3D Shear

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

3D Shear

The 3D generalization of "push sideways" — now any axis can push any other.

Example: Z-shear (x and y shift based on z): [x'] [1 0 shx 0] [x] [y'] = [0 1 shy 0] [y] [z'] [0 0 1 0] [z] [1 ] [0 0 0 1] [1] Worked example: shear (1,1,2) with shx=1, shy=2 x' = 1 + 1(2) = 3 y' = 1 + 2(2) = 5 z' = 2 (unchanged) New point: (3, 5, 2)

z-shear: x' = x + shx·z, y' = y + shy·z, z' = z With shx = 1, shy = 0, x' = 3, y' = 1.

Set shx = 1 and shy = 2 to reproduce the worked example above: (1,1,2) becomes (3,5,2).

💡 Recognize the pattern from 2D shear — the "shift factor × the unchanged axis's value" logic is identical, just with an extra axis available to shear against.

What is shearing against what

In 2D there was one choice to make — shear x against y, or y against x. In 3D any axis can push either of the other two, so the matrix has six possible off-diagonal shear terms rather than two.

The z-shear above is the usual worked example: x and y each shift in proportion to z, while z itself is unchanged. Points on the plane z = 0 do not move at all, and that invariant plane is what tilts the solid rather than sliding it.

In 2D the unmoved set was a line; in 3D it is a plane. Identifying which plane stays fixed is the quickest way to say which shear was applied.

💡 Shear preserves volume — its determinant is 1, since the matrix is triangular with ones on the diagonal. It does not preserve angles or lengths. This is exactly why oblique projections, which are shears, keep proportions along each axis while making the shape look slanted.

Syllabus points

3D shear transformation

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