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Composite rotation and shear in 2D

Short answer

Composite rotation: successive rotations add angles, R(θ1)·R(θ2)=R(θ1+θ2). For shear with xref=-1, shy=0.5, x'=x, y'=y+0.5(x+1). New coordinates: (0,0.5), (1,1), (0,1.5), (1,2).

Explain the composite rotation in 2D transformation. Given the points (0,0), (1,0), (0,1) and (1,1) where Xref = -1 and Shy = 0.5, calculate and sketch the shearing transformation new coordinates.

Composite rotation

When two or more rotations are applied one after another, they can be combined into one matrix by multiplication. Two successive rotations by θ1 and θ2 about the origin give

R(θ2)·R(θ1) = R(θ1 + θ2)

because the angles add. The order does not matter for rotations about the same point. Rotation about an arbitrary pivot (xr, yr) is a composite of a translation, a rotation and a translation back: T(xr,yr)·R(θ)·T(−xr,−yr).

Shear (y-direction shear relative to the reference line x = xref)

x' = x
y' = y + shy·(x − xref)

| 1    0   0          |
| shy  1  −shy·xref   |
| 0    0   1          |

With xref = −1 and shy = 0.5: y' = y + 0.5(x + 1).

PointCalculationNew point
(0,0)y' = 0 + 0.5(0 + 1) = 0.5(0, 0.5)
(1,0)y' = 0 + 0.5(1 + 1) = 1(1, 1)
(0,1)y' = 1 + 0.5(0 + 1) = 1.5(0, 1.5)
(1,1)y' = 1 + 0.5(1 + 1) = 2(1, 2)
 y
 2.0 |          * (1,2)
     |        / |
 1.5 * (0,1.5)  |
     |          |
 1.0 |          * (1,1)
     |        /
 0.5 * (0,0.5)
     |
 0.0 +----------+---- x
     0          1

The unit square becomes a parallelogram with vertices (0,0.5), (1,1), (1,2), (0,1.5). The vertical sides stay vertical and keep length 1, while the horizontal sides tilt upward with slope 0.5.

Answer: (0,0.5), (1,1), (0,1.5), (1,2).

✓ Answer checked by Claude Opus 5.5 (high effort)

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