2D Shear

Theory of Computation & Computer Graphics — Two-Dimensional Transformation, NEC licence examination syllabus (Nepal Engineering Council).

2D Shear: pushing a shape sideways, like a slanted stack of cards

The transformation that turns a rectangle into a parallelogram.

X-direction shear matrix (shift x based on y, factor sh): [x'] [1 sh 0] [x] [y'] = [0 1 0] [y] [1 ] [0 0 1] [1] Worked example: shear point (2,3) with shx=2 x' = 1(2) + 2(3) = 8 y' = 3 (unchanged) New point: (8, 3)

x' = x + shx·y, y' = y + shy·x With shx = 1, shy = 0, x' = 2, area = 1.

Set shx = 2 and the corner at (0,2) slides to x = 4, while the corners on the x-axis stay exactly where they were.

Y-direction shear works the mirror-opposite way — shifts y based on x, leaving x unchanged.

💡 One-liner: "What visually happens to a square under x-shear?" → It turns into a parallelogram — the top and bottom edges stay horizontal, but the top slides sideways relative to the bottom.

What shear is doing to each point

An x-shear shifts a point sideways in proportion to its height. Points on the x-axis, where y = 0, do not move at all; a point at y = 3 moves three times as far as a point at y = 1. That proportionality is what tilts the shape rather than sliding it.

The line of points that does not move is called the invariant line — the x-axis for an x-shear, the y-axis for a y-shear. Identifying it is usually the fastest way to answer a question about which shear was applied.

💡 Shear preserves area, which surprises most students: the determinant of a pure x-shear is 1×1 − sh×0 = 1. A parallelogram made by shearing a square has the same area as the square, because it has the same base and the same perpendicular height. What shear does not preserve is angles or lengths.

Where it is actually used

Shear is how italic text is generated from an upright font, and it is the transformation applied in oblique projections to show depth on a flat screen — which is why it reappears in the 3D projection topic rather than being an isolated curiosity.

Syllabus points

Shear along x and y

Create a free account to tick topics off, take notes as you read, watch the video lessons and get a day-by-day study plan built around your exam date.