2D Rotation

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

2D Rotation: spinning a shape around the origin

The trigonometry you learned years ago, quietly running every time something spins on screen.

Rotation matrix (about the origin, angle θ, counter-clockwise): [x'] [cosθ -sinθ 0] [x] [y'] = [sinθ cosθ 0] [y] [1 ] [0 0 1] [1] Worked example: rotate point (1,0) by θ=90° cos90°=0, sin90°=1 x' = 0(1) - 1(0) + 0 = 0 y' = 1(1) + 0(0) + 0 = 1 New point: (0, 1) — makes sense: (1,0) rotated 90° lands on the y-axis!

x' = x·cosθ − y·sinθ, y' = x·sinθ + y·cosθ With th = 45 °, x' = 1.41, y' = 1.41.

Set θ to 90° and watch the corner at (1,0) land on (0,1) — the worked example above, happening.

💡 Practice rotating a point/triangle by common angles (30°, 45°, 90°) — memorize sin/cos values for these standard angles to move fast in the exam.

Why rotation swings rather than spins

The matrix rotates about the origin, not about the shape. A shape sitting far from the origin does not turn on the spot — it travels along an arc, like a stone on the end of a string, and can end up somewhere completely unexpected.

This is the single most common mistake in exam questions on rotation. "Rotate the square about its own centre" is not what the plain rotation matrix does. You must translate the centre to the origin, rotate, then translate back — the same three-step pattern used for fixed-point scaling.

💡 Sign convention: positive θ is anticlockwise. A question asking for a clockwise rotation of 30° needs θ = −30°, which flips the sign of both sine terms (cosine is unaffected, since cos(−θ) = cosθ).

Properties worth stating

Rotation preserves lengths, angles and areas — it is rigid, like translation. Its determinant is cos²θ + sin²θ = 1, which is the formal statement that area is unchanged. The inverse of a rotation by θ is a rotation by −θ, and because of that identity the inverse matrix equals the transpose, which is a favourite one-line exam question.

Syllabus points

Rotation matrix; rotation about origin

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