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My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant line & shear-factor of 3

http://i.stack.imgur.com/ysr4y.png

My question is if you are provided the original polygon & asked to do shear with y=3 as invariant & shear-factor 3 or -3 how would I know whether to slide segment AD to right or left? Same confusion thus occurs with segment BC?

Moreover shear-factor is defined via (object-image dist)/(object-invariant dist) thus if the object polygon ABCD & its image say the blue one is given & you are asked to completely define the transformation I can give the invariant line & the factor's magnitude but I can't tell the sign (+ or -) of the shear-factor because distances are always positive? Plz help this teacher

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1 Answer 1

When you have the y axis or the x axis as the invariant line then it is relatively simple to use the transformational matrices for shear to determine the directions. So if the invariant line is either one of the axes you can simply use a transformational matrix to determine the shear factor if it is positive or negative.

Generally if the invariant line is horizontal, and the object point is in: Quadrants 1 and 2: Images are to the right if positive shear factor, to the left if negative shear factor. Quadrants 3 and 4: Images are to the left if positive shear factor, to the left if negative shear factor.

If the invariant line is vertical, and the object point is in: Quadrants 1 and 4: Images move up if positive shear factor and down if negative shear factor. Quadrants 2 and 3: Images move up if negative shear factor and down if positive shear factor.

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