GRAFIKA KOMPUTER ~ M. Ali Fauzi. Drawing 2 D Graphics.

GRAFIKA KOMPUTER~ M. Ali Fauzi

Drawing

2D Graphics

VIEWPORTTRANSFORMATION

Recall :Coordinate SystemglutReshapeFunc(reshape);

void reshape(int w, int h){

glViewport(0,0,(GLsizei) w, (GLsizei) h);glMatrixMode(GL_PROJECTION);glLoadIdentity();gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

World Coordinate System

World Coordinate System

00 1 2-2 -1

1

2

-1

World Coordinate System

~ The representation of an object is measured in some physical or abstract units.~ Space in which the object geometry is defined.

World Window

0 1 2-2 -1

1

2

-1

World Window

0 1 2-2 -1

1

2

-1

World Window

~ Rectangle defining the part of the world we wish to display.~ Area that can be seen (what is captured by the camera), measured in OpenGL coordinates.

World Window

~ Known as clipping-area

void reshape(int w, int h){

glViewport(0,0,(GLsizei) w, (GLsizei) h);glMatrixMode(GL_PROJECTION);glLoadIdentity();gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

The Default

The default OpenGL 2D clipping-area is an orthographic view with x and y in the range of -1.0

and 1.0, i.e., a 2x2 square with centered at the origin.

World Window

~ Reset to the default

void reshape(int w, int h){

glViewport(0,0,(GLsizei) w, (GLsizei) h);glMatrixMode(GL_PROJECTION);glLoadIdentity();gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

ViewportScreen

Screenwindow

InterfaceWindow

Viewport

Screen Coordinate System

~ Space in which the image is displayed

~ For example : 800x600 pixels

Interface Window

~ Visual representation of the screen coordinate system for windowed displays

glutInitWindowSize(320, 320); glutInitWindowPosition(50, 50);

Vewport

~ A rectangle on the interface window defining where the image will appear, ~ The default is the entire screen or interface window.

Viewport

Interface Window

~ Set the viewport to the entire screen / windowvoid reshape(int w, int h){

glViewport(0,0,(GLsizei) w, (GLsizei) h);glMatrixMode(GL_PROJECTION);glLoadIdentity();gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

Interface Window

~ Set the viewport to half of the screen / window

glutInitWindowSize(320, 320); glutInitWindowPosition(50, 50);

--------------------------------

glViewport(0,0,160,160);

ViewportScreen

Screenwindow

InterfaceWindow

Viewport

Viewport

Viewport Transformation

~ The Process called viewport transformation

THE ASPECT RATIOPROBLEM

The distortionScreen

Screenwindow

InterfaceWindow

Viewport

The distortionScreen

Screenwindow

InterfaceWindow

Viewport

Ratio

~ R = Aspect Ratio of World

Ratio

Keeping the RatioScreen

Screenwindow

InterfaceWindow

Viewport

Keeping the RatioScreen

Screenwindow

InterfaceWindow

Viewport

The Startegy

~ To avoid distortion, we must change the size of the world window accordingly.~ For that, we assume that the initial world window is a square with side length L

The Startegy

~ DefaultglOrtho2D (-L, L, -L, L);

~ For example L=1, glOrtho2D (-1, 1, -1, 1);

The Startegy

if (w <= h)glOrtho2D(-L, L, -L * h/w, L * h/w);

elseglOrtho2D(-L * w/h, L * w/h, -L, L);

if (w <= h)glOrtho2D(-1, 1, -1 * h/w, 1 * h/w);

elseglOrtho2D(-1 * w/h, 1 * w/h, -1, L);

The Startegy

The Startegy

~ A possible solution is to change the world window whenever the viewport of the interface window were changed. ~ So, the callback Glvoid reshape(GLsizei w, GLsizei h) must include the following code :

void reshape(GLsizei width, GLsizei height) {

if (height == 0) height = 1; GLfloat aspect = (GLfloat)width/(GLfloat)height; glViewport(0, 0, width, height); glMatrixMode(GL_PROJECTION); glLoadIdentity(); if (width >= height) {

gluOrtho2D(-1.0 * aspect, 1.0 * aspect, -1.0, 1.0);

} else {

gluOrtho2D(-1.0, 1.0, -1.0 / aspect, 1.0 / aspect);

