GRAFIKA KOMPUTER - Universitas Brawijaya · World Coordinate System ~ The representation of an...

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


00 1 2-2 -1

1

2

-1


~ The representation of anobject is measured in somephysical or abstract units.~ Space in which the objectgeometry is defined.

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


{



glLoadIdentity();

gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

The Default

The default OpenGL 2Dclipping-area is anorthographic view with x

and y in the range of -1.0and 1.0, i.e., a 2x2 square

with centered at the origin.

World Window

~ Reset to the default


{



glLoadIdentity();

gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

Viewport

Screen

Screenwindow

InterfaceWindow

Viewport

Screen Coordinate System

~ Space in which the image is displayed

~ For example : 800x600pixels

Interface Window

~ Visual representation of thescreen coordinate system forwindowed displays

glutInitWindowSize(320, 320);

glutInitWindowPosition(50, 50);

Vewport

~ A rectangle on the interfacewindow defining where theimage will appear,~ The default is the entirescreen or interface window.

Viewport

Interface Window

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

{



glLoadIdentity();

gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

}

Interface Window

~ Set the viewport to half ofthe screen / window

glutInitWindowSize(320, 320);

glutInitWindowPosition(50, 50);

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

glViewport(0,0,160,160);

Viewport

Screen

Screenwindow

InterfaceWindow

Viewport

Viewport

Viewport Transformation

~ The Process called viewporttransformation

THE ASPECT RATIOPROBLEM

The distortion

Screen

Screenwindow

InterfaceWindow

Viewport

Ratio

~ R = Aspect Ratio of World

Keeping the Ratio

Screen

Screenwindow

InterfaceWindow

Viewport

The Startegy

~ To avoid distortion, we mustchange the size of the worldwindow accordingly.~ For that, we assume that theinitial world window is asquare 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);

else

glOrtho2D(-L * w/h, L * w/h, -L, L);

if (w <= h)

glOrtho2D(-1, 1, -1 * h/w, 1 * h/w);

else

glOrtho2D(-1 * w/h, 1 * w/h, -1, L);

The Startegy

The Startegy

~ A possible solution is to changethe world window whenever theviewport of the interfacewindow were changed.~ So, the callback Glvoidreshape(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);


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


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);

}

}

2D

TRANSFORMATION

Transformation

Is The geometrical changesof an object from a currentstate to modified state.

Why Needed?

To manipulate the initiallycreated object and todisplay the modified objectwithout having to redraw it

Transformation Pipeline

Vertex ModelviewMatrix

ProjectionMatrix

PerspectiveDivision

ViewportTransformation

ObjectCoordinates

EyeCoordinates

ClipCoordinates

Normalizeddevice

Coordinates

WindowCoordinates

GL_MODELVIEW mode

glTranslate()

glRotate()

glScale()

glLoadMatrix()

glMultMatrix()

gluLookAt()

GL_PROJECTION mode

glOrtho()

gluOrtho2D()

glFrustum()

gluPerspective()

glViewport()

Types of Transformation

~ Modeling.In 3D graphics, handles movingobjects around the scene.

~ Viewing.In 3D graphics, specifies thelocation of the camera.


~ Projection.Defines the viewing volume andclipping planes from eyecoordinate to clip coordinates.


~ Viewport.Maps the projection of the sceneinto the rendering window.

~ Modelview.Combination of the viewing andmodeling 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 Pipeline

Vertex ModelviewMatrix

ProjectionMatrix

PerspectiveDivision

ViewportTransformation

ObjectCoordinates

EyeCoordinates

ClipCoordinates

Normalizeddevice

Coordinates

WindowCoordinates

GL_MODELVIEW mode

glTranslate()

glRotate()

glScale()

glLoadMatrix()

glMultMatrix()

gluLookAt()

GL_PROJECTION mode

glOrtho()

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

da c

b d=


1-3

56+ =


1-3

56+ =

63


1-3

+ =3 1-1 4


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


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


1 2 1 23 1. =


1 2 1 23 1. = 6 6


12

1 23 1. =


12

1 23 1. = Tak boleh

Matrix Types

efe f

Row Vector Column Vector


a bc d

ef. =

e fa bc d. =

e fa cb d. =


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.

Matrix Math

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

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

6

4= +

Translation

x’y’

xy

tx

ty= +

(2, 2) tx= 6

ty=4

?

86

22

6

4= +

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

Rotation

Find the transformed point, P’,

caused by rotating P= (5, 1) about the

origin through an angle of 90.

Rotation

Scaling

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

Scaling

Scales are about the origin.

P

P’

p'x = sx • pxp'y = sy • py

Scaling

The transformation using Scale Matrix

New Position Scale Matrix Current Position

P' = S • P

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(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(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(1, 2)

P’

Scaling

What does scaling by 1 do?

Sx=1 , Sy=1

Scaling


Sx=1 , Sy=1 Nothing changedWhat is that matrix called?

Scaling


Sx=1 , Sy=1 Nothing changedWhat is that matrix called?Identity Matrix

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

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 threecoordinates. So our matrix is needs to be 3x3.

Homogenous Coord

Translation:

x’y’

xy

tx

ty= +

Homogenous Coord

Rotation:

Homogenous Coord

Scaling:

Homogenous Coord

P (Px,Py)=(4 , 6) : Translate(6 , -3) ->

Rotation(60˚) -> Scaling (0.5 , 2.0)

Homogenous Coord

We can represent any sequence of transformations as a single matrix

Homogenous Coord

Does the order of operations matter?

Homogenous Coord

Yes, the order does matter!

1. Translate2. Rotate

1. Rotate2. Translate

Homogenous Coord

Yes, the order does matter!A . B = B . A?

Homogenous Coord

Yes, the order does matter!A . B B . A

Homogenous Coord

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

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

++++++

++++++=

++

++=

dhlcfldgjcejdhkcfkdgicei

bhlaflbgjaejbhkafkbgiaei

lk

ji

dhcfdgce

bhafbgae

lk

ji

hg

fe

dc

ba

++++++

++++++=

++

++

=

dhldgjcflcejdhkdgicfkcei

bhlbgjaflaejbhkbgiafkaei

hlgjhkgi

flejfkei

dc

ba

lk

ji

hg

fe

dc

ba

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 sinsin 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

Sebelum Sesudah

Sebelum Sesudah

Shearing

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