Shear — Definition, Formula & Examples

A shear is a geometric transformation that slants a figure in one direction, shifting each point by an amount proportional to its distance from a fixed line. The shape tilts like a deck of cards being pushed sideways, but its area stays the same.

A shear mapping is an affine transformation in which all points are displaced parallel to a fixed direction by a distance proportional to their signed distance from a line (the invariant line) parallel to that direction. In two dimensions, a horizontal shear is given by $(x, y) \mapsto (x + ky,\, y)$ where $k$ is the shear factor, and a vertical shear is given by $(x, y) \mapsto (x,\, y + kx)$ .

Key Formula

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

Where:

$k$ = Shear factor — controls how much the figure slants
$(x, y)$ = Original point
$(x', y')$ = Image point after the horizontal shear

How It Works

Pick the direction you want to shear (horizontal or vertical) and a shear factor

k

. For a horizontal shear, every point moves left or right by

k

times its

y

-coordinate, while its

y

-coordinate stays fixed. Points on the

x

-axis don't move at all because their

y

-value is zero. The farther a point is from the invariant line, the more it slides. Rectangles become parallelograms, but the base and height remain the same, so the area is preserved.

Worked Example

Problem: Apply a horizontal shear with shear factor k = 2 to the point (3, 4).

Step 1: Use the horizontal shear formula: the new x-coordinate is x + ky.

x' = 3 + 2(4) = 11

Step 2: The y-coordinate is unchanged in a horizontal shear.

y' = 4

Answer: The image of the point is (11, 4).

Why It Matters

Shear transformations appear in computer graphics when italicizing text or skewing images. In physics, shear describes how layers of material slide past each other under stress. Understanding shear also helps in linear algebra courses when studying how matrices encode different types of geometric change.

Common Mistakes

Mistake: Assuming a shear changes the area of a figure.

Correction: The shear matrix has determinant 1, so it always preserves area. A rectangle becomes a parallelogram with the same base and height.