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Lecture 9

2D Transformations IIBasic Transformations

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Basic Transformations:

Translation

Rotation

Scaling

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Translation:

Moving an point from one location to another location without altering

the point in a straight line is called translation

Translation is the displacement of the point along a straight line We translate a 2D point (x, y) by adding translation distances tx, ty, i.e.,

x = x + tx

y = y + ty

Where tx is the displacement in x direction and ty is the displacement in y

direction

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

vector

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Translation:

We can use the matrices for the translated equations like:

Or we can write

xPy

!

txTty

!

''

'xPy

!

' P P T !

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Translation:

Translating an object

means moving an object

without deforming theobject i.e., every point of

the object is translated by

the same amount

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Translation:

-

1

3

-

5

5

tx = 2

ty = 4

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Rotation:

A rotation is applied to an object by reposition it along a circular path in

the xy plane

To generate a rotation we specify

The rotation angle and

The rotation point or the pivot point about which the object is to be rotated

If the angle is taken positive, the object is rotated counter clockwise If the angle is taken negative, the object is rotated clockwise

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Rotation:

Suppose the Point P = (x, y) is

the pivot point

Suppose the point is r unitsfrom the origin and making an

angle , then the parametric

equation for the P is

x = r cos

y = r sin

Now if we want to rotate this point

by degree, then

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Rotation:

The new angle will be ( + )

And the new point will be given by

x = r cos ( + ) = r cos cos r sin sin

y = r sin ( + ) = r cos sin + r sin cos

But we know that:

r cos = x and r sin = y

x = x cos y sin and

y = x sin + y cos

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Rotation:

In the matrix format, we get

i.e., P = R . P

'''

xPy

!

cos sinsin cos

R U U

U U

!

xPy

!

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Scaling:

A scaling transformation alters

the size of the object

It could be increasing the sizeof the object

Or decreasing the size of the

object

-1

2

6

3

-

Sx = 3

Sy= 3

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Scaling:

Suppose P = (x, y) is the point

and Sx is the scaling factor

along x-axis and Sy along y-axis,then the scaled point can be

expressed as

x = x . Sx

Y = y . Sy

To scale a polygon, we

calculated the scaled points for

each of the vertices

-1

2

6

3

-

Sx = 3

Sy= 3

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Scaling:

In the matrix format, we get

i.e., P = S . P

If the scaling factor is < 1, the size of the object will decrease

If the scaling factor is > 1, the size will increase

''

'xPy

!

0

0

SxS

Sy !

xPy

!

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Scaling:

There are two types of scaling

Uniform Scaling

Differential Scaling

In uniform scaling, the Sx and Sy values are always equal. By scaling an

object uniformly, the shape of the object remains intact

In differential scaling the Sx and Sy factors are unequal. By applying the

differential scaling, the object loses its original shape

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Scaling:

The scaling can be applied to lines, circles, polygons and ellipses

For a line, the scaling is applied at end points

For polygons, the scaling is applied at each vertex For circles and ellipses, the scaling is applied to only radii