Transcript Unit 7.5: Shear

Unit 7.5: Shear
A transformation that changes the
shape but not the size of the figure.
Shear
The transformation consists of:
• The invariant line (x-axis or y-axis or any line
parallel to the x- and y-axes)
• Shear factor, k
Shear factor,
k = displacement of image point from object point
displacement of object point from L (invariant line)
Find the shear factor, k given the figures of
object and image under shear with x-axis as
the invariant line.
8
4
CC '
K
BC
8

4
K 2
All points move a distance parallel to the invariant line,
except points which are on the invariant line (x-axis)
POINTS ON INVARIANT LINE DO NOT MOVE !!
Shear factor =
Distance a point moves due to shear
Perpendicular distance of point from the invariant line
Shear this
triangle by
the shear
factor 1. The
line AB is the
invariant line
C’
C
A
Invariant line
B
Shear factor =
Distance a point moves due to shear
Perpendicular distance of point from the invariant line
Shear this
triangle by
the shear
factor -1.
The line AB
is the
invariant line
C’
C
A
Invariant line
B
Shear factor =
Distance a point moves due to shear
Perpendicular distance of point from the invariant line
Shear this
triangle by
the shear
factor 1. The
x-axis is the
invariant line
C
A
C’
A’
Invariant line
B
B’
Shear factor =
Distance a point moves due to shear
Perpendicular distance of point from the invariant line
Shear this
triangle by
the shear
factor 2. The
x-axis is the
invariant line
C
C’
A
B
Invariant line
Shear this
triangle by
the shear
factor 1. The
line y = 4 is
the invariant
line
If invariant line is x-axis
or parallel to x-axis:
• Above invariant line
 If k = +ve
(shear to the right)
 If k = -ve
(shear to the left)
• Below invariant line
 If k = +ve
(shear to the left)
 If k = -ve
(shear to the right)
C
C’
Invariant line
A’
A B’
B
Shear factor =
Distance a point moves due to shear
Perpendicular distance of point from the invariant line
Shear this
triangle by
the shear
factor 1½ .
The y-axis is
the invariant
line
C’
B
Invariant line
A
C
Shear factor =
Distance a point moves due to shear
Perpendicular distance of point from the invariant line
Shear this
triangle by
the shear
factor -2.
The y-axis is
the invariant
line
A
C
A’
B
Invariant line
C’
B’
Shear this
triangle by
the shear
factor 1. The
line x = 4 is
the invariant
line
If invariant line is y-axis
or parallel to y-axis:
• Right of invariant line
 If k = +ve
(shear upwards)
 If k = -ve
(shear downwards)
• Left of invariant line
 If k = +ve
(shear downwards)
 If k = -ve
(shear upwards)
Invariant line
C
B’
C’
A
B
A’
Describe fully the transformation
under shear.
Important points required to describe a shear:
• the word ‘shear’
• invariant line
• shear factor, k
y
Describe fully the transformation that takes triangle A onto triangle B
•shear
•invariant line is the x axis
6
•shear factor is
8=2
4
8
4
A
2
B
4
2
4
6
8
10
x
y
Describe fully the transformation that takes triangle A onto triangle B
•shear
•invariant line is the y axis
•shear factor is
4=1
8 2
8
6
B
4
4
2
A
2
4
6
8
10
x
y
Describe fully the transformation that takes triangle A onto triangle B
6
6
4
A
B
•shear
6
•invariant line is y = 1
•shear factor is
2
2
4
6
8
6=1
6
10
x