Euler Angles Calculator
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Understanding Euler Angles
What are Euler Angles?
Euler angles are three angles used to describe the orientation of a rigid body in 3D space. They represent:
- Precession (ψ) - rotation about the z-axis
- Nutation (θ) - rotation about the new x-axis
- Intrinsic rotation (φ) - rotation about the new z-axis
Rotation Conventions
There are several conventions for Euler angles:
- Proper Euler angles (z-x-z, x-y-x, etc.)
- Tait-Bryan angles (x-y-z, z-y-x, etc.)
- Aerospace sequence (yaw-pitch-roll)
- Gimbal sequence (heading-elevation-bank)
Mathematical Foundation
R = R₃(γ)R₂(β)R₁(α)
where R₁, R₂, R₃ are basic rotation matrices
Important Properties
Non-commutative
R₁R₂ ≠ R₂R₁
Gimbal Lock
Loss of one degree of freedom when β = ±90°
Periodicity
Full rotation at 360°
Applications
Aerospace
Aircraft orientation and navigation
Robotics
Robot arm manipulation and control
Computer Graphics
3D object rotation and camera control
Virtual Reality
Head tracking and motion control