the -axis move in the opposite direction, while points on the axis stay fixed. Straight lines parallel to the -axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the -axis. Vertical lines, in particular, become oblique lines with slope . Therefore, the shear factor is the cotangent of the shear angle between the former verticals and the -axis. (In the example on the right the square is tilted by 30°, so the shear angle is 60°.) If the coordinates of a point are written as a column vector (a 2×1 matrix), the shear mapping can be written as multiplication by a 2×2 matrix: A vertical shear (or shear parallel to the -axis) of lines is similar, except that the roles of and are swapped. It corresponds to multiplying the coordinate vector by the transposed matrix: The vertical shear displaces points to the right of the -axis up or down, depending on the sign of . It leaves vertical lines invariant, but tilts all other lines about the point where they meet the -axis. Horizontal lines, in particular, get tilted by the shear angle to become lines with slope . General shear mappings For a vector space V and subspace W, a shear fixing W translates all vectors in a direction parallel to W. To be more precise, if V is the direct sum of W and W′, and we write vectors as v = w + w′ correspondingly, the typical shear L fixing W is L(v) = (Lw + Lw′) = (w + Mw′) + w′, where M is a linear mapping from W′ into W. Therefore in block matrix terms L can be represented as Applications The following applications of shear mapping were noted by William Kingdon Clifford: "A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area." "... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle." The area-preserving property of a shear mapping can be used for results involving area. For instance, the Pythagorean theorem has been illustrated with shear mapping as well as the related geometric