Unit 7.5: Shear

1. Unit 7.5: Shear A transformation that changes the shape but not the size of the figure.

2. 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)

3. Find the shear factor, k given the figures of object and image under shear with x-axis as the invariant line. 8 4 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 !!

4. 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 B Invariant line

5. 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 B Invariant line

6. Shear factor = Distance a point moves due to shear Perpendicular distance of point from the invariant line C C’ Shear this triangle by the shear factor 1. The x-axis is the invariant line A B’ A’ B Invariant line

7. 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

8. Shear this triangle by the shear factor 1. The line y = 4 is the invariant line C C’ • If invariant line is x-axis or parallel to x-axis: • Above invariant line Invariant line • If k = +ve (shear to the right) B A’ A B’ • If k = -ve (shear to the left) • Below invariant line • If k = +ve(shear to the left) • If k = -ve(shear to the right)

9. 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

10. 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’

11. Shear this triangle by the shear factor 1. The line x = 4 is the invariant line Invariant line C B’ C’ • If invariant line is y-axis or parallel to y-axis: • Right of invariant line A B • If k = +ve (shear upwards) • If k = -ve (shear downwards) A’ • Left of invariant line • If k = +ve(shear downwards) • If k = -ve(shear upwards)

12. Describe fully the transformation under shear. Important points required to describe a shear: the word ‘shear’ invariant line shear factor, k

13. y 6 4 2 2 4 8 10 x 6 Describe fully the transformation that takes triangle A onto triangle B • shear • invariant line is the x axis • shear factor is 8 = 2 4 8 A B 4

14. y 6 4 2 2 4 8 10 x 6 Describe fully the transformation that takes triangle A onto triangle B • shear • invariant line is • shear factor is 4 = 1 8 2 the y axis 8 B 4 A

15. y 6 4 2 2 4 8 10 x 6 Describe fully the transformation that takes triangle A onto triangle B 6 A B • shear 6 • invariant line is y = 1 • shear factor is 6 = 1 6