T-1(dx, dy, dz)= T(-dx, -dy, -dz)

Rotation: R-1

() = R(-) Holds for any rotation matrix

Note that since cos(-) = cos() and sin(-)=-sin()

R-1() = R T()

Scaling: S-1(sx, sy, sz)= S(1/sx, 1/sy, 1/sz)


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19

Concatenation

We can form arbitrary affine transformationmatrices by multiplying together rotation,

translation, and scaling matrices

Because the same transformation is applied to

many vertices, the cost of forming a matrixM=ABCDis not significant compared to the cost of

computing Mpfor many vertices p

The difficult part is how to form a desired

transformation from the specifications in theapplication


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20

Order of Transformations

Note that matrix on the right is the first applied

Mathematically, the following are equivalent

p = ABCp= A(B(Cp))

Note many references use column matrices to

represent points. In terms of column matricespT= pTCTBTAT


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21

General Rotation About the Origin

x

z

y

v

A rotation by about an arbitrary axis

can be decomposed into the concatenation

of rotations about thex, y, and zaxes

R() = Rz(z) Ry(y) Rx(x)

x y z are called the Euler anglesNote that rotations do not commute

We can use rotations in another order but

with different angles


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22

Rotation About a Fixed Point other than the

Origin

Move fixed point to originRotate

Move fixed point back

M= T(pf) R() T(-pf)


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23

Shear

Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions


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24

Shear Matrix

Consider simple shear along xaxis

x = x + y cot

y = y

z = z

1000

0100

0010

00cot1

H() =


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Transformasi Pada OpenGL

In OpenGL matrices are part of the stateMultiple types

Model-View (GL_MODELVIEW)

Projection (GL_PROJECTION)

Texture (GL_TEXTURE) (ignore for now)Color(GL_COLOR) (ignore for now)

Single set of functions for manipulation

Select which to manipulated byglMatrixMode(GL_MODELVIEW);

glMatrixMode(GL_PROJECTION);


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Tugas..Menggambarkan Orbit Bumi dan Bulan


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Tugas..Menggambarkan Gerakan Tutup Gelas Terbuka

Gelas tertutup Gelas terbukaGerakan terbuka dantertutup

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