2 d transformations 2
TRANSCRIPT
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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