komfik_geometri

Komputer GrafikKuliah 1: Prinsip Dasar Komputer Grafik

Dr. Trias Aditya

Rute• Computer Graphic Pipeline Proses Operasi Komputer Grafik Teknik Geodesi & Geomatika & Komputer Grafik Contoh Aplikasi

• Geometri 2D & Geometri 3D Sistem Koordinat Transformasi Koordinat Proyeksi

Computer Graphics Pipeline

Definisi 101

• Komputer Grafik: mencakup tampilan,manipulasi dan penyimpanan gambar dandata eksperimental untuk penyajianmenarik memanfaatkan komputer

• Komputer grafik terdiri dari komputer,memori dan: monitor berwarna, pirantikeras penunjang pemgolahan data dantampilan serta piranti lunak data grafis

Definisi 101

• Komputer Grafik: mencakup tampilan,manipulasi dan penyimpanan gambar dandata eksperimental untuk penyajianmenarik memanfaatkan komputer

• Komputer grafik terdiri dari komputer,memori dan: monitor berwarna, pirantikeras penunjang pemgolahan data dantampilan serta piranti lunak data grafis

• Proses memproduksi gambar atau citramenggunakan komputer proses menulis program untukmenghasilkan/mengolah grafik gambar yang dihasilkan tertayang dilayar komputer sebagai element grafikdisebut “pixel”

ObjectSpecification

SceneDescription

ModelTransformation

SceneDescription

Clipping &Hidden Surface

Removal

ViewTransformation

View & LightSpecification

Shading Image

Image DisplayTransformation

Output

MODELING

RENDERING

DISPLAY

McConnell 2006

Komputer Grafik untuk TeknikGeodesi & Geomatika

• Ragam AplikasiKomputer Grafik:- 3D object modeliing- animation- visualization

3D surface model

Depth image Triangle mesh Texture image

Textured 3DWireframe model

• intermap_harrier2.avigallery_30.mpeg

Supporting Disciplines

• Computer science (algorithms, datastructures, software engineering, …)

• Mathematics (geometry, numerical, …)• Physics (Optics, mechanics, …)• Psychology (Colour, perception)• Art and design

Applications

• Computer Aided Design (CAD)• Computer Aided Geometric Design

(CAGD)• Entertainment (animation, games, …)• Geographic Information Systems (GIS)• Visualization (Scientific Vis., Inform. Vis.)• Medical Visualization• …

Interactive Computer Graphics

User

Application

Screen

input

image

A detailedpolygonmesh(thank

you, WetaDigital)

Graphics pipeline

User

Screen

Model

Geometry, colour

Map

Display

Edit

Manipulation of geometry andcolor…

Monsters, Inc

Manipulation of geometry andcolor…

Manipulation of geometry andcolor…

Manipulation of geometry andcolor…

Manipulation of geometry andcolor…

Representations in graphics

Vector Graphics• Image is represented by continuous

geometric objects: lines, curves, etc.

Raster Graphics• Image is represented as an rectangular

grid of coloured squares

Vector graphics

• Graphics objects: geometry + colour• Complexity ~ O(number of objects)• Geometric transformation possible without

loss of information (zoom, rotate, …)• Diagrams, schemes, ...• Examples: PowerPoint, CorelDraw, ...

Raster graphics• Generic• Image processing techniques• Geometric Transformation: loss of

information• Complexity ~ O(number of pixels)• Jagged edges, anti-aliasing• Realistic images, textures, ...• Examples: Paint, PhotoShop, ...

Conversion

Vector graphics

Rasterization, Pattern recognition

Scan conversion

Raster graphics

Hardware

• Vector graphics• Raster graphics• Colour lookup table• 3D rendering hardware

I/O Devices

3D Scanner 2D ScannerCCDCharge couple device

Vector Graphics Hardware

V E C T O R

Display Controller

move 10 20

line 20 40

...

char O

char R

Display list

continuous & smoothlines

no filled objects

random scan

refresh speed dependson complexity of thescene

Raster Graphics Hardware

Video Controller

jaggies (stair casing)

filled objects

(anti)aliasing

R A S T E R

refresh speed independent ofscene complexity

pixel

scan conversion

resolution

bit planes

0 0 0 0 0 00 7 7 7 60 7 7 70 0 00 00

Frame buffer

Colour Lookup Table

0 0 0 0 0 00 7 7 7 6

0 7 7 70 0 0

0 00

Frame buffer

0 0 0

102 255 53

255 255 204

255 102 153

102 0 51

RR GG BB

0

1

2

4

7

...

colourindex

CLUT:

pixel = code

True colour:

pixel = R,G,B

Geometri Komputer Grafik

2D geometric modelling

• Coordinates• Transformations

Coordinates

• Point: position on planep = (px , py)x = (x, y)x = (x1 , x2)x = x1 e1 + x2 e2, e1 = (1, 0), e2 = (0, 1)

• Vector: direction and magnitudev = (vx , vy), etc.

x

yp

v

Vector arithmetic

• Addition of two vectors:v + w = (vx + wx , vy + wy)

• Multiplication vector-scalar:v = (vx , vy)

x

yw

v

v+w

x

y

v

2v

Coordinate systems

imagewheel

train

world

vanwijk@win.tue.nl

Why??

Pendefinisian Sistem Koordinatpenting karena….

• Terdapat banyak sistem koordinat padaproses/operasi komputer grafik: WorldCoordinates, Model Coordinates, ViewCoordinates

• Perlu kerangka referensi untukmemposisikan obyek dalam dunia nyata,model maupun dalam tampilan.

Contoh

• Penggambaran obyek 2D pada komputer:

x

y

x

y

Koordinat Peta

Koordinat Layar

Transformasi Geometri

Transformasi Geometri padaKomputer Grafik

• Transformasi geometri memegang perananpenting di komputer grafik karena : obyekdipindah, diperbesar untuk menghasilkanefek/modifikasi

• Graphics API (Application ProgrammingInterface) melakukan transformasi untukmenempatkan/merepresentasikan obyeksebagai tampilan

• Perlu dilakukan transformasi 3D ke 2D (obyekdengan geometri 3D ke layar 2D)

Transformation Pipeline

• World Transformation– Model coordinates World coordinates

• View Transformation– World coordinates Camera space

• Projection Transformation– Camera space View Plane

• These are a series of matrix multiplications

World Transformation

• Translation• Rotation• Scaling

+x

+z

+y

World origin

World Coordinates

Local model coordinates

Local model coordinates

View Transformation

+x

+z

+y

World origin

World Coordinates

+y

+x

+z

• Cameraposition

• Look vector

• Perbesaran (scaling), Rotation (rotasi) dantranslasi (translation) merupakan jenistransformasi yang penting pada komputergrafik.

Skala

Rotasi

Pemotongan (Shear)

Translasi (Translation)

NOT LINEAR!!!Perkalian matrix tidak dapat digunakan

Homogeneous Coordinates

• Pada komputer grafik, transformasi yangkompleks (menggunakan matrix)diperlukan.

• Dengan pendekatan Kartesian, translasidan rotasi, perbesaran tidak dapatdigabung menjadi satu proses

Xn

Yn=

X

YT *

Koordinat Homogen

• Untuk merepresentasikan transformasisebagai matrix diperlukan sistem koordinatyang mampu mewadahi rotasi, perbesarandan translasi sebagai matrix multiplikasi.

• X,y,z pada koord. Homogen = x/z, y/zpada koord. Kartesian

• (X0, y0) pada koord. Kartesian = (x0,y0,1)pada koord. Homogen = (z.x0, z.y0, z)dimana z # 0

Koord. Homogen

Translasi pada Koord. Homogen

Frank Klawonn, Intro to Computer Graphics

Koordinat Homogen• Kartu grafis pada komputer bisa bekerja

cepat karena komposisi matrix utuktransformasi dapat dirangkai menjadi satu

• urutan transformasi penting!• secara default dari kanan ke kiri

contoh

Aplikasi Transformasi (contoh)

Frank Klawonn, Intro to Computer Graphics

3D coordinate system

• Pada koordinat 2D : semua datar• Untuk merepresentasikan kedalaman

diperlukan koordinat z• 3 sumbu koordinat!!• Semua titik pada ruang 3D dapat

dinyatakan sebagai x,y,z

3D coordinate system

x

y

z

(5,0,0)

(5,5,0)

(5,5,5)

(0,0,0)

(0,0,5)

(0,5,5)

(5,0,5)

(0,5,5)

a

b

c

d

e

fg

h

i

j

k

l

Koordinat Homogen

• Melibatkan tidak hanya koordinat x,y dan ztetapi juga nilai w (matrix 4 x 4)

• Titik 3D (x,y,z) direpresentasikan sebagai(x,y,z,1).

• Jika nilai w tidak 1 (x/w, y/w, z/w, 1)

Terms and definitions

C

x

y

z A

B

A’

B’

screenz extent

x extent

y extent

view volume

centre of projection

2D -> 3D• good news is that we can still use our drawing, shape, line

and point classes.

• bad news is that they are all going to require some (hopefullysmall) alterations to work in 3d.

3D Points

1zyx•Still using homogenous

coordinates

•Need 4 coords )1,,,( zyx

3D Transformations

– Since we now have an additional coordinateto cope with, our transformation matricesmust also expand to cope.

– each transformation matrix with require anextra row and an extra column to become4x4 matrices.

Scaling

x

y

z

1

.

1000000000000

1zyx

SS

S

zyx

z

y

x

Translation

x

y

z

1

.

1000100010001

1zyx

TTT

zyx

z

y

x

Rotation (about axes)• In 2 dimensions, rotation were defined to take place about

some stationary point - the centre of rotation.

• In 3d, all rotation take place about one of the axes.

x

y

z

x

y

z

x

y

z

+ve

+ve

+ve

Rotation (about axes)‏

x

y

z

=-90°

-90 about x axis

Rotation

• x-axis

• y-axis

• z-axis

1

.

10000cossin0

0sincos00001

1z

yx

z

yx

1

.

10000cos0sin

00100sin0cos

1z

yx

z

yx

1

.

10000100

00cossin00sincos

1z

yx

z

yx

Local Rotation

• translate the object tobe rotated to theorigin, perform therotation and translateit back to where itstarted from

x

y

z

=-90°

Proyeksi

• Representasi object 3D di atas layar 2D• Asal muasalnya dan seringnya digunakan

oleh juru gambar dan arsitek dalammenggambar desain

• Several different projections existBergantung dari arah penglihatanpengamat

Terms and definitions

C

x

y

z A

B

A’

B’

screenz extent

x extent

y extent

view volume

centre of projection

(also called plane)

Macam Proyeksi

…so?

• We need to calculate the screen coordinates(xsys) of each projected point from the 3dcoordinates– (they’re not the same!)‏

• Usually!

• Fortunately – we can consider this calculationas a transformation.

• Can use matrices to represent the variousways of doing it.

Parallel Projection

1

.

10000000

00100001

10 z

yx

yx

s

s

Perspective Projection

1

.

1/100000000100001

1

0 zyx

ddz

yx

s

s

Perspective Projection

p (x,y,z)‏

z

x

screen

ps (xs,ys)‏C

d

By similar triangles:

dx

dzx s

dz

xxs

1

dz

yys

1

Perspective Projection

10

1

1

.

dz

ydz

x

want So….Divide by

dz

1

Perspective Projection

1

.

1/100000000100001

1

0 zyx

ddz

yx

s

s

Oblique Parallel Projections• A parallel projection is one where all the lines that are

parallel in 3d space remain parallel when projected.

• An oblique parallel projection is one where the object inquestion is viewed “from the side” - unlike the (plain)parallel projection.

• The z axis is drawn at some angle () to the x axis

• A number of standard oblique parallelprojections are used in engineering drawing:

– cavalier

– cabinet

– orthogonal

Cavalier Projection

The length of a line onthe screen is equal to itslength in the model.

This causes a distortionby over emphasising thez-axis

Cabinet Projection

The foreshortening of the zaxis is increased to provide amore “realistic” view.

Orthogonal Projection

The orthogonal projection isjust the simple parallelprojection

Oblique Parallel Projections

x

y

(xs,ys)‏

‏(0,0,1)P

Consider the point P:

P can be represented in 3Dspace - ‏(0,0,1)

P can be represented in 2D(screen coords) - (xs,ys)‏

P can be represented in 2Dvia polar coordinates - (,)‏

Oblique Parallel Projections

• At ‏(0,0,1)xs = cos ys = sin

• Generally– multiply by z and allow for (non-zero) x and y

xs= x + z..cos ys = y + z..sin

Oblique Parallel Projection

x

y

z

(0,0,1)

Cavalier cabinetand orthogonalprojections can allbe specified interms of ()

-or () since

tan = 1/

Oblique Parallel Projection

= 0 – 360Orthogonal projection= 90=0

= 0 – 360Cabinet projection= 63.4=0.5

= 0 - 360Cavalier projection= 45=1

Oblique Parallel Projectionas a matrix

1

.

100000000sin100cos01

10 z

yx

yx

s

s

Isometric Projection

1000060cos060sin0010060cos060sin

All 3 axes are equally foreshortened allowingmeasurements along the axes to made with the same scale.