transformation.doc

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-axis

-axis

P

P

T

-axis

2D Transformations

Translation

Repositioning an object along a straight line path from one coordinate location to another

Add translational distance tx ,ty to original coordinate position (x,y) to move the point to a new position (x,y)x = x + tx y = y + ty

where the pair (tx , ty) is called the translation vector.

We can write equation as a single matrix equation by using column vectors to represent coordinate points and

translation vectors. i.e.P = x T = tx P = x

y ty y .so we can write

P = P + T

.Scaling

The transformation changes thesize of an object

Such that we can magnify or reduce its sizeIn case of polygons scaling is done by multiplying coordinate values (x,y) of each vertex by scaling factors sx sy

produce the final transformed coordinates (x,y).sx scales object in x directionsy scales object in y direction

or

P = x T = sx 0 P = xy 0 sy y

or

x sx 0 x

y = 0 sy . yor

P = S . P

Values greater than 1 for s x s y produce enlargementValues greater than 1 for s x s yreduce size of object

s x = s x = 1 leaves the size of the object unchanged

When s x, s y are assigned the same value s x = s x = 3 or 4 etc then a Uniform Scalingis produced.

Rotation

Repositioning an object along a circular path in x,y plane

Specify rotation angle and position (x r,y r) of rotation point about which the object is to be rotated+ value for define counter-clockwise rotation about a point

- value for define clockwise rotation about a point

If (x,y) is the original point r the constant distance from origin, the original angular displacement from x-

axis.Now the point (x,y) is rotated through angle in a counter clock wise direction

Express the transformed coordinates in terms of and asx = r cos( + ) = r cos. cos - r sin. sin .(i)

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

We know that original coordinates of point in polar coordinates arex = r cos

y = r sin

substituting these values in (i) and (ii)

Anil Verma, IOE Pulchowk Page

rr

x

x

x -axis

x -axis


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-axis

-axis

-axis

we get,

x = x cos - y sin

y = x sin + y cos

so using column vector representation for coordinate points the matrix form would beP = R . P

where the rotation matrix is

R = cos -sin

sin cos

Reflection

Generates a mirror image of the original object

(i).Reflection about x axis or about line y = 0

Keeps x value same but flips y value of coordinate pointsSo x = x

y = -yi.e.

x = 1 0 x

y 0 -1 y

(ii).Reflection about y axis or about line x = 0

Keeps y value same but flips x value of coordinate points

So x = -x

y = yi.e.

x = -1 0 x

y 0 1 y

(iii).Reflection about origin

Flip both x and y coordinates of a pointSo x = -x

y = -y

i.e.

x = -1 0 xy 0 -1 y

(iv).Reflection about line y = x

Steps required:i. Rotate about origin in clockwise direction by 45

degree which rotates line y = x to x-axisii. Take reflection against x-axis

iii. Rotate in anti-clockwise direction by same angle

here,R = cos sin R = cos -sin

-sin cos sin cos

For reflection against x-axis,


x -axis

x -axis

x -axis

x -axis


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-axis

-axis

Rf = 1 0

0 1

so for = 45 we get

R(y = x) = 0 11 0

(v).Reflection about line y = -x

Steps required:i. Rotate about origin in clockwise direction by 45

ii. Take reflection against y-axisiii. Rotate in anti-clockwise direction by same angle

here,

R = cos sin R = cos -sin

-sin cos sin cosFor reflection against x-axis,

Rf = 1 0

0 1so for = 45 we get

R(y = -x) = 0 -1-1 0

Shearing

Distorts the shape of object in either x or y or both direction

In case of single directional shearing (e.g. in x direction can be viewed as an object made up of very thin layer

and slid over each other with the base remaining where it is).

in x direction,

x = x + shx . yy = y

x = 1 shx x

y 0 1 . yin y direction,

x = x

y = y + shy . x

x = 1 0 xy shy 1 . y

in both directions,

x = x + shx . y

y = y + shy . xx = 1 shx x

y shy 1 . y

Homogenous CoordinatesThe matrix representations for translation, scaling and rotation are respectively:

P = T + P

P = S . P


bi direction sheary direction shear

x direction shear

x -axis

x -axis


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P = R . P

Translation is treated differently (as addition) from scaling and rotation (as multiplication)

We need to treat all three transformations in a consistent way so they can be combined easily.

Graphics applications involve sequence of geometric transformations. e.g. animation may require an object to betranslated , rotated etc in a sequence.

For an animation that requires scaling, rotation, translation, instead of applying each transformation one at a tim

separately we can combine them so that final coordinates are obtained directly from initial coordinates thus

eliminating intermediate steps.Here, incase of homogenous coordinates we add a third coordinate h to a point (x,y) so that each point is

represented by (hx, hy, h).h is normally set to 1.

If points are expressed in homogenous coordinates, all geometrical transformation equations can be represented

matrix multiplications.

Coordinates of a point are represented as three element column vectors, transformation operations are written asx 3 matrices.

So, for translation we have

x 1 0 tx xy = 0 1 ty . y

1 0 0 1 1or, P = T( tx , ty ) . PWith T( tx , ty ) as translation matrix, inverse of this translation matrix is obtained by representing tx, ty with -tx-ty.

Similarly, the rotation transformation equation about the coordinate origin arex cos -sin 0 x

y = sin cos 0 . y

1 0 0 1 1

or, P = R() . PHere, rotation operator R() is a 3 x 3 matrix and inverse rotation is obtained with .

Similarly, scaling relative to coordinate origin is

x shx 0 0 xy = 0 shy 0 . y

1 0 0 1 1

or, P = S( shx , shy ) . PInverse scaling matrix is obtained with 1 / shx and 1 / shx .Composite Transformation

With the matrix representation of transformation equations it is possible to setup a matrix for any sequence of

transformations as a composite transformation matrix by calculating the matrix product of individualtransformation.

For column matrix representation of coordinate positions we form composite transformation by multiplying

matrices in order from right to left.

i. Two Successive Translation are Additive

Let two successive translation vectors ( tx1 , ty1 ) and ( tx2 , ty2 ) are applied to a coordinate position P thenor, P = T( tx2 , ty2 ) . {T( tx1 , ty1 ) . P} and P = {T( tx2 , ty2 ) . T( tx1 , ty1 ) } . P

Here the composite transformation matrix for this sequence of translation is

1 0 tx2 1 0 tx1 1 0 tx1 + tx2

0 1 ty2 . 0 1 ty1 = 0 1 ty1+ ty2

0 0 1 0 0 1 0 0 1



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or, T( tx2 , ty2 ) . T( tx1 , ty1 ) = T( tx1 + tx2 , ty1 + ty2 )

ii. Two successive Scaling operations are MultiplicativeLet ( sx1 , sy1 ) and ( sx2 , sy2 ) be two successive vectors applied to a coordinate position P then the composite

scaling matrix thus produced is

or, P = S( sx2 , sy2 ) . {S( sx1 , sy1 ) . P} and P = {S( sx2 , sy2 ) . S( sx1 , sy1 ) } . P

Here the composite transformation matrix for this sequence of translation is

sx2 0 0 sx1 0 0 sx2 . sx1 0 00 sy2 0 . 0 sy1 0 = 0 sy2 . sy1 0

0 0 1 0 0 1 0 0 1

or, S( sx2 , sy2 ) . S( sx1 , sy1 ) = S( sx1 . sx2 , sy1 . sy2 )

iii. Two successive Rotation operations are Additive

Let R(1) and R(2) be two successive rotations applied to a coordinate position P then the composite scalingmatrix thus produced is

or, P = R(2) . {R(1) . P} and P = {R(2) . R(1) } . PHere the composite transformation matrix for this sequence of translation isor,

cos2 -sin2 cos 1 -sin 1

sin2 cos2 . sin 1 cos 1

or,

cos2 . cos1 - sin2 . sin1 -cos2 . sin1 - sin2 . cos1sin2 . cos1 + sin1 . cos2 . cos2 . cos1 - sin2 . sin1

or,

cos(2 + 2) -sin(2 + 2)

sin(2 + 2) . cos(2 + 2)

or, R(2) . R(1) = R(1 + 2)

Affine Transformation

Linear 2D geometric transformation which maps variables (e.g. pixel intensity values located at point (x1,y1) in a

input image) into new variables (e.g. (x2y2) in an output image) by applying a linear combination of translation,

rotation , scaling and/or shearing operations.

**NOTE:

- See class notes for pivot and fixed point transformations

- See class notes for reflection about arbitrary line (y = mx + c)


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transformation.doc