cgmb214_topic 6_ geometric transformation

7/30/2019 CGMB214_topic 6_ Geometric Transformation

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T O P I C 6 :

G E O M E T R I C T R A N S F O R M A T I O N

CGMB214: Introduction to

Computer Graphics


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Objectives

To provide an understanding on the geometrictransformation


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So far.

We have been discussing the basic elements ofgeometric programming. We have discussed points,

vectors and their operations and coordinate frames

and how to change the representation of points andvectors from one frame to another.

Next topic involves how to map points from oneplace to another (transformation).


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Affine Transformation

In this topic, we will concentrate on one particulartransformation called affine transformations.Examples of affine transformations are:

translations

Rotations

Uniform and non-uniform scaling

reflections (flipping objects about a line)

shearing ( which deform squares into parallelogram)


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Examples of affine transformation


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Transformation in 2-D and 3-D


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Affine Transformation-Characteristics

These transformations all have a number of thingsin common.

They all map lines to lines ( parallel lines will still be parallelafter the transformations)

Translation, rotation and reflection preserve the lengths ofline segments and the angles between segments

Uniform scaling preserve angles but not length

Non-uniform scaling and shearing do not preserve angles or

lengths


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Affine Transformation-Usage

Transformations are useful in a number ofsituations:

1. We can compose a scene out a number of objects


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2. Can design a single motif and manipulate themotif to produce the whole shape of an objectespecially if the object has certain symmetries.

Example of snowflake after reflections, rotations and translations of themotif


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3. To view an object from a different angle


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4. To produce animation


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The two special types of affine transformation are

1. Rigid transformation: These transformationspreserve both angles and lengths (I.e. translations,

rotations and reflections) 2. Orthogonal transformation: These

transformations preserve angles but not necessarilylength.


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Affine Transformation -why?

The most common transformations used incomputer graphics.

They simplify operations to scale, rotate and

reposition objects. Several affine transformations can be combined into

a simple overall affine transformation in terms of acompact matrix representation


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Transformation Matrix (2-D)

For a point P that map to Q after affinetransformation has a matrix representation asshown:

11001

232221

131211

y

x

y

x

PP

mmmmmm

QQ

NB: the third of the resultant matrix will always be 1 fora point


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Transformation Matrix (2-D)

For a vector V that maps to W after affinetransformation has the matrix representation asshown:

01000

232221

131211

y

x

y

x

V

V

mmm

mmm

W

W

NB: the third row of the resultant matrix will always be 0for a vector


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Transformation Matrix

The scalar m that we have in the transformationmatrix will determine which affine transformationthat we want to perform.


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OpenGL graphics pipeline

Normally in OpenGL, many transformationprocesses occur before the final objects are displayedon the screen. Basically the object that we define inour world will go through the same procedure.


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Transformations

Transformations: changes in size, shape andorientation that alter the coordinate descriptions ofobjects.

Usually, transformations are represented andcalculated using matrices.

OpenGL also uses the same approach to performtransformations


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Sine and Cosine

Sin = b/c if c = 1 then b = sin

Cos = a/c if c = 1 then c = cos

c

a

b

Recall our algebra lesson!!


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Sine and Cosine (cont.)

90

0180

270

y

x

z

-


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Sine and Cosine (cont.)

Cos = x/z if z = 1 then cos = x

Sin = y/z if z = 1 then sin = y

Cos (- ) = x/z = cosSin (- ) = -y/z = -sin

Cos ( ) = cos cos - sin sinSin ( ) = sin cos + cos sin


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Matrices

In graphics most of the time matrix operation that we willdeal with is matrix multiplication. The formula formultiplication of matrix A with m xp dimension and matrixB withp x n dimension is:

mnm

ij

n

pnpjp

nj

mpm

ipi

p

cc

c

cc

bbb

bbb

aa

aa

aa

...

.....

..

.....

...

......

.........

.........

.........

......

...

.....

...

.....

...

1

111

1

1111

1

1

111

where

p

k

kjikpjipjijiij babababac1

2211...


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Example

987

654

321

A

10

21

01

B

251

161

71

AB


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Characteristics of Matrix

Multiplication of matrices is not commutative

AB != BA

Multiplication of several matrices is associative

A(BC) = (AB)C


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Identity Matrix

Set of matrices that when they multiply anothermatrix which reproduce that matrix is called identitymatrices i.e:

1000

0100

0010

0001

100

010

001

10

011

1D 2D 3D 4D

Obj T f i C di


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Object Transformation vs CoordinateTransformation

Object Transformation: alters the coordinates ofeach point on the object according to some rule. Nochange of coordinate system

Coordinate Transformation: defines a newcoordinate system in terms of the old one and thenrepresents all of the objects point in the newcoordinate system


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Translation

Reposition an object along a straight line path fromone coordinate location to another

2D point is translated by adding translation

distances to the x and y coordinates When translating (x,y) to (x,y) by value t x = x + tx y = y + ty


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Translation

The (tx,ty) is the translation distances calledTRANSLATION VECTOR

In matrix form

Then point P = P + T

yxP

'

''yxP

y

x

ttT

y

x

y

x

tytx

tt

yxP'


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Translation

For 2-D translation the transformation matrix Thasthe following form:

100

10

01

y

x

m

m

Where mxand myare the translation values in x and y

axis

T=


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Translation (cont.)

