Lecture 12

Shearing

Chapter 5: 2-D Geometric Transformations







Shearing, Composite Transformations, Properties of transformations

Distorts the shape such that the transformed shape appears as if the object were composed of layers that had been caused to slide over each other. Shearing is done in x-direction (shx) and in y-direction (shy).





Shearing w.r.t a reference line

Shearing in x




Shearing w.r.t. a reference lines shears and translates an object too by a shift amount.



 x '  1  y ' =  sh    0y  1   0

shx 1 0

0  x  0 ⋅  y  1  1 

+ive values of shx causes a shift in the object towards right. -ive values of shx causes a shift in the object towards left. +ive values of shy causes an upward shift in the object. -ive values of shy causes a downward shift in the object.

Shift = -shx.yref  for x-direction shearing. Shift = -shy.xref  for y-direction shearing.

MATRIX FOR SHEARING IN Y - ?

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Result of shearing an object in y-direction

Result of shearing an object in xdirection w.r.t a reference line

Reference line here is xref = -1 WHAT IS A REFERENCE LINE?

2D Composite Transformations



Sequence of transformations: composite transformation matrix Matrices listed in reverse order

  x   x  x'   y ' = M ⋅  M ⋅  y   = (M ⋅ M ) ⋅  y  2  1   2 1        1    1   1  

P' = M 2 ⋅ M 1 ⋅ P = M ⋅ P

Composite 2D Translations P ' = T (t 2 x , t 2 y ) ⋅ {T (t1 x , t1 y ) ⋅ P} = {T (t 2 x , t 2 y ) ⋅ T (t1x , t1 y )}⋅ P

1 0 t 2 x  1 0 t1 x  1 0 t1x + t 2 x  0 1 t  ⋅  0 1 t  = 0 1 t + t  2y   1y  1y 2y    0 0 1  0 0 1  0 0 1  T (t 2 x , t 2 y ) ⋅ T (t1x , t1 y ) = T (t1x + t 2 x , t1 y + t 2 y )

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Composite 2D Rotations P ' = R (θ 2 ) ⋅ {R (θ1 ) ⋅ P} = {R (θ 2 ) ⋅ R (θ1 )}⋅ P cos θ 2  sin θ 2   0

− sin θ 2 cos θ 2 0

0 cos θ1 0 ⋅  sin θ1 1  0

− sin θ1 0 cos(θ1 + θ 2 ) − sin (θ1 + θ 2 ) 0 cos θ1 0 =  sin (θ1 + θ 2 ) cos(θ1 + θ 2 ) 0 0 1  0 0 1

R (θ 2 ) ⋅ R (θ1 ) = R (θ 2 + θ1 )

Composite 2D Scalings P ' = S ( s2 x , s2 y ) ⋅ {S ( s1x , s1 y ) ⋅ P} = {S ( s2 x , s2 y ) ⋅ S ( s1x , s1 y )}⋅ P s2 x 0   0

0 s2 y 0

0  s1x 0 ⋅  0   1  0

0 s1 y 0

0  s1x ⋅ s 2 x 0 =  0   1  0

0 s1 x ⋅ s2 y 0

0 0  1

S ( s2 x , s2 y ) ⋅ S ( s1x , s1 y ) = S ( s1x s2 x , s1 y s2 y )

General 2D Pivot-Point Rotation

R (xr , y r , θ ) = T (xr , y r ) ⋅ R (θ ) ⋅ T (− xr ,− y r )

General 2D Fixed-Point Scaling

S (x f , y f , s x , s y ) = T (x f , y f )⋅ S (s x , s y )⋅ T (− x f ,− y f )

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Matrix Concatenation Properties


Associative:




Not commutative:

M 3 ⋅ M 2 ⋅ M 1 = (M 3 ⋅ M 2 ) ⋅ M 1 = M 3 ⋅ (M 2 ⋅ M 1 ) M 2 ⋅ M1 ≠ M1 ⋅ M 2

Order is important!

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