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Geometrical Transformations
3
...
They
also have many important applications
...
For example, structural engineers use stretches,
shears, and rotations to understand the deformation of materials
...
Many of
these simple geometrical transformations in R2 and R3 are linear
...
(projil and perpil belong to the list of geometrical transformations, too, but they were
discussed in Chapter 1 and so are not included here
...
See Figure 3
...
1
...
Thus, Re(t1) = tRe(1) for any t E R
...
Since the shape
of a parallelogram is not altered by a rotation, the picture for the parallelogram rule of
addition should be unchanged under a rotation, so Re(1 + y) = Re(1) + Re(Y)
...
3
...
Counterclockwise rotation through angle e in the plane
...
Assuming that
calculate
[Re]
...
3
...
3
...
What is the matrix of rotation of�
...
/3/2
-
v'3/2
-1/2
]
X1
EXERCISE 1
Determine
[Rrr14]
and use it to calculate
Rrr14(1, 1)
...
Rotation Through Angle(} About the x3-axis in IR
...
3
...
This rotation leaves x3 unchanged, so that if the transformation is de
R, R(O,
noted
then
0, 1)
(0, 0, 1)
...
Is it
3
possible to determine the matrix of a rotation about an arbitrary axis in JR ? We shall
see how to do this in Chapter 7
...
3
...
Stretches Imagine that all lengths in the x1 -direction in the plane are stretched by a
scalar factor t > 0, while lengths in the x2-direction are left unchanged (Figure 3
...
4)
...
) It should be obvious that
stretches can also be defined in the x2-direction and in higher dimensions
...
Contractions and Dilations If a linear operator T : JR2
[� �l
with t > 0, then for any 1, T(x)
=
�
JR2 has matrix
tx, so that this transformation stretches
vectors in all directions by the same factor
...
If 0 < t < 1, such a
transformation is called a contraction; if t > l, it is a dilation
...
3
...
Sometimes a force applied to a rectangle will cause it to deform into a paral
lelogram, as shown in Figure 3
...
5
...
Although the deformation
=
of a real solid may be more complicated, it is usual to assume that the transformation S
is linear
...
Since the action of
...
(0, 1)
(5, 1)
(2, 1)
(2 + 5, 1)
------
(2,0)
Figure 3
...
5
A shear in the direction of
x1
by amounts
...
2 or Coordinates Planes in
IR;
...
2 � JR
...
3
...
Then each vector
corresponding to a point above the axis is mapped by R to the mirror image vector
below
...
...
...
...
mu larly, a re ect1on mt
[- OJ
l
0
1
...
Points above the plane are reflected to points below the plane
...
-1
l
Figure 3
...
6
EXERCISE 2
A reflection in JR;
...
W rite the matrices for the reflections in the other two coordinate planes in JR
...
General Reflections
We consider only reflections in (or "across") lines in JR
...
3 that pass through the origin
...
Consider the plane in JR
...
Since a reflection is related to
proj,7, a reflection in the plane with normal vector it will be denoted refl;t
...
Figure 3
...
7 shows reflection in a line
...
reft;t a
Figure 3
...
7
A reflection in R2 over the line with nonnal vector n
...
The calcu
lations for reflection in a line in IR2 are similar to those for a plane, provided that the
equation of the line is given in scalar form it· 1 0
...
=
-
=
EXAMPLE2
Consider a r·efiection refl,1
:
1!
...
Determine the matrix [refl,1]
...
The equation
for refl;t(p) can be written as
refl;t(P)
=
Id(jJ)
-
2 proj;t(P )
=
(Id +(-2) projit )(jJ)
Thus,
PROBLEMS 3
...
(b) 1T
can
(d) 6f'
(c) -�
A2 (a) In the plane, what is the matrix of a stretch S
by a factor 5 in the x2-direction?
(b) Calculate the composition of S followed by a
rotation through angle e
...
A3 Determine the matrices of the following reflections
2
in JR
...
3
3
AS (a) Let D : JR � JR be the dilation with factor
3
� JR4 be defined by
t = 5 and let inj : JR
inj(x1,x2,x3)
(x1,X2,0,x3)
...
2
3
(b) Let P : JR � JR be defined by P(x1,x2,X3)
3
(x2, x3) and let S be the shear in JR such that
=
x2
...
=
A4 Determine the matrix of the reflections in the fol
3
lowing plane in JR
...
Determine the
2
(c) Can you define a shear T : JR
To P
�
2
JR such that
Po S, where P and S are as in part (b)?
2
3
(d) Let Q : JR � JR be defined by Q(x1,x2,x3) =
0
=
(xt, x2)
...
Homework Problems
Bl Determine the matrices of the rotations in the plane
B4 Determine the matrix of the reflections in the fol
through the following angles
...
(a)-
(a) Xt - 3x2
�
(c) �
(b)
(d)
-n
-�
(b) 2X1
B2 (a) In the plane, what is the matrix of a stretch S
by a factor 0
...
B3 Determine the mat1ices of the following reflections
2
in JR
...
=
X2- X3
=
0
=
0
3
3
BS (a) Let C : JR � JR be contraction with fac
3
tor 1 /3 and Jet inj : JR � JR5 be defined by
(b) Calculate the composition of S followed by a
rotation through angle e
...
Determine the
matrix of inj oC
...
Determine the
matrices Co S and S o C, where C is the con
traction in part (a)
...
=
(xt
+
3x2,x2,X3)
...
Conceptual Problems
Dl Verify that for rotations in the plane [Ra o Re]
[Ra][Re] = [Ra+e]
...
are refiect10n matrices
...
Determine the angle of the
rotation
...
D3 In R3, calculate the matrix of the composition of a
reflection in the x2x3-plane followed by a reflection
in the x1 x2-plane and identify it as a rotation about
some coordinate axis
...
(Hint: Think geometrically
...
DS From geometrical considerations, we know that
reflt1 o refl,1 = Id
...
(Hint: [reft,1] = I 2[projn] and proj,1 sat
isfied the projection property from Section 1
...
)
-