A shear slides one axis by a multiple of the other. The area of the unit square stays 1, and the corner angle changes.
The shear slides x. It adds k times y to x. Horizontal lines stay horizontal.
Area1.00
Angle—
The red square is the unit square. Its area is the absolute value of the determinant. For this shear, that value stays 1.
2×2 shear. Read each row from left to right.
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Where this is used
Type designers make oblique letters by shearing the upright forms. A true italic is usually drawn again, not only sheared.
Technical drawings use an oblique view. That view shears the front of an object so the side can show.
A document scanner deskews a crooked page. That correction removes a shear.
In geology and in structural engineering, shear is a related idea. Layers or parts slide past each other. The matrix on this page is not that full model, but the picture is the same kind of slide.
Common mistakes
A shear does not change the area. The absolute value of the determinant stays 1.
A shear is not a rotation. The right angle is lost, and the slanted side grows longer.
An oblique letter is a sheared upright. A true italic is usually drawn again, not only sheared.
In geology, shear is a related idea. The matrix on this page is the slide, not that full model.