Angle Rotation Calculator

Understanding Angle Rotation

2D Rotation

For point (x, y) rotated by angle θ:

  • x' = x cos θ - y sin θ
  • y' = x sin θ + y cos θ

Rotation Matrix:

[cos θ -sin θ]

[sin θ cos θ]

3D Rotation

Rotation Matrices:

X-axis rotation:

[1 0 0 ]

[0 cos θ -sin θ]

[0 sin θ cos θ]

Y-axis rotation:

[ cos θ 0 sin θ]

[ 0 1 0 ]

[-sin θ 0 cos θ]

Z-axis rotation:

[cos θ -sin θ 0]

[sin θ cos θ 0]

[ 0 0 1]

Properties of Rotations

Composition

  • Multiple rotations can be combined
  • Order matters (non-commutative)
  • Matrix multiplication represents composition

Preservation Properties

  • Preserves distances
  • Preserves angles
  • Preserves orientation
  • Determinant = 1

Advanced Topics

Euler Angles

  • Roll (φ): Rotation about x-axis
  • Pitch (θ): Rotation about y-axis
  • Yaw (ψ): Rotation about z-axis
  • Different conventions exist

Quaternions

  • 4D representation of rotations
  • Avoids gimbal lock
  • Efficient composition
  • Used in computer graphics