Transcript Transformations 6 (Shear and Stretch)

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Shear
Example 1: 6/6 = 1 = shear factor
A
A’
6
Points on the invariant line do
not move with the shear, they
remain fixed.
6
B
A shear is a transformation of
an object in the plane in
relation to an invariant line.
It is a type of slide that is
parallel to the invariant line.
A shear is fully described only
when both the shear factor
and the invariant line are given.
C
Invariant Line BC
To calculate the shear factor
we note the distance moved by
a point and divide it by the
perpendicular distance of the
point from the invariant line.
Comment on the area of
both object and image.
Shear
Shear
Example 2: Calculate the shear factor when xy is the invariant line.
Shear factor = 6/2 = 3
6
2
y
x
Comment on the area of
both object and image.
Shear
Example 3: Calculate the shear factor when xy is the invariant line.
Shear factor = 6/4 = 1½
6
4
y
x
Comment on the area of
both object and image.
Shear
Question 1: Calculate the shear factor when xy is the invariant line in each case.
(a)
(b)
Shear Factor = 1
Image
(c)
Shear Factor = 1½
object
object
Shear Factor = 2
object
Image
Image
y
x
y
x
(d)
(e)
Shear Factor = 1½
object
y
x
Shear Factor = 3
Image
Image
object
x
x
y
y
Performing a Shear
Shear
Example 1: If xy is an invariant line, draw the image of the triangle under a
shear factor of 2 to the right.
A
x
These points on xy remain
fixed during the shear.
A’
A  xy = 8 units so
A  A’ = 16 units
parallel to xy.
y
Shear
Performing a Shear
Example 2: If A moves to A’ under a shear, draw the image and state the
shear factor. xy is the invariant line.
y
A’  A = 3 units and
b’
A’
A
x
c
b
c’
A’  xy = 1 units so
shear factor = 3
Since these points on xy
remain fixed we just need
to move points b and c
parallel to xy applying the
shear factor of 3.
b  xy = 3 so
b  b’ = 9
c  xy = 1½ so
c  c’ = 4½
Shear
Question 2: Using the shear factors given in each case, draw the image
after the shear against the invariant line xy.
(a)
Shear Factor = 1
(b)
Shear Factor = 1½
Shear Factor = 2
(c)
y
x
y
x
(d)
x
(e)
Shear Factor = 1½
x
x
y
y
Shear Factor = 3
y
Shear
Question 3: In each case below, A moves to A’ under a shear. Draw the image
and state the shear factor (xy is the invariant line).
(a)
y
(b)
x
A’
A’
A
Shear Factor = 1½
x
A
Shear Factor = 2
y
Stretch
Example 1:
6/3 = 2 = scale factor
A’
6
Points on the invariant line do
not move, they remain fixed.
To calculate the scale factor
we note the perpendicular
distances from xy of a point
and its image and divide image
distance by point distance.
A
3
x
A stretch is a transformation
of an object in the plane in one
direction only. It is fully
described only when both the
scale factor and the invariant
line are given.
y
Invariant Line xy
Stretch
Stretch
Example 2: Calculate the scale factor when xy is the invariant line.
Scale factor = 14/4 = 3½
x
4
y
A’
A
14
Stretch
Example 3: Calculate the scale factor when xy is the invariant line.
Scale factor = 10/5 = 2
x
5
10
y
Stretch
Example 4: Calculate the scale factor when xy is the invariant line.
Scale factor = 6/3 = 2
6
x
3
y
Stretch
Question 4: Calculate the scale factor when xy is the invariant line in each case.
(a)
(b)
Scale Factor = 3
x
Scale Factor = 2½
(c)
Scale Factor = 2
Image
Image
x
Object
Object
y
Image
Object
y
x
y
(d)
Image
x
Scale Factor = 3
(e)
Scale Factor = 1½
Image
Object
Object
y
x
y
Stretch
Question 5: Draw the image for the given scale factor and invariant line.
(a)
(b)
Scale Factor = 3
x
Scale Factor = 2½
(c)
Scale Factor = 1½
y
x
y
x
y
(d)
Scale Factor = 2
(e)
Scale Factor = 4
x
y
y
x
Stretch
Question 6: Find the invariant line in each case below.
(a)
(b)
Scale Factor = 2
Scale Factor = 2½
(c)
Scale Factor = 1½
x
Image
Image
x
Object
Object
Image
Object
y
x
y
y
y
x
(d)
Scale Factor = 3
(e)
Scale Factor = 2
Image
Object
x
Object
Image
y
Shear
Question 1: Calculate the shear factor when xy is the invariant line in each case.
(a)
(b)
Image
(c)
object
object
object
Image
Image
y
x
y
x
y
x
(e)
(d)
object
Image
Image
object
x
x
y
y
Worksheet
Shear
Question 2: Using the shear factors given in each case, draw the image after
the shear against the invariant line xy.
(a)
Shear Factor = 1
(b)
Shear Factor = 1½
Shear Factor = 2
(c)
y
x
y
x
(d)
x
(e)
Shear Factor = 1½
x
x
y
y
Shear Factor = 3
y
Shear
Question 3: In each case below, A moves to A’ under a shear. Draw the image
and state the shear factor (xy is the invariant line).
(a)
y
(b)
x
A’
A
A’
A
x
y
Stretch
Question 4: Calculate the scale factor when xy is the invariant line in each case.
(a)
(b)
x
(c)
Image
Image
x
Object
Object
y
Image
Object
y
x
y
(e)
(d)
Image
x
Image
Object
Object
y
x
y
Stretch
Question 5: Draw the image for the given scale factor and invariant line.
(a)
(b)
Scale Factor = 3
x
Scale Factor = 2½
(c)
Scale Factor = 1½
y
x
y
x
y
(d)
Scale Factor = 2
(e)
Scale Factor = 4
x
y
y
x
Stretch
Question 6: Find the invariant line in each case below.
(a)
Scale Factor = 2
(b)
Scale Factor = 2½
(c)
Scale Factor = 1½
Image
Image
Object
Image
Object
(d)
Object
Scale Factor = 3
(e)
Scale Factor = 2
Image
Object
Object
Image