Rotation Matrix Calculator — 2D

2D Rotations at a Glance

A counter-clockwise rotation by angle θ uses the matrix R(θ) = \[\[ \cos θ,\; -\sin θ\],\; [\sin θ,\; \cos θ]\]. To rotate a point p = ⟨x,y⟩ about the origin, compute p' = R(θ)\,p. To rotate about a center c = ⟨cₓ,cᵧ⟩, translate to the origin, rotate, then translate back: p' = R(θ)(p − c) + c.

Clockwise rotation is the same as using −θ. In degrees, convert to radians with θᵣ = θ·π/180. This tool accepts degrees or radians, handles clockwise/CCW, and lets you paste a list of points for batch rotation—handy for geometry, robotics, or simple SVG/prototyping workflows.

Understanding the 2D Rotation Matrix

A 2D rotation is a linear transformation that turns every point in the plane by an angle θ about a chosen center. In standard counter-clockwise (CCW) orientation, the rotation matrix is R(θ) = [[cos θ, −sin θ],[sin θ, cos θ]]. If you supply a point p = ⟨x, y⟩ as a column vector and rotate about the origin, the rotated point is p' = R(θ) p. Rotating about an arbitrary center c = ⟨cₓ, cᵧ⟩ uses a translate-rotate-translate sequence: p' = R(θ)(p − c) + c.

Degrees vs Radians and Direction (CW vs CCW)

Most formulas are defined in radians. Convert degrees with θᵣ = θ·π/180. Our calculator accepts both units and handles clockwise (CW) by simply negating the angle: CW(θ) ≡ CCW(−θ). This is equivalent to R(−θ) = R(θ)ᵀ, which is also the inverse of R(θ).

Why the Matrix Looks Like That

The first column of R(θ) is where the unit vector along +x (⟨1,0⟩) ends up after rotation: ⟨cos θ, sin θ⟩. The second column is where +y (⟨0,1⟩) ends up: ⟨−sin θ, cos θ⟩. Because the columns are orthonormal, R(θ) is orthogonal: R(θ)ᵀ R(θ) = I, det R(θ) = 1, and R(θ)^{-1} = R(θ)ᵀ = R(−θ).

Composing Rotations and Stability

A key property is angle additivity: R(α) R(β) = R(α + β). That means you can chain multiple rotations in any order and replace them with a single rotation whose angle is the sum of the angles. Numerically, if you apply many tiny rotations, floating-point rounding can drift the matrix away from being perfectly orthogonal. A quick fix is to recompute cos/sin from an accumulated angle rather than repeatedly multiplying matrices, or to re-orthogonalize using the transpose.

Rotating Batches of Points

If you have a list of points, stack them as columns in a 2×n matrix P and compute P' = R(θ) P. Rotating about a center c becomes P' = R(θ)(P − c·1ᵀ) + c·1ᵀ, where 1 is a vector of ones. In graphics and robotics, a homogeneous 3×3 matrix is often used to bundle translation and rotation: T = [[cos θ, −sin θ, tₓ],[sin θ, cos θ, tᵧ],[0,0,1]] with [tₓ, tᵧ]ᵀ = c − R(θ)c. Then apply [x',y',1]ᵀ = T [x,y,1]ᵀ.

Common Pitfalls

• Wrong direction: CW vs CCW swapped. Use CW(θ) = CCW(−θ).
• Unit mismatch: Feeding degrees into sin/cos that expect radians.
• About the origin only: Forgetting to translate when rotating about a point c.
• Rounding: Rounding too aggressively can distort results; keep 4–6 decimals for display.

Worked Example

Rotate p = ⟨3, 1⟩ by θ = 30° CCW about the origin. cos 30° = √3/2 ≈ 0.8660, sin 30° = 0.5. Then p' = ⟨3·0.8660 − 1·0.5, 3·0.5 + 1·0.8660⟩ ≈ ⟨2.098, 2.366⟩. About a center c = ⟨1,−2⟩, compute p − c = ⟨2, 3⟩, rotate to ⟨2·0.8660 − 3·0.5, 2·0.5 + 3·0.8660⟩ ≈ ⟨0.232, 3.098⟩, then add back c to get p' ≈ ⟨1.232, 1.098⟩.

Summary: use R(θ) for CCW, −θ for CW, and wrap any non-origin center with translate-rotate-translate. Our calculator handles all of this for single points or batches, entirely in your browser.