Rotation2d

Represents a 2D rotation using the unit complex number representation.

A rotation in 2D can be represented as a unit complex number \(z = \cos\theta + i\sin\theta\), where \(\theta\) is the angle of rotation. This representation is stored as the pair (real, imag) = \((\cos\theta, \sin\theta)\).

Why Unit Complex Numbers?

This representation has several advantages:

• Efficient composition: Multiplying two rotations \(z_1 \cdot z_2\) composes them

• No gimbal lock: Unlike Euler angles, no singularities

• Numerically stable: No wrap-around issues at \(2\pi\)

• Direct vector transformation: Rotating a vector is a simple complex multiplication

Mathematical Operations

Composition (multiplication): \((\cos\theta_1, \sin\theta_1) \cdot (\cos\theta_2, \sin\theta_2) = (\cos(\theta_1+\theta_2), \sin(\theta_1+\theta_2))\)

Inverse (conjugate): \((\cos\theta, \sin\theta)^{-1} = (\cos\theta, -\sin\theta)\)

Vector rotation: To rotate a vector \((x, y)\) by angle \(\theta\): \((x', y') = (\cos\theta \cdot x - \sin\theta \cdot y, \sin\theta \cdot x + \cos\theta \cdot y)\)

Example Usage

// Create a 90-degree rotation
val rot90 = Rotation2d.exp(Math.PI / 2)

// Create from an Angle
val rot = Rotation2d.fromAngle(45.0.degrees)

// Compose rotations
val composed = rot90 * rot90  // 180-degree rotation

// Rotate a vector
val v = Vector2d(1.0.inches, 0.0.inches)
val rotated = rot90 * v  // Results in approximately (0, 1)

// Get angle back
val angle = rot90.log()  // π/2 radians