2 d transformations 2

8/20/2019 2 D Transformations 2

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Topics to be Covered

Transformation Geometrical

Coordinate

Matrix Representations and Homogenous Coordinates

Basic Transformations

Translation

Rotation

Scaling

Reflection

Shearing

Composite Transformations

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Transformation

Simulated spatial manipulation is referred as

Transformation

Two types

Geometric

Coordinate


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Translation Displacement of an object in a given distance and direction from its original

position.

Rigid body transformation that moves object without deformation Initial Position point P (x, y)

The new point P’ (x’, y’)

where

x’ = x + tx

y’ = y + tytx and ty is the displacement in x and y respectively.

The translation pair (tx, ty) is called a translation vector or shift vector

P(x,y)

P’(x’,y’)


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TRANSLATION

Matrix representation

y

x P

'

'

' y

x

P

ty

tx

T


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Rotation

Rotation is applied to an object by repositioning it

along a circular path in the xy plane.

To generate a rotation, we specify

Rotation angle θ Pivot point ( xr , yr)

Positive values of θ for counterclockwise rotation

Negative values of θ for clockwise rotation.


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2-D Rotation

(x, y)

(x’, y’)

Ф


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2-D Rotation

x = r cos (f)

y = r sin (f)

x’ = r cos (f + )

y’ = r sin (f + )

Trig Identity…

x’ = r cos(f) cos() – r sin(f) sin()

y’ = r sin(f) sin() + r cos(f) cos()

Substitute…

x’ = x cos() - y sin()

y’ = x sin() + y cos()


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2-D Rotation

P R P '

cossin

sincos R

x’ = x cos() - y sin()

y’ = x sin() + y cos()

Matrix representation



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Scaling

Scaling alters the size of an object .

Operation can be carried out by multiplying each of its

components by a scalar

Uniform scaling means this scalar is the same for all

components:

2


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Scaling

Non-uniform scaling: different scalars per component:

X 2,

Y 0.5


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Scaling

x’ = x* sx

y’ = y * sy

In matrix form:

y

x

sy

sx

y

x

0

0

'

'


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Homogenous Coordinate System

Allows us to express all transformation equations as

matrix multiplications , providing that we also

expand the matrix representations for coordinatepositions.


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Reflection

A reflection is a transformation that produces a

mirror image of an object

Generated relative to an axis of reflection

1. Reflection along x axis

2. Reflection along y axis

3. Reflection relative to an axis perpendicular to the xy plane and

passing through the coordinate origin

4. Reflection of an object relative to an axis perpendicular to the xy

plane and passing through point P

5. Reflection of an object with respect to the line y=x


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Reflection About x-Axis

P1

P3P2

P1’

P2’ P3’

x

yOriginalImage

Reflected Image


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100

010

001

M

Reflection about x-axis

Reflection About x-Axis



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Reflection About y-axis

Original

Image

x

y

Reflected

Image


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100

010

001

M

Reflection about y-axis

Reflection About y-axis



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Reflection relative to an axis perpendicular to the xy plane and passing through the

coordinate origin

x

y

ReflectedImage

OriginalImage


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100

010

001

M

Reflection about the origin point

Reflection relative to an axis perpendicular to the xy plane and passing through the

coordinate origin



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Reflection of an object with respect to the line y=x

x

y

Reflected

Image

Original

Image


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100

001

010

M

Reflection about with respect to line y=x

Reflection of an object with respect to the line y=x



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Shearing

A transformation that distorts the shape of an

object such that the transformed object appears as

if the object were composed of internal layers that

had been caused to slide over each other.


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Shearing

Shear relative to the x-axis • Shear relative to the y-axis

100

010

01 y

sh

100

01001 x sh


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Composite Transformations

For a sequence of transformations , composite

transformation matrix could be setup by the matrix

product of the individual transformations

Also referred as Concatenation or Composition ofMatrices

2 d transformations 2

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