} }

/* Handler for window re-size event. Called back when the window first appears and whenever the window is re-sized with its new width and height */ void reshape(GLsizei width, GLsizei height) {

// GLsizei for non-negative integer // Compute aspect ratio of the new window if (height == 0) height = 1; // To prevent divide by 0 GLfloat aspect = (GLfloat)width / (GLfloat)height; // Set the viewport to cover the new window glViewport(0, 0, width, height); // Set the aspect ratio of the clipping area to match the viewport glMatrixMode(GL_PROJECTION); glLoadIdentity(); // Reset the projection matrix if (width >= height) {

// aspect >= 1, set the height from -1 to 1, with larger width gluOrtho2D(-1.0 * aspect, 1.0 * aspect, -1.0, 1.0);

} else {

// aspect < 1, set the width to -1 to 1, with larger height gluOrtho2D(-1.0, 1.0, -1.0 / aspect, 1.0 / aspect);

} }

2DTRANSFORMATION

Transformation

Is The geometrical changes of an object from a current state to modified state.

Why Needed?

To manipulate the initially created object and to display the modified object without having to redraw it

Transformation PipelineVertex Modelview

MatrixProjection

MatrixPerspective

DivisionViewport

Transformation

ObjectCoordinates

EyeCoordinates

ClipCoordinates

Normalizeddevice

Coordinates

WindowCoordinates

GL_MODELVIEW modeglTranslate()glRotate()glScale()glLoadMatrix()glMultMatrix()gluLookAt()

GL_PROJECTION modeglOrtho()gluOrtho2D()glFrustum()gluPerspective()

glViewport()

Types of Transformation

~ Modeling. In 3D graphics, handles moving objects around the scene.~ Viewing.In 3D graphics, specifies the location of the camera.

Types of Transformation

~ Projection.Defines the viewing volume and clipping planes from eye coordinate to clip coordinates.

Types of Transformation

~ Viewport. Maps the projection of the scene into the rendering window.~ Modelview.Combination of the viewing and modeling transformations.

Transformation Matrix

Transformations in OpenGL using matrix concept

Matrix Modes

~ ModelView Matrix (GL_MODELVIEW)

These concern model-related operations such as translation, rotation, and scaling, as well as viewing transformations.

Matrix Modes

~ Projection Matrix (GL_PROJECTION)

Setup camera projection and cliiping-area

Transformation PipelineVertex Modelview

MatrixProjection

MatrixPerspective

DivisionViewport

Transformation

ObjectCoordinates

EyeCoordinates

ClipCoordinates

Normalizeddevice

Coordinates

WindowCoordinates

GL_MODELVIEW modeglTranslate()glRotate()glScale()glLoadMatrix()glMultMatrix()gluLookAt()

GL_PROJECTION modeglOrtho()gluOrtho2D()glFrustum()gluPerspective()

glViewport()

Why do we use matrix?

~ More convenient organization of data.~ More efficient processing~ Enable the combination of various concatenations

THE MATRIX

Matrix addition and subtraction

a

b

c

d a c

b d=

Matrix addition and subtraction

1-3

56+

=

Matrix addition and subtraction

1-3

56+

=63

Matrix addition and subtraction

1-3

+ =

3 1-1 4

Matrix addition and subtraction

1-3

+ =

3 1-1 4

Tak boleh

Matrix Multiplication

a bc d

e fg h. =

a.e + b.g a.f + b.hc.e + d.g c.f + d.h

Matrix Multiplication

a be fg h. = a.e + b.g a.f + b.h

Matrix Multiplication

1 2 1 23 1. =

Matrix Multiplication

1 2 1 23 1. = 6 6

Matrix Multiplication

12

1 23 1. =

Matrix Multiplication

12

1 23 1. = Tak boleh

Matrix Types

efe f

Row Vector Column Vector

Matrix Multiplication

a bc d

ef. =

e fa bc d. =

e fa cb d. =

Matrix Multiplication

a bc d

ef. =

a.e + b.fc.e + d.f

e fa bc d. = a.e + c.f b.e + d.f

e fa cb d. = a.e + b.f c.e + d.f

Matrix Math

We’ll use the column-vector representation for a point.

y

x

Matrix Math

Which implies that we use pre-multiplication of the transformation – it appears before the point to be transformed in the equation.

DyCx

ByAx

y

x

DC

BA

THETRANSFORMATION

Translation

A translation moves all points in an object along the same straight-line path to new positions.

Translation

The path is represented by a vector, called the translation or shift vector.

x’y’

xy

tx

ty

= +

New Position Current Position Translation Vector

P' = P + T

Translation

x’y’

xy

tx

ty

= +

(2, 2) tx= 6

ty=4

?

x’y’

22

64

= +

Translation

x’y’

xy

tx

ty

= +

(2, 2) tx= 6

ty=4

?

86

22

64

= +

Rotation

A rotation repositions all points in an object along a circular path in the plane centered at the pivot point.

Rotation

P

P’

p'x = px cos – py sin

p'y = px sin + py cos

Rotation

The transformation using Rotation Matrix

New Position Rotation Matrix Current Position

P' = R . P

y

x

y

x

cossin

sincos

'

'

Rotation

The transformation using Rotation Matrix

New Position Rotation Matrix Current Position

P' = R . P

y

x

yx

yx

cossin

sincos

cossin

sincos

Rotation

Find the transformed point, P’, caused by rotating P= (5, 1) about the origin through an angle of 90.

Rotation

cossin

sincos

cossin

sincos

yx

yx

y

x

90cos190sin5

90sin190cos5

0115

1105

5

1

Scaling

Scaling changes the size of an object and involves two scale factors, Sx and Sy for the x- and y- coordinates respectively.

Scaling

Scales are about the origin.

P

P’

p'x = sx • px

p'y = sy • py

Scaling

The transformation using Scale Matrix

New Position Scale Matrix Current Position

P' = S • P

y

x

Sy

Sx

y

x

0

0

'

'

Scaling

The transformation using Scale Matrix

New Position Scale Matrix Current Position

P' = S • P

y

x

Sy

Sx

ySy

xSx

0

0

.

.

Scaling

If the scale factors are in between 0 and 1 the points will be moved closer to the origin the object will be smaller.

Scaling

P(2, 5), Sx = 0.5, Sy = 0.5Find P’ ?

p'x = sx • px

p'y = sy • py

P(2, 5)

P’

Scaling

If the scale factors are larger than 1 the points will be moved away from the origin the object will be larger.

Scaling

P(2, 5), Sx = 2, Sy = 2Find P’ ?

p'x = sx • px

p'y = sy • py

P(2, 5)

P’

Scaling

If the scale factors are not the same, Sx Sy differential scalingChange in size and shape

Scaling

If the scale factors are the same, Sx = Sy uniform scaling

Scaling

P(1, 3), Sx = 2, Sy = 5square rectangle

p'x = sx • px

p'y = sy • py

P(1, 2)

P’

Scaling

What does scaling by 1 do?Sx=1 , Sy=1

Scaling

What does scaling by 1 do?Sx=1 , Sy=1 Nothing changedWhat is that matrix called?

Scaling

What does scaling by 1 do?Sx=1 , Sy=1 Nothing changedWhat is that matrix called?Identity Matrix

y

x

y

x

10

01

Scaling

What does scaling by a negative value do? Sx=-2 , Sy=-2

Scaling

What does scaling by a negative value do? Sx=-2 , Sy=-2 Moved to Different Quadran

COMBINING TRANSFORMATIONS

Combining Transf

For example, we want to rotate/scale then we want to do translation

P' = M • P + A

Combining Transf

For example, we want to translate, then we want to rotate and sclae

P' = S • R • (A + P)

Combining Transf

P (Px,Py)=(4 , 6) : Translate(6 , -3) -> Rotation(60˚) -> Scaling

(0.5 , 2.0)P' = S • R • (A + P)

Combining Transf

3

10

3

6

6

4

'

'

Ty

Tx

Py

Px

y

x

2.10

4.2

3

10

5.087.0

87.05.0

60cos60sin

60sin60cos

'

'

Py

Px

y

x

3.20

2.1

2.10

4.2

20

05.0

0

0

'

'

Py

Px

Sy

Sx

y

x

Combining Transf

To combine multiple transformations, we must explicitly compute each transformed point.P' = S • R • (A + P)

Combining Transf

It’d be nicer if we could use the same matrix operation all the time. But we’d have to combine multiplication and addition into a single operation.P' = S • R • (A + P)

Combining Transf

Let’s move our problem into one dimension higher

HOMOGENOUS COORDINATES

Homogenous Coord

Let point (x, y) in 2D be represented by point (x, y, 1) in the new space.

y y

x

x

w

Homogenous Coord

~ Scaling our new point by any value a puts us somewhere along a particular line: (ax, ay, a).~ A point in 2D can be represented in many ways in the new space.~ (2, 4) ----------- > (2, 4, 1) or (8, 16, 4) or (6, 12, 3) or (2, 4, 1) or etc.

Homogenous Coord

We can always map back to the original 2D point by dividing by the last coordinate(15, 6, 3) (5, 2).(60, 40, 10) ?.

Homogenous Coord

The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate system its name – homogeneous coordinates.

Homogenous Coord

With homogeneous coordinates, we can combine multiplication and addition into a single operation.

Homogenous Coord

Point in column-vector:

Our point now has three coordinates. So our matrix is needs to be 3x3.

1

y

x

y

x

Homogenous Coord

Translation:

x’y’

xy

tx

ty

= +

1100

10

01

1

y

x

t

t

y

x

y

x

Homogenous Coord

Rotation:

y

x

y

x

cossin

sincos

'

'

1100

0cossin

0sincos

1

y

x

y

x

Homogenous Coord

Scaling:

1100

00

00

1

y

x

s

s

y

x

y

x

y

x

Sy

Sx

y

x

0

0

'

'

Homogenous Coord

P (Px,Py)=(4 , 6) : Translate(6 , -3) -> Rotation(60˚) -> Scaling

(0.5 , 2.0)

1

3.20

2.1

1

6

4

100

310

601

100

060cos60sin

060sin60cos

100

020

005.0

1

Qy

Qx

Homogenous Coord

We can represent any sequence of transformations as a single matrix

1

3.20

2.1

1

6

4

100

310

601

100

060cos60sin

060sin60cos

100

020

005.0

1

Qy

Qx

Homogenous Coord

Does the order of operations matter?

Homogenous CoordYes, the order does matter!

1. Translate2. Rotate

1. Rotate2. Translate

Homogenous CoordYes, the order does matter!A . B = B . A?

Homogenous CoordYes, the order does matter!A . B B . A

dhcfdgce

bhafbgae

hg

fe

dc

ba

hdgbhcga

fdebfcea

dc

ba

hg

fe

Homogenous CoordYes, the order does matter!(A.B).C = A.(B.C)?

Homogenous CoordYes, the order does matter!(A.B).C = A.(B.C)

ûëúù

êé

++++++++++++

=

úû

ùêë

é·ú

û

ùêë

é++++

=

dhlcfldgjcejdhkcfkdgiceibhlaflbgjaejbhkafkbgiaei

lkji

dhcfdgcebhafbgae

úû

ùêë

é·÷÷

ø

öççè

æúû

ùêë

é·ú

û

ùêë

é

lkji

hgfe

dcba

úû

ùêë

é++++++++++++

=

úû

ùêë

é++++

·úû

ùêë

é=

dhldgjcflcejdhkdgicfkceibhlbgjaflaejbhkbgiafkaei

hlgjhkgiflejfkei

dcba

÷÷ø

öççè

æúû

ùêë

é·ú

û

ùêë

é·ú

û

ùêë

é

lkji

hgfe

dcba

Composite Transformation Matrix• Arrange the transformation matrices in order from

right to left.• General Pivot- Point Rotation

• Operation :-1. Translate (pivot point is moved to origin)2. Rotate about origin3. Translate (pivot point is returned to original

position)

T(pivot) • R() • T(–pivot)

T(pivot) • R() • T(–pivot)

1 0 -tx

0 1 -ty

0 0 1

cos -sin 0sin cos 0 0 0 1

1 0 tx

0 1 ty

0 0 1 . .

cos -sin -tx cos+ ty sin + tx

sin cos -tx sin - ty cos + ty

0 0 1

cos -sin -tx cos+ ty sin sin cos -tx sin - ty cos 0 0 1

1 0 tx

0 1 ty

0 0 1 .

Composite Transformation Matrix

Reflection: x-axis y-axis

100

010

001

100

010

001

Other Transf

Reflection: origin line

x=y

100

010

001

100

001

010

Other Transf

• Shearing-x

• Shearing-y

1100

010

01

1

)( Py

PxS

Qy

Qx

SxShear

1100

01

001

1

)( Py

Px

SQy

Qx

SyShear

Sebelum Sesudah

Sebelum Sesudah

Shearing