Rotation transforms orient geometry in 2D and 3D space about an origin or axis without altering size or shape.

2D Rotation

A 2D rotation depends on a single parameter, the angle $\theta$. Rotating the standard Cartesian basis vectors $\mathbf{i} = [1, 0]^T$ and $\mathbf{j} = [0, 1]^T$ counterclockwise by $\theta$ yields transformed basis vectors $\mathbf{p}$ and $\mathbf{q}$:

$$ \begin{aligned} \mathbf{p} &= \cos\theta \mathbf{i} + \sin\theta \mathbf{j} = \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix} \\ \mathbf{q} &= -\sin\theta \mathbf{i} + \cos\theta \mathbf{j} = \begin{bmatrix} -\sin\theta \\ \cos\theta \end{bmatrix} \end{aligned} $$

Placing $\mathbf{p}$ and $\mathbf{q}$ into matrix columns forms the 2D rotation matrix:

$$ \mathbf{R}(\theta) = \begin{bmatrix} \mathbf{p} & \mathbf{q} \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $$

Applying $\mathbf{R}(\theta)$ to an arbitrary vector $\mathbf{v} = [v_x, v_y]^T$ expresses the transformed vector $\mathbf{v}^\prime$ as a linear combination of the rotated basis:

$$ \mathbf{v}^\prime = \mathbf{R}(\theta)\mathbf{v} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} v_x \\ v_y \end{bmatrix} = \begin{bmatrix} v_x \cos\theta - v_y \sin\theta \\ v_x \sin\theta + v_y \cos\theta \end{bmatrix} $$

See also complex numbers .

3D Rotation

Cardinal Axes

Rotating around the primary coordinate axes in a right-handed system produces elementary $3 \times 3$ rotation matrices:

$$ \mathbf{R}_x(\alpha) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & -\sin\alpha \\ 0 & \sin\alpha & \cos\alpha \end{bmatrix} $$

$$ \mathbf{R}_y(\beta) = \begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix} $$

$$ \mathbf{R}_z(\gamma) = \begin{bmatrix} \cos\gamma & -\sin\gamma & 0 \\ \sin\gamma & \cos\gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Arbitrary Axis (Rodrigues’ Rotation Formula)

Given an arbitrary unit axis $\hat{\mathbf{n}}$ ($\lVert \hat{\mathbf{n}} \rVert = 1$) and a rotation angle $\theta$, we seek a transformation $\mathbf{v}^\prime = \mathbf{R}(\hat{\mathbf{n}}, \theta)\mathbf{v}$.

Decompose the target vector $\mathbf{v}$ into parallel and perpendicular components relative to $\hat{\mathbf{n}}$:

$$ \begin{aligned} \mathbf{v}_{\parallel} &= (\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} \\ \mathbf{v}_{\perp} &= \mathbf{v} - \mathbf{v}_{\parallel} = \mathbf{v} - (\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} \end{aligned} $$

Under rotation about $\hat{\mathbf{n}}$, the parallel component remains unchanged ($\mathbf{v}^\prime_{\parallel} = \mathbf{v}_{\parallel}$). The perpendicular component $\mathbf{v}_{\perp}$ rotates entirely within the plane orthogonal to $\hat{\mathbf{n}}$.

Construct an in-plane perpendicular basis vector $\mathbf{w}$ using the cross product:

$$ \mathbf{w} = \hat{\mathbf{n}} \times \mathbf{v}_{\perp} = \hat{\mathbf{n}} \times (\mathbf{v} - \mathbf{v}_{\parallel}) = \hat{\mathbf{n}} \times \mathbf{v} $$

Because $\hat{\mathbf{n}} \perp \mathbf{v}_{\perp}$ and $\lVert \hat{\mathbf{n}} \rVert = 1$, the length $\lVert \mathbf{w} \rVert = \lVert \hat{\mathbf{n}} \rVert \lVert \mathbf{v}_{\perp} \rVert \sin(90^\circ) = \lVert \mathbf{v}_{\perp} \rVert$.

The orthogonal vectors $\mathbf{v}_{\perp}$ and $\mathbf{w}$ span the 2D rotation plane. Rotating $\mathbf{v}_{\perp}$ by $\theta$ gives:

$$ \mathbf{v}^\prime_{\perp} = \cos\theta \mathbf{v}_{\perp} + \sin\theta \mathbf{w} = \cos\theta (\mathbf{v} - (\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}}) + \sin\theta (\hat{\mathbf{n}} \times \mathbf{v}) $$

Recombining parallel and perpendicular components gives Rodrigues’ rotation formula:

$$ \begin{aligned} \mathbf{v}^\prime &= \mathbf{v}^\prime_{\parallel} + \mathbf{v}^\prime_{\perp} \\ &= (\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} + \cos\theta (\mathbf{v} - (\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}}) + \sin\theta (\hat{\mathbf{n}} \times \mathbf{v}) \\ &= \cos\theta \mathbf{v} + \sin\theta (\hat{\mathbf{n}} \times \mathbf{v}) + (1 - \cos\theta)(\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} \end{aligned} $$

Evaluating Rodrigues’ formula on each standard basis vector $\mathbf{e}_1 = [1, 0, 0]^T$, $\mathbf{e}_2 = [0, 1, 0]^T$, and $\mathbf{e}_3 = [0, 0, 1]^T$ produces the column vectors of the full $3 \times 3$ matrix:

$$ \mathbf{R}(\hat{\mathbf{n}}, \theta) = \begin{bmatrix} n_x^2(1 - \cos\theta) + \cos\theta & n_y n_x(1 - \cos\theta) - n_z \sin\theta & n_z n_x(1 - \cos\theta) + n_y \sin\theta \\ n_x n_y(1 - \cos\theta) + n_z \sin\theta & n_y^2(1 - \cos\theta) + \cos\theta & n_z n_y(1 - \cos\theta) - n_x \sin\theta \\ n_x n_z(1 - \cos\theta) - n_y \sin\theta & n_y n_z(1 - \cos\theta) + n_x \sin\theta & n_z^2(1 - \cos\theta) + \cos\theta \end{bmatrix} $$

Connection to Quaternions

Composing multiple 3D rotations using Euler angles or full $3 \times 3$ matrices often introduces gimbal lock and matrix orthogonalization drift.

Quaternions represent rotations compactly using a 4D unit rotor $q = [\cos\frac{\theta}{2}, \sin\frac{\theta}{2}\hat{\mathbf{n}}]$. The quaternion sandwich product $p^\prime = q p q^\ast$ algebraically produces the exact same vector transformation as Rodrigues’ formula without trigonometric evaluations per vertex:

$$ \mathbf{v}^\prime = \cos\theta \mathbf{v} + \sin\theta (\hat{\mathbf{n}} \times \mathbf{v}) + (1 - \cos\theta)(\mathbf{v} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} $$

For full algebraic proofs of rotor construction, sandwich product invariance, rotation composition, and spherical linear interpolation, see Quaternions .