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AML710 CAD
LECTURE 4
Geometric Transformations
Two dimensional Transformations
Representation of Points and Lines
A vertex or point denotes location
A point is represented as a position vector
In two dimensions as [x y] and in three
dimensions [x y z] or alternatively by column
vectors as [x y]T and [x y z]T respectively.
y
P(x,y)
x
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GeometricTransformations
• A geometric object is represented by its vertices (as
position vectors)
A geometric transformation is an operation that modifies
its shape, size, position, orientation etc with respect to its
current configuration operating on the vertices (position
vectors).
• Mathematically a transformation P*=L(P) where P* is called
the image of P
• It can be seen as a mapping from R2 to R2
• Therefore P=L-1(P*), where L-1 is an inverse operator of L
Transformation as Matrix
multiplication
• Given two matrices [A] and [B] find the solution matrix [T]
such that
[ B] = [ A][T ]
• We know that in the above case the solution works out to
be:
[T ] = [ A]−1[ B]
where [A]-1 is the inverse of the square matrix [A]
• An alternate way is to see the matrix [T] as a geometric
operator and the matrices [A] and [T] are assumed known
where matrix [A] contains set of position vectors (vertices)
w.r.t to some coordinate system that need to be
transformed
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Geometric Transformation
• Consider a 2-D position vector of an arbitrary point as [x y]
• Let us take a 2 x 2 matrix [T] (a geometrical operator) as
given below for studying the effect of each element on the
transformed coordinates of the point [x y]
[ X *] = [ X ][T ]
[x *
y *] = [x
⎡a b ⎤
y ]⎢
= [ax + cy bx + dy ]
⎥
⎣c d ⎦
Transformation of Points and Lines
•
•
•
•
Let us consider some typical cases
Case 1: a=d=1 and b=c=0 – No Change (identity)
Case 2: d=1, b=c=0 – Scaling in x coordinate
Case 3: b=c=0 – Scaling in both x and y coordinates
[x *
y *] = [x
[x *
y *] = [x
[x *
y *] = [x
⎡1
y ]⎢
⎣0
⎡a
y ]⎢
⎣0
0⎤
= [x y ]
1⎥⎦
0⎤
= [ax y ]
1⎥⎦
⎡ a 0⎤
y ]⎢
⎥ = [ax by ]
0
b
⎣
⎦
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Transformation of Points and Lines
• Case 4; a=d =|s|>1 – Enlargement of the original entity
• Case 5: 0<a=d=|s|<1 – Compression of the entity
[x *
y *] = [x
⎡ a 0⎤
y ]⎢
⎥ = [ax by ]
0
b
⎣
⎦
• Note that scaling with respect to origin involves translation
Transformation of Points and Lines
• Case 6: b=c=0, a=1,d=-1 – Reflection about x- axis
• Case 7: b=c=0, a=-1,d=1 – Reflection about y- axis
• Case 8: b=c=0, a=d<0 – Reflection about the origin
⎡1 0 ⎤
y ]⎢
⎥ = [x − y ]
0
−
1
⎣
⎦
− 1 0⎤
[x * y *] = [x y ]⎡⎢
⎥ = [− x y ]
0
1
⎣
⎦
−1 0 ⎤
[x * y *] = [x y ]⎡⎢
⎥ = [− x − y ]
0
1
−
⎣
⎦
[x *
y *] = [x
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Transformation of Points and Lines
• Case 9: a=d=1, c=0 – Shear along y
• Case 10: a=d=1, b=0 – Shear along x
• Case 11: a=d=1- Two-dimensional shear
⎡1
y ]⎢
⎣0
1
[x * y *] = [x y ]⎡⎢
⎣c
1
[x * y *] = [x y ]⎡⎢
⎣c
[x *
y *] = [x
b⎤
= [x (bx + y )]
1⎥⎦
0⎤
= [( x + cy ) y ]
1⎥⎦
b⎤
= [x + cy bx + y ]
1⎥⎦
Morphing: An Application of Shear
Shear results in material deformations of mechanical problems and it is
exploited in motion pictures and animated movies.
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