For 3-D translation the transformation matrix T hasthe following form:

1000

100

010

001

z

y

x

mm

m

Where mx, myand mzare the translation values in x, y andz axis

T=


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Translation

Translation is a rigid body transformation the object is not being deformed

all points are moved in the same distance

How to translate? Straight lines based on end points

Polygons - based on vertices

Circles - based on centre


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Rotations

Reposition an object along a circular path in aspecified plane

Rotations are specified by:

a rotation angle a rotation point (pivot point) at position (x,y)

Positive rotation angle means rotate counter-clockwise negative rotation angle means rotate

clockwise


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Rotations

(x,y)

(x,y)

r

r

The original point (x,y) can be represented in polarcoordinates form:

x = r cos (1)y = r sin (2)


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Rotations

The new point (x,y) can be expressed as x = r cos ( )

y = r sin ( )

These equation can be written as x = r cos cos - r sin sin

y = r cos sin + r sin cos


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Rotations

Substituting the equations with equation (1) and (2),we get

x = x cos - y sin

y = x sin + ycos

Therefore, the rotation matrixR can be expressed as

cossin

sincosR=


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Rotations

For 2-D rotation the transformation matrixR has thefollowing form (in homogeneous coordinate system):

100

0cossin

0sincos

NB: for counter-clockwise rotation


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Rotations

In 3-D world, the rotation is more complex since wehave to consider rotation about three different axis;x, y and z axis.

Therefore we have three different transformationmatrices in 3-D world.

1000

0100

00cossin

00sincos

Rz( ) =

Rotation about z-axis (ccw)


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Rotations

This rotation matrix is for case where the rotation isat the origin.

For a rotation at any other points, we need to

perform the following transformations: Translating the object so the rotation point is at the origin.

Rotating the object around the origin

Translating the object back to its original position

Recall that rotation is a rigid affine transformation


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Scaling

Scaling changes the size of an object

Scaling can also reposition the object but not always

Uniform scaling is performed relative to some

central fixed point (I.e at the origin). The scalingvalue (scale factor) for uniform scaling must be bothequal.

Non-uniform scaling has different scaling factors.

Also refers as differential scaling


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Scaling

The value of the scale factors (Sx, Sy, Sz) determinethe size of the scaled object.

if equal to 1 -> no changes

if greater than 1 -> increase in size (magnification)

if 0 < scale factors < 1 -> decrease in size

(demagnification)

if negative value -> reflection !!!


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Scaling (cont.)

For 2-D scaling the transformation matrix S willhave the following form

100

00

00

y

x

S

S

S =


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Scaling

For 3-D scaling the transformation matrix S has thefollowing form:

1000

000

000

000

z

y

x

S

S

S

S =


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Reflection

For 3-D reflection, the transformation matrixFhasthe following form

1000

0100

0010

0001

F(z) =

Reflection about z-axis


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Reflection

Reflection can be generalized by concatenatingrotation and reflection matrices.

Example: If reflection at y=x axis (45 degree), thetransformations involved are:

1. Clockwise rotation of 45 degree

2. Reflection about x axis

3. Counter clockwise rotation of 45 degree


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Shearing

Distort an object by moving one side relative toanother

It neither rigid body nor orthogonal transformation.i.e. changing a square into parallelogram in 2-D orcube into parallelepiped in 3-D space

It is normally used to display italic text using regularones


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Shearing

For 2-D shearing transformation the transformationmatrix has the following form

X direction y direction

100

010

01 xsh

100

01

001

ysh


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Shearing

The values can be positive or negative numbers

Positive: moves points to the right

Negative: moves points to the left


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Shearing

For 3-D space shearing transformations the numberof transformation matrix are many.

Basically, to get one transformation matrix forshearing, we can substitute any zero term in identitymatrix with a value like example below:

1000

010

01

0001

b

ca


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Example

Let say we want to produce an object that goesthrough the following transformation operations:

translate by (3, -4),M1 then rotate through 30 degree,M2 then scale by (2, -1),M3 then translate by (0, 1.5),M4 and finally, rotate through 30 degree,M5

All of these transformation can be represented by asingle matrix,M

M = M5M4M3M2M1


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Affine transformation in OpenGL

Recall our lesson on OpenGL graphics pipeline.

CT = Current Transformation


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OpenGL

All the transformations will be performed in CT bychanging the current transformation matrix. In other

way, the CT matrix is an example of compositematrix.

Typically, in OpenGL, we will have the followingfragment of code in our program before we startperforming transformations.

glMatrixMode(GL_MODELVIEW);glLoadIdentity();


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OpenGL

By having this code in our program, we set ourmatrix mode to be GL_MODELVIEW and initializethe CT matrix to be identity

CT I

Once we have set this, we can performtransformation which means we modify the CT bypost-multiplication by a matrix


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OpenGL

CT


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OpenGL

The three transformation supported in most graphicssystem(including OpenGL) are translation, rotation

with a fixed point of the origin and scaling with afixed point of the origin.

The functions for these transformation in OpenGLare: glTranslatef(dx, dy, dz);

glRotatef (angle, vx, vy, vz);

glScalef(sx, sy, sz);


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OpenGL

If we want to perform rotation at any other point usingthe provided functions in OpenGL, (I.e. 45 degreerotation aboutthe line through the origin and the point(1,2,3) with a fixed point of (4,5,6).) the following codefragment shows us how toperform this transformation.

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glTranslatef(4.0,5.0,6.0);

glRotatef(45.0,1.0,2.0,3.0);glTranslatef(-4.0,-5.0,-6.0);

NB: Take note at the reverse order implementation


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OpenGL

For most purposes, rotation, translation and scalingcan be used to form our desired object. However, insome cases the provided transformation matrices arenot enough. For example, if we want to form a

transformation matrix for shearing and reflection.

For these transformations, it is easier if we set up thematrix directly. We can load a 4 x 4 homogeneous-

coordinate matrix as the current matrix (CT)


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O


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OpenGL

To define a user-defined matrix (for shearing andreflection) we can follow the same way as shown

below:

Glfloat myarray[16];

for (i=0;i

cgmb214_topic 6_ geometric transformation